GCSE Maths: Roots, intercepts, symmetry and turning points (Foundation)
Part of the GCSE Maths quadratic and other graphs chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Roots, intercepts, symmetry and turning points
A quadratic graph is full of useful information. There are four things you must be able to read from the graph.
The roots are the x-values where the curve meets the x-axis. On the x-axis every point has y = 0, so the roots are the solutions of the equation y = 0. A quadratic can have two roots, one root (the curve just touches the axis) or no roots (the curve never reaches the axis).
The y-intercept is where the curve crosses the y-axis. On the y-axis every point has x = 0, so the y-intercept is the number on its own in the rule (for y = x² − 2x − 3 it is −3).
The turning point is the lowest point of a U-shaped curve (a minimum) or the highest point of an n-shaped curve (a maximum). Give it as a coordinate pair (x, y).
The line of symmetry is the vertical mirror line through the turning point. Its equation is x = (the x-value of the turning point). It is exactly halfway between the two roots, which is a handy way to find it.
Finding roots by algebra. You can also find roots without a graph. Put y = 0. If the rule is written as two brackets multiplied together, such as y = (x − 3)(x + 2), then (x − 3)(x + 2) = 0. Two numbers multiplied together give 0 only when one of them is 0, so x − 3 = 0 or x + 2 = 0. That gives x = 3 or x = −2. Notice that the sign flips. If the rule is y = x² − 16, put y = 0, so x² = 16, and x = 4 or x = −4 (both square to give 16).
Key ideas
- Roots: where the curve crosses or touches the x-axis (y = 0). Give them as x = ... values.
- y-intercept: where the curve crosses the y-