GCSE Maths Quadratic and Other Graphs (Higher): check-up and exam-style questions
Part of the GCSE Maths quadratic and other graphs chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Exercise 13Z Check-up (non-calculator)
Non-calculator| x | −2 | −1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y |
| x | −2 | −1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y | 1 | −2 | −3 | −2 | 1 |
Key words. To substitute means to replace the letter by a number. A table of values lists x-values and the y-values that go with them. Each pair (x, y) is a coordinate that can be plotted.
Method. Put each x-value into the rule, one at a time, and work out y.
- Write the rule and the x-values. A table of values pairs every x-value with its y-value. Each y-value comes from putting that x-value into the rule.y = x2 − 3
x-values: −2, −1, 0, 1, 2 - Substitute the x-values one at a time. Do the same calculation every time, only the number changes. Use brackets round negative numbers.x = −2: y = (−2)2 − 3 = 4 − 3 = 1
x = −1: y = (−1)2 − 3 = 1 − 3 = −2
x = 0: y = 02 − 3 = 0 − 3 = −3
x = 1: y = 12 − 3 = 1 − 3 = −2
x = 2: y = 22 − 3 = 4 − 3 = 1 - Write the y-values in the table. Each pair of numbers is a coordinate, ready to be plotted on a graph.y-values: 1, −2, −3, −2, 1
Coordinates: (−2, 1), (−1, −2), (0, −3), (1, −2), (2, 1)
Check. The gaps between neighbouring y-values are −3, −1, 1, 3. The gaps change by 2 every time (second differences are constant), which is what must happen for a quadratic. ✓
Watch out. Work out the x² part first. A negative number squared is positive.
Key words. A quadratic is a rule where the highest power of x is x² (x squared). Its graph is a smooth curve called a parabola. The turning point is where the curve changes direction (the lowest or highest point). A root is an x-value where the curve meets the x-axis (where y = 0). The y-intercept is where the curve crosses the y-axis (where x = 0).
Method. The roots are the x-values where the curve crosses the x-axis, so read them off the graph.
- Remember what a root is. A root is an x-value where the curve meets the x-axis. On the x-axis every point has y = 0, so roots are the solutions of y = 0.
- Find where the curve crosses the x-axis. Look along the horizontal axis for the places where the curve meets it. Read the x-coordinate of each place from the numbers along the bottom.The curve crosses the x-axis at (0, 0) and (4, 0).
- Write the roots. We only need the x-values (the roots are x-values, not coordinates).x = 0 or 4
Check. Substitute each root into the rule; y should be 0.
y = 02 − 4 × 0 = 0 + 0 = 0
y = 42 − 4 × 4 = 16 − 16 = 0 ✓
Watch out. Give x-values only: say x = 3, not (3, 0).
Key words. A quadratic is a rule where the highest power of x is x² (x squared). Its graph is a smooth curve called a parabola. The turning point is where the curve changes direction (the lowest or highest point). A root is an x-value where the curve meets the x-axis (where y = 0). The y-intercept is where the curve crosses the y-axis (where x = 0).
Method. The turning point is the lowest point of a U-shaped curve, or the highest point of an n-shaped curve. Read both coordinates.
- Remember what a turning point is. The turning point is where the curve stops going down and starts going up (or the other way round). A U-shaped curve has a lowest point.
- Find the lowest point of the curve. Read the x-coordinate along the bottom and the y-coordinate up the side.The turning point is (2, −4).
- Write the answer as a coordinate pair. It is a minimum (lowest point).(2, −4)
Check. Substitute x = 2: y = 22 − 4 × 2 = 4 − 8 = −4. ✓
Watch out. Write both coordinates, x first: (x, y).
Key words. A quadratic is a rule where the highest power of x is x² (x squared). Its graph is a smooth curve called a parabola. The turning point is where the curve changes direction (the lowest or highest point). A root is an x-value where the curve meets the x-axis (where y = 0). The y-intercept is where the curve crosses the y-axis (where x = 0).
Method. The y-intercept is where the curve crosses the y-axis, so read the y-value where x = 0.
- Remember what the y-intercept is. The y-intercept is where the curve crosses the y-axis (the vertical axis). Every point on the y-axis has x = 0.
- Find where the curve crosses the y-axis. Look up or down the vertical axis for the place where the curve meets it. Read the number.The curve crosses the y-axis at (0, 8).
- Write the answer. The y-intercept is the y-value at that point.y-intercept = 8
Check. Substitute x = 0 into the rule: y = −(0)2 + 2 × 0 + 8 = 0 + 0 + 8 = 8. ✓
Watch out. The y-intercept is a y-value (a height). The roots are x-values. Do not mix them up.
Key words. A quadratic is a rule where the highest power of x is x² (x squared). Its graph is a smooth curve called a parabola. The turning point is where the curve changes direction (the lowest or highest point). A root is an x-value where the curve meets the x-axis (where y = 0). The y-intercept is where the curve crosses the y-axis (where x = 0).
Method. The line of symmetry is the vertical line through the turning point: x = (the x-coordinate of the turning point).
- Remember what the line of symmetry is. A parabola is a mirror image of itself. The line of symmetry is the vertical mirror line that goes through the turning point.
- Find the turning point (the highest point). The mirror line goes straight through it, so it has the same x-value as the turning point.Turning point = (1, 9)
- Write the equation of the line. A vertical line has an equation of the form x = number. Use the x-coordinate of the turning point.x = 1
Check. The line of symmetry is halfway between the roots: (−2 + 4) ÷ 2 = 1. ✓
Watch out. A vertical line is written x = number (not y = number). y = number is a horizontal line.
Key words. A quadratic is a rule where the highest power of x is x² (x squared). Its graph is a smooth curve called a parabola. The turning point is where the curve changes direction (the lowest or highest point). A root is an x-value where the curve meets the x-axis (where y = 0). The y-intercept is where the curve crosses the y-axis (where x = 0).
Method. The maximum value of y is the y-coordinate of the turning point.
- Remember what a turning point is. The turning point is where the curve stops going down and starts going up (or the other way round). An n-shaped curve has a highest point.
- Find the highest point of the curve. Read the x-coordinate along the bottom and the y-coordinate up the side.The turning point is (1, 9).
- Write the answer. The maximum (highest point) value of y is the y-coordinate of the turning point.maximum value of y = 9
Check. Substitute x = 1: y = −(1)2 + 2 × 1 + 8 = −1 + 2 + 8 = 9. ✓
Watch out. The minimum value is the y-value (height), not the x-value.
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. Draw the horizontal line y = (the number on the right). Where it crosses the curve, read the x-values.
- Spot the graph being used. The graph shows y = x2 − 4x. The equation we want to solve is the same expression set equal to 5.x2 − 4x = 5 means y = 5 on this graph
- Draw the horizontal line y = 5. Every point on this line has a y-value of 5. Where the line meets the curve, the curve has y = 5, which is exactly when the equation is true.Start at 5 on the y-axis and draw straight across (a ruler helps).
- Find where the line crosses the curve. Mark each crossing point.The line meets the curve at (−1, 5) and (5, 5).
- Read the x-values. The solutions of the equation are the x-values of the crossing points (the numbers along the bottom axis).x = −1 or 5
Check. Substitute back: y = (−1)2 − 4 × (−1) = 1 + 4 = 5; y = 52 − 4 × 5 = 25 − 20 = 5 ✓
Watch out. Read the x-values of the crossing points, not the y-values. The y-value is just the number you started with.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Look at the highest power of x (or whether x is on the bottom of a fraction) to name the graph, then use the sign to describe its direction.
- Look for the highest power of x. If the highest power is x³, the graph is a cubic.y = x3 + 1: the highest power is x³.
- Use the sign of the x³ term. A positive x³ term makes the graph go from bottom-left to top-right. A negative one makes it go from top-left to bottom-right.The number in front of x³ is 1, which is positive.
- Describe the shape. Put the two facts together.Cubic: cubic curve rising from bottom-left to top-right.
Check. Test the idea with a point: when x = 1, y = 2, and when x = −1, y = 0. These agree with the shape described. ✓
Watch out. A graph with x² is a curve with one turning point; do not call a cubic a quadratic.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Look at the highest power of x (or whether x is on the bottom of a fraction) to name the graph, then use the sign to describe its direction.
- Look for the highest power of x. If the highest power is x², the graph is a quadratic (a parabola).y = −x2: the highest power is x².
- Use the sign of the x² term. A positive x² term gives a U-shape. A negative one gives an n-shape.The number in front of x² is −1, which is negative.
- Describe the shape. Put the two facts together.Quadratic: n-shaped parabola (has a highest point).
Check. Test the idea with a point: when x = 1, y = −1, and when x = −1, y = −1. These agree with the shape described. ✓
Watch out. A graph with x² is a curve with one turning point; do not call a cubic a quadratic.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Look at the highest power of x (or whether x is on the bottom of a fraction) to name the graph, then use the sign to describe its direction.
- Look at where x is. In this rule x is on the bottom of a fraction. A rule of the form number ÷ x is a reciprocal graph.y = 3x
- Use the sign of the number on top. A positive number on top gives branches in the top-right and bottom-left. A negative number gives branches in the top-left and bottom-right.The number on top is 3, which is positive.
- Describe the shape. Put the two facts together.Reciprocal: reciprocal curve in two branches, in the top-right and bottom-left parts of the grid.
Check. Test the idea with a point: when x = 1, y = 3, and when x = −1, y = −3. These agree with the shape described. ✓
Watch out. A graph with x² is a curve with one turning point; do not call a cubic a quadratic.
| x | −2 | −1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y |
| x | −2 | −1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y | −8 | −1 | 0 | 1 | 8 |
Key words. To substitute means to replace the letter by a number. A table of values lists x-values and the y-values that go with them. Each pair (x, y) is a coordinate that can be plotted.
Method. Put each x-value into the rule, one at a time, and work out y.
- Write the rule and the x-values. A table of values pairs every x-value with its y-value. Each y-value comes from putting that x-value into the rule.y = x3
x-values: −2, −1, 0, 1, 2 - Substitute the x-values one at a time. Do the same calculation every time, only the number changes. Use brackets round negative numbers.x = −2: y = (−2)3 = −8
x = −1: y = (−1)3 = −1
x = 0: y = 03 = 0
x = 1: y = 13 = 1
x = 2: y = 23 = 8 - Write the y-values in the table. Each pair of numbers is a coordinate, ready to be plotted on a graph.y-values: −8, −1, 0, 1, 8
Coordinates: (−2, −8), (−1, −1), (0, 0), (1, 1), (2, 8)
Check. First gaps: 7, 1, 1, 7. Gaps of the gaps: −6, 0, 6. Gaps of those: 6, 6, always the same, which is the pattern for a cubic. ✓
Watch out. A negative number cubed is negative: (−2)³ = −8.
Key words. To substitute means to replace the letter by a number. A table of values lists x-values and the y-values that go with them. Each pair (x, y) is a coordinate that can be plotted.
Method. Substitute the x-value into the rule and work out y, using BIDMAS.
- Write down the rule. The rule tells us how to get y from x. Here y comes from 12x.y = 12x
- Replace x by (−3). Substituting means putting the number in place of the letter. Put a negative number in brackets: (−2)² means (−2) × (−2) = 4.y = 12 ÷ (−3)
- Work it out in the right order. Powers first, then multiplying and dividing, then adding and subtracting (BIDMAS).y = −4
- Write the answer. This tells us the curve passes through the point (−3, −4).When x = −3, y = −4
Check. Type it into a calculator using brackets for negatives. The display shows −4. ✓
Watch out. You cannot divide by zero, so x = 0 has no y-value on a reciprocal graph.
Key words. A distance-time graph shows how far something has travelled from its start (up the side) as time passes (along the bottom). The gradient (steepness) of the line is the speed, because gradient = change up ÷ change across = distance ÷ time. A flat line means stationary (not moving).
Method. Speed is the gradient of a distance-time graph: speed = change in distance ÷ change in time.
- Pick the part of the graph. Speed is the gradient of the line. We need the change in distance and the change in time for this part.From (3, 60) to (4, 0)
- Find the change in distance and the change in time. Subtract the start values from the end values.Change in distance = 0 − 60 = −60 km
Change in time = 4 − 3 = 1 hours - Divide distance by time. Speed = distance ÷ time.−60 ÷ 1 = −60 km/h
- Interpret the sign. The line goes down, so the distance from the start is getting smaller: the person is travelling back towards the start. The speed itself is the size of the gradient.Speed = 60 km/h (heading back)
Check. Distance = speed × time = 60 × 1 = 60 km, the change we read from the graph. ✓
Watch out. Use the change in distance and the change in time for the same part of the journey, not the totals from the start.
Key words. A speed-time graph shows the speed (up the side) as time passes (along the bottom). The gradient is the acceleration (how fast the speed changes), and the area under the graph is the distance travelled. Negative acceleration (the speed is dropping) is called deceleration.
Method. Acceleration is the gradient of a speed-time graph: change in speed ÷ change in time.
- Pick the part of the graph. Acceleration is the gradient of the line, so we use the change in speed and the change in time.From (0, 0) to (10, 20)
- Find the change in speed and the change in time. Subtract the start values from the end values.Change in speed = 20 − 0 = 20 m/s
Change in time = 10 − 0 = 10 s - Divide. Acceleration = change in speed ÷ change in time.20 ÷ 10 = 2
- Write the answer with its unit. The unit is metres per second per second, written m/s².Acceleration = 2 m/s²
Check. The speed changes by 2 m/s every second, for 10 s: 2 × 10 = 20 m/s, the change we found. ✓
Watch out. Acceleration is the gradient. The height of the line is the speed, not the acceleration.
Key words. A speed-time graph shows the speed (up the side) as time passes (along the bottom). The gradient is the acceleration (how fast the speed changes), and the area under the graph is the distance travelled. Negative acceleration (the speed is dropping) is called deceleration.
Method. Distance travelled = area under the speed-time graph. Split it into rectangles, triangles and trapezia.
- Remember: distance is the area under the graph. Distance = speed × time. On the graph, speed is the height and time is the width, so distance is the area between the line and the time axis.
- Split the area into simple shapes. Use a rectangle where the line is flat, a triangle where it starts or ends at 0, and a trapezium otherwise.From 0 s to 6 s, a rectangle: 12 × 6 = 72
From 6 s to 10 s, a triangle: ½ × 4 × 12 = 24 - Add the areas. The total area is the total distance travelled.72 + 24 = 96 m
Check. Sense check: the vehicle never goes faster than 12 m/s and travels for 10 s, so the distance must be less than 120 m. Our 96 m is smaller. ✓
Watch out. The unit of the area is the unit of distance (m), not m/s. Area = height × width means speed × time.
Total 52 marks. Allow about 52 minutes.
Exercise 13X Exam-style questions
Non-calculator| x | −1 | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|---|
| y | 1 | −3 | 1 |
Key words. To substitute means to replace the letter by a number. A table of values lists x-values and the y-values that go with them. Each pair (x, y) is a coordinate that can be plotted.
Method. Make a table of values, plot the points, then join them with a smooth curve.
- Make a table of values. Each y-value comes from putting an x-value into the rule. Use brackets round negative numbers.x = −1: y = (−1)2 − 4 × (−1) + 1 = 1 + 4 + 1 = 6
x = 0: y = 02 − 4 × 0 + 1 = 0 + 0 + 1 = 1
x = 1: y = 12 − 4 × 1 + 1 = 1 − 4 + 1 = −2
x = 2: y = 22 − 4 × 2 + 1 = 4 − 8 + 1 = −3
x = 3: y = 32 − 4 × 3 + 1 = 9 − 12 + 1 = −2
x = 4: y = 42 − 4 × 4 + 1 = 16 − 16 + 1 = 1
x = 5: y = 52 − 4 × 5 + 1 = 25 − 20 + 1 = 6 - List the coordinates. Each x with its y makes a point (x, y) to plot.(−1, 6), (0, 1), (1, −2), (2, −3), (3, −2), (4, 1), (5, 6)
- Choose the scales for the axes. The x-values go from −1 to 5 and the y-values go from −3 to 6. The axes must cover all of these, and each square must always stand for the same amount.x-axis: −1 to 5. y-axis: −4 to 7.
- Plot each point carefully. Go along the x-axis first, then up or down to the y-value. Mark each point with a small cross.Plot (−1, 6), (0, 1), (1, −2), (2, −3), (3, −2), (4, 1), (5, 6).
- Join the points with a smooth curve. The graph of this rule is a smooth curve, not a set of straight lines. Draw it freehand in one flowing line, rounded at the turning point (no sharp corner).Label the curve y = x2 − 4x + 1.
Check. Every plotted point should lie on the curve, and the curve should pass through (−1, 6) and (5, 6) at the ends. It should be symmetrical about the line x = 2. ✓
Watch out. Do not use a ruler to join the points, and do not draw pointed corners. A quadratic is smooth and rounded at the turning point.
Key words. A quadratic is a rule where the highest power of x is x² (x squared). Its graph is a smooth curve called a parabola. The turning point is where the curve changes direction (the lowest or highest point). A root is an x-value where the curve meets the x-axis (where y = 0). The y-intercept is where the curve crosses the y-axis (where x = 0).
Method. The roots are the x-values where the curve crosses the x-axis, so read them off the graph.
- Remember what a root is. A root is an x-value where the curve meets the x-axis. On the x-axis every point has y = 0, so roots are the solutions of y = 0.
- Find where the curve crosses the x-axis. Look along the horizontal axis for the places where the curve meets it. Read the x-coordinate of each place from the numbers along the bottom.The curve crosses the x-axis at (0, 0) and (4, 0).
- Write the roots. We only need the x-values (the roots are x-values, not coordinates).x = 0 or 4
Check. Substitute each root into the rule; y should be 0.
y = 02 − 4 × 0 = 0 + 0 = 0
y = 42 − 4 × 4 = 16 − 16 = 0 ✓
Watch out. Give x-values only: say x = 3, not (3, 0).
Key words. A quadratic is a rule where the highest power of x is x² (x squared). Its graph is a smooth curve called a parabola. The turning point is where the curve changes direction (the lowest or highest point). A root is an x-value where the curve meets the x-axis (where y = 0). The y-intercept is where the curve crosses the y-axis (where x = 0).
Method. The line of symmetry is the vertical line through the turning point: x = (the x-coordinate of the turning point).
- Remember what the line of symmetry is. A parabola is a mirror image of itself. The line of symmetry is the vertical mirror line that goes through the turning point.
- Find the turning point (the lowest point). The mirror line goes straight through it, so it has the same x-value as the turning point.Turning point = (2, −4)
- Write the equation of the line. A vertical line has an equation of the form x = number. Use the x-coordinate of the turning point.x = 2
Check. The line of symmetry is halfway between the roots: (0 + 4) ÷ 2 = 2. ✓
Watch out. A vertical line is written x = number (not y = number). y = number is a horizontal line.
Key words. A quadratic is a rule where the highest power of x is x² (x squared). Its graph is a smooth curve called a parabola. The turning point is where the curve changes direction (the lowest or highest point). A root is an x-value where the curve meets the x-axis (where y = 0). The y-intercept is where the curve crosses the y-axis (where x = 0).
Method. The turning point is the lowest point of a U-shaped curve, or the highest point of an n-shaped curve. Read both coordinates.
- Remember what a turning point is. The turning point is where the curve stops going down and starts going up (or the other way round). A U-shaped curve has a lowest point.
- Find the lowest point of the curve. Read the x-coordinate along the bottom and the y-coordinate up the side.The turning point is (2, −4).
- Write the answer as a coordinate pair. It is a minimum (lowest point).(2, −4)
Check. Substitute x = 2: y = 22 − 4 × 2 = 4 − 8 = −4. ✓
Watch out. Write both coordinates, x first: (x, y).
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. Draw the horizontal line y = (the number on the right). Where it crosses the curve, read the x-values.
- Spot the graph being used. The graph shows y = x2 − 2x − 3. The equation we want to solve is the same expression set equal to 2.x2 − 2x − 3 = 2 means y = 2 on this graph
- Draw the horizontal line y = 2. Every point on this line has a y-value of 2. Where the line meets the curve, the curve has y = 2, which is exactly when the equation is true.Start at 2 on the y-axis and draw straight across (a ruler helps).
- Find where the line crosses the curve. Mark each crossing point.The line meets the curve at (−1.4, 2) and (3.4, 2).
- Read the x-values. The solutions of the equation are the x-values of the crossing points (the numbers along the bottom axis).x ≈ −1.4 or 3.4 (accept answers within 0.1)
Check. Substitute back: at x = −1.4 the rule gives 2.00; at x = 3.4 the rule gives 2.00 ✓
Watch out. Read the x-values of the crossing points, not the y-values. The y-value is just the number you started with.
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. The solutions are the x-values of the points where the two graphs cross.
- Say which two graphs are being used. The equation has the rule of the curve on one side and the rule of the line on the other. Each side is drawn as its own graph.Curve: y = x2 − 1
Line: y = x + 1 - Explain why crossing points give the answers. At a crossing point both graphs have the same x and the same y, so the two rules give the same value. That is what the equation says.Solve x² − 1 = x + 1.
- Find the crossing points. Look for every place where the two graphs meet.They cross at (−1, 0) and (2, 3).
- Read the x-values. The solutions are the x-coordinates of the crossing points.x = −1 or 2
Check. Substitute each x-value into both sides.
x = −1: left = 0, right = 0. Equal. ✓
x = 2: left = 3, right = 3. Equal. ✓
Watch out. Write only the x-values. (The y-values at the crossings are not the answer.)
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Look at the highest power of x (or whether x is on the bottom of a fraction) to name the graph, then use the sign to describe its direction.
- Look for the highest power of x. If the highest power is just x (power 1), the graph is a straight line.y = −x + 4: the highest power is x.
- Use the number in front of x (the gradient). A positive gradient slopes up as you move right; a negative gradient slopes down.The gradient is −1, which is negative.
- Describe the shape. Put the two facts together.Linear: straight line going down as you move right.
Check. Test the idea with a point: when x = 1, y = 3, and when x = −1, y = 5. These agree with the shape described. ✓
Watch out. A graph with x² is a curve with one turning point; do not call a cubic a quadratic.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Look at the highest power of x (or whether x is on the bottom of a fraction) to name the graph, then use the sign to describe its direction.
- Look for the highest power of x. If the highest power is x², the graph is a quadratic (a parabola).y = x2 + 2: the highest power is x².
- Use the sign of the x² term. A positive x² term gives a U-shape. A negative one gives an n-shape.The number in front of x² is 1, which is positive.
- Describe the shape. Put the two facts together.Quadratic: U-shaped parabola (has a lowest point).
Check. Test the idea with a point: when x = 1, y = 3, and when x = −1, y = 3. These agree with the shape described. ✓
Watch out. A graph with x² is a curve with one turning point; do not call a cubic a quadratic.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Look at the highest power of x (or whether x is on the bottom of a fraction) to name the graph, then use the sign to describe its direction.
- Look for the highest power of x. If the highest power is x³, the graph is a cubic.y = x3: the highest power is x³.
- Use the sign of the x³ term. A positive x³ term makes the graph go from bottom-left to top-right. A negative one makes it go from top-left to bottom-right.The number in front of x³ is 1, which is positive.
- Describe the shape. Put the two facts together.Cubic: cubic curve rising from bottom-left to top-right.
Check. Test the idea with a point: when x = 1, y = 1, and when x = −1, y = −1. These agree with the shape described. ✓
Watch out. A graph with x² is a curve with one turning point; do not call a cubic a quadratic.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Look at the highest power of x (or whether x is on the bottom of a fraction) to name the graph, then use the sign to describe its direction.
- Look at where x is. In this rule x is on the bottom of a fraction. A rule of the form number ÷ x is a reciprocal graph.y = −4x
- Use the sign of the number on top. A positive number on top gives branches in the top-right and bottom-left. A negative number gives branches in the top-left and bottom-right.The number on top is −4, which is negative.
- Describe the shape. Put the two facts together.Reciprocal: reciprocal curve in two branches, in the top-left and bottom-right parts of the grid.
Check. Test the idea with a point: when x = 1, y = −4, and when x = −1, y = 4. These agree with the shape described. ✓
Watch out. A graph with x² is a curve with one turning point; do not call a cubic a quadratic.
| x | −2 | −1.5 | −1 | −0.5 | 0 | 0.5 | 1 | 1.5 | 2 |
|---|---|---|---|---|---|---|---|---|---|
| y | −1.875 | 0.375 | −0.375 | 1.875 |
Key words. To substitute means to replace the letter by a number. A table of values lists x-values and the y-values that go with them. Each pair (x, y) is a coordinate that can be plotted.
Method. Make a table of values, plot the points, then join them with a smooth curve.
- Make a table of values. Each y-value comes from putting an x-value into the rule. Use brackets round negative numbers.x = −2: y = (−2)3 − (−2) = −8 + 2 = −6
x = −1.5: y = (−1.5)3 − (−1.5) = −3.375 + 1.5 = −1.875
x = −1: y = (−1)3 − (−1) = −1 + 1 = 0
x = −0.5: y = (−0.5)3 − (−0.5) = −0.125 + 0.5 = 0.375
x = 0: y = 03 − 0 = 0 + 0 = 0
x = 0.5: y = 0.53 − 0.5 = 0.125 − 0.5 = −0.375
x = 1: y = 13 − 1 = 1 − 1 = 0
x = 1.5: y = 1.53 − 1.5 = 3.375 − 1.5 = 1.875
x = 2: y = 23 − 2 = 8 − 2 = 6 - List the coordinates. Each x with its y makes a point (x, y) to plot.(−2, −6), (−1.5, −1.875), (−1, 0), (−0.5, 0.375), (0, 0), (0.5, −0.375), (1, 0), (1.5, 1.875), (2, 6)
- Choose the scales for the axes. The x-values go from −2 to 2 and the y-values go from −6 to 6. The axes must cover all of these, and each square must always stand for the same amount.x-axis: −2 to 2. y-axis: −7 to 7.
- Plot each point carefully. Go along the x-axis first, then up or down to the y-value. Mark each point with a small cross.Plot (−2, −6), (−1.5, −1.875), (−1, 0), (−0.5, 0.375), (0, 0), (0.5, −0.375), (1, 0), (1.5, 1.875), (2, 6).
- Join the points with a smooth curve. The graph of this rule is a smooth curve, not a set of straight lines. Draw it freehand in one flowing line, passing through every point.Label the curve y = x3 − x.
Check. Every plotted point should lie on the curve, and the curve should pass through (−2, −6) and (2, 6) at the ends. Check one point by substituting. ✓
Watch out. Join the points with a smooth curve, not with straight segments.
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. Draw the horizontal line y = (the number on the right). Where it crosses the curve, read the x-values.
- Spot the graph being used. The graph shows y = 8x. The equation we want to solve is the same expression set equal to 3.2.8x = 3.2 means y = 3.2 on this graph
- Draw the horizontal line y = 3.2. Every point on this line has a y-value of 3.2. Where the line meets the curve, the curve has y = 3.2, which is exactly when the equation is true.Start at 3.2 on the y-axis and draw straight across (a ruler helps).
- Find where the line crosses the curve. Mark each crossing point.The line meets the curve at (2.5, 3.2).
- Read the x-values. The solutions of the equation are the x-values of the crossing points (the numbers along the bottom axis).x = 2.5
Check. Substitute back: y = 8 ÷ 2.5 = 3.2 ✓
Watch out. Read the x-values of the crossing points, not the y-values. The y-value is just the number you started with.
Key words. A distance-time graph shows how far something has travelled from its start (up the side) as time passes (along the bottom). The gradient (steepness) of the line is the speed, because gradient = change up ÷ change across = distance ÷ time. A flat line means stationary (not moving).
Method. Speed is the gradient of a distance-time graph: speed = change in distance ÷ change in time.
- Pick the part of the graph. Speed is the gradient of the line. We need the change in distance and the change in time for this part.From (0, 0) to (1, 15)
- Find the change in distance and the change in time. Subtract the start values from the end values.Change in distance = 15 − 0 = 15 km
Change in time = 1 − 0 = 1 hours - Divide distance by time. Speed = distance ÷ time.15 ÷ 1 = 15 km/h
Check. Distance = speed × time = 15 × 1 = 15 km, the change we read from the graph. ✓
Watch out. Use the change in distance and the change in time for the same part of the journey, not the totals from the start.
Key words. A distance-time graph shows how far something has travelled from its start (up the side) as time passes (along the bottom). The gradient (steepness) of the line is the speed, because gradient = change up ÷ change across = distance ÷ time. A flat line means stationary (not moving).
Method. A horizontal line shows a stop. Its length along the time axis is how long the stop lasts.
- Spot the flat part. A horizontal line means the distance is not changing, so the person is stationary (not moving).The line is flat from 1 to 1.5.
- Work out how long it lasts. Subtract the start time from the end time.1.5 − 1 = 0.5 hours
Check. The stop starts at 1 and ends at 1.5, so it lasts 0.5 hours. ✓
Watch out. A flat line does not mean "going at constant speed"; it means not moving at all.
Key words. A distance-time graph shows how far something has travelled from its start (up the side) as time passes (along the bottom). The gradient (steepness) of the line is the speed, because gradient = change up ÷ change across = distance ÷ time. A flat line means stationary (not moving).
Method. Average speed = total distance travelled ÷ total time taken.
- Find the total distance travelled. Add the distance of every part where the person moved, in either direction (going back still counts as distance travelled).15 + 20 = 35 km
- Find the total time taken. Use the whole time from the start to the end, including any stops.2.5 − 0 = 2.5 hours
- Divide total distance by total time. Average speed = total distance ÷ total time.35 ÷ 2.5 = 14 km/h
Check. Distance = average speed × time = 14 × 2.5 = 35 km. ✓
Watch out. Include the time spent stopped. And do not average the speeds of the parts: use total distance ÷ total time.
Key words. A speed-time graph shows the speed (up the side) as time passes (along the bottom). The gradient is the acceleration (how fast the speed changes), and the area under the graph is the distance travelled. Negative acceleration (the speed is dropping) is called deceleration.
Method. Acceleration is the gradient of a speed-time graph: change in speed ÷ change in time.
- Pick the part of the graph. Acceleration is the gradient of the line, so we use the change in speed and the change in time.From (0, 0) to (8, 16)
- Find the change in speed and the change in time. Subtract the start values from the end values.Change in speed = 16 − 0 = 16 m/s
Change in time = 8 − 0 = 8 s - Divide. Acceleration = change in speed ÷ change in time.16 ÷ 8 = 2
- Write the answer with its unit. The unit is metres per second per second, written m/s².Acceleration = 2 m/s²
Check. The speed changes by 2 m/s every second, for 8 s: 2 × 8 = 16 m/s, the change we found. ✓
Watch out. Acceleration is the gradient. The height of the line is the speed, not the acceleration.
Key words. A speed-time graph shows the speed (up the side) as time passes (along the bottom). The gradient is the acceleration (how fast the speed changes), and the area under the graph is the distance travelled. Negative acceleration (the speed is dropping) is called deceleration.
Method. Distance travelled = area under the speed-time graph. Split it into rectangles, triangles and trapezia.
- Remember: distance is the area under the graph. Distance = speed × time. On the graph, speed is the height and time is the width, so distance is the area between the line and the time axis.
- Split the area into simple shapes. Use a rectangle where the line is flat, a triangle where it starts or ends at 0, and a trapezium otherwise.From 0 s to 8 s, a triangle: ½ × 8 × 16 = 64
From 8 s to 18 s, a rectangle: 16 × 10 = 160
From 18 s to 22 s, a triangle: ½ × 4 × 16 = 32 - Add the areas. The total area is the total distance travelled.64 + 160 + 32 = 256 m
Check. Sense check: the vehicle never goes faster than 16 m/s and travels for 22 s, so the distance must be less than 352 m. Our 256 m is smaller. ✓
Watch out. The unit of the area is the unit of distance (m), not m/s. Area = height × width means speed × time.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. The maximum is at the turning point of the curve.
- Find the highest point of the curve. This is the turning point. A U-shaped curve has a lowest point; an n-shaped curve has a highest point.The turning point is (2, 20).
- Read both coordinates. The horizontal coordinate is the time at which it happens. The vertical coordinate is the maximum height.Time = 2 s, height = 20 m
- Write the answer in words. Always say what the numbers mean and give units.The maximum height is 20 m, at 2 s.
Check. Use the rule: t = 2: h = −5 × (2)2 + 20 × 2 = −20 + 40 = 20. ✓
Watch out. The question may ask for the maximum value (a height) or for when it happens (a time). Read the right coordinate.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Go across from the height to the curve, then down to the time. Check for more than one crossing.
- Find 15 m on the vertical axis. The vertical axis shows the height.Locate h = 15.
- Draw a horizontal line across to the curve. Use a ruler. Every point on this line has a height of 15 m. Mark every place where it meets the curve (there may be more than one).The line meets the curve at (1, 15) and (3, 15).
- Go straight down from each point to the horizontal axis and read the time. The horizontal axis shows the time.1 and 3 s
Check. Use the rule: t = 1: h = −5 × (1)2 + 20 × 1 = −5 + 20 = 15; t = 3: h = −5 × (3)2 + 20 × 3 = −45 + 60 = 15. ✓
Watch out. A horizontal line can cross a curve twice. Look for both crossing points.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. The time in the air is the time between the two points where the height is 0.
- Find where the height is 0. The curve meets the horizontal axis at the start and at the end.At 0 s and 4 s.
- Subtract to find how long it lasts. The time between the two is the end minus the start.4 − 0 = 4 s
Check. The curve is symmetrical about time 2 s, the middle of 0 and 4. ✓
Watch out. Give the length of time, not the two readings.
Key words. A distance-time graph shows how far something has travelled from its start (up the side) as time passes (along the bottom). The gradient (steepness) of the line is the speed, because gradient = change up ÷ change across = distance ÷ time. A flat line means stationary (not moving).
Method. They meet where the lines cross.
- Find the crossing point. Look where the two lines meet.(2, 40)
- Read the time. The horizontal coordinate.2 hours after Priya leaves
Check. At t = 2 both are at 40 km. ✓
Watch out. Time is horizontal.
Key words. A distance-time graph shows how far something has travelled from its start (up the side) as time passes (along the bottom). The gradient (steepness) of the line is the speed, because gradient = change up ÷ change across = distance ÷ time. A flat line means stationary (not moving).
Method. They meet where the lines cross.
- Find the crossing point. (2, 40)
- Read the distance. The vertical coordinate.40 km from A
Check. Priya: 20 × 2 = 40. ✓
Watch out. Distance is vertical.
Key words. To sketch a graph means to draw its shape with the key points labelled, without plotting every point. The intercepts are where the graph crosses the axes.
Method. Find the shape, the y-intercept, the roots and the turning point, then draw a smooth curve through them.
- Decide the shape. The sign of the x² term tells us: positive gives a U-shape, negative gives an n-shape.The number in front of x2 is 1, so the curve is U-shaped.
- Find the y-intercept. The curve crosses the y-axis where x = 0, which is the constant term.x = 0: y = 02 − 2 × 0 − 8 = 0 + 0 − 8 = −8
The y-intercept is (0, −8). - Find the roots. The roots are where the curve meets the x-axis, so put y = 0 and solve. Factorise, then use "if two brackets multiply to 0, one of them is 0".Spot the pattern. This is a quadratic of the form x² + bx + c, with b = -2 and c = -8.
We need two numbers that multiply to -8 (the last number) and add to -2 (the middle number). List the pairs that multiply to -8:
1 × -8 = -8, add: 1 + -8 = -7 ✗
-1 × 8 = -8, add: -1 + 8 = 7 ✗
2 × -4 = -8, add: 2 + -4 = -2 ✓ this pair works
Put the pair into two brackets: (x + 2)(x − 4).
Check by expanding: x² − 4x + 2x − 8 = x2 − 2x − 8. ✓
Set (x − 4)(x + 2) = 0, so x = −2 or x = 4
The curve crosses the x-axis at (−2, 0) and (4, 0). - Find the turning point. The turning point is on the line of symmetry. Complete the square, or use the midpoint of the roots.y = (x − 1)2 − 9
Turning point = (1, −9) - Draw the axes and the curve. A sketch does not need an accurate scale. It must show the right shape and label the key points: the y-intercept, any roots and the turning point.Label (0, −8), the roots, and the turning point (1, −9).
Check. The curve is symmetrical: the turning point lies halfway between the two roots, and the y-intercept is on the curve (substitute x = 0 to see). ✓
Watch out. A sketch needs labelled coordinates, not a perfect drawing. Make sure the curve is smooth and symmetrical.
Key words. Completing the square means writing x² + bx + c as (x + p)² + q. The turning point is the lowest point (minimum) of a U-shape or the highest point (maximum) of an n-shape.
Method. Complete the square: write the quadratic as a(x + p)² + q. Then the turning point is (−p, q).
- Write the quadratic and look at the numbers. We want it in the form a(x + p)² + q. That form shows the turning point at a glance.y = x2 + 8x + 3
- Halve the number in front of x. In (x + p)² = x² + 2px + p², the middle term is 2p. So p is half of the number in front of x.8 ÷ 2 = 4, so the bracket is (x + 4)
- Square the bracket and see what it gives. Expanding the bracket gives the x² and x terms we want, but it also adds an extra number p².(x + 4)2 = x2 + 8x + 16
- Take away the extra number and add the original constant. We have an extra 16, so subtract it to keep the value the same. Then put back the constant term.x2 + 8x + 3 = (x + 4)2 − 16 + 3
- Tidy up the numbers. Combine the two plain numbers.y = (x + 4)2 − 13
- Find the turning point. A squared bracket is never negative, and it equals 0 when the bracket is 0. So y is at its minimum when (x + 4) = 0, that is when x = −4. Then y = −13.Turning point = (−4, −13) (minimum value of y is −13)
Check. Expand the answer: (x + 4)2 − 13 = x2 + 8x + 3, which is the original. ✓
Watch out. The turning point has the opposite sign to the number in the bracket: (x − 3)² + 1 has its turning point at x = +3.
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. Rearrange the equation so that one side is the rule of the graph you are given. The other side is the line to draw.
- Compare the equation with the graph that is drawn. The graph is y = x2 − 2x − 3. We try to make the left side of the equation look like x2 − 2x − 3.x2 − x − 5 = 0
- Split the left side into the graph's expression plus what is left over. Use the x² and x terms so that the graph's rule appears exactly. Whatever is left is the extra part.x2 − x − 5 = (x2 − 2x − 3) + (x − 2)
- Move the left-over part to the other side. Do the same to both sides: take away the left-over part (every term changes sign when it crosses to the other side).x2 − 2x − 3 = −x + 2
- Decide which line to draw on the grid. The curve already shows y = x2 − 2x − 3. The right-hand side gives the second graph.Draw the line y = −x + 2.
- Read the crossing points. The line and curve cross where the two sides are equal, which is where the original equation is true.They cross at (−1.8, 3.8) and (2.8, −0.8).
- Write the x-values. The solutions are the x-values of the crossing points.x ≈ −1.8 or 2.8 (accept answers within 0.1)
Check. Substituting the unrounded values makes the left side exactly 0; the rounded readings make it very nearly 0. ✓
Watch out. Watch the signs when you move terms to the other side: each one changes sign.
Key words. A surd is a root that does not simplify to a whole number or fraction, such as √5. Exact means leaving it as √5 instead of a rounded decimal.
Method. Complete the square, then undo it: square root both sides (positive and negative).
- Put y = 0. Roots are where the curve meets the x-axis.0 = x2 − 4x + 1
- Complete the square. Halve the number in front of x (−4 ÷ 2 = −2), square the bracket, and correct the constant.0 = (x − 2)2 − 3
- Get the bracket on its own. Move the 3 to the other side (do the inverse operation).(x − 2)2 = 3
- Take the square root of both sides. There are two square roots: positive and negative. Leave the answer as a surd (an exact root), simplified if possible.x − 2 = ±√3
- Get x on its own. Do the inverse of subtracting 2.x = 2 ± √3
Check. Decimals: 2 + 1.73 = 3.73 and 2 − 1.73 = 0.27. Substituting 3.73 into the rule gives about 0. ✓
Watch out. Do not forget the ±. Leave surds exact unless the question asks for decimals.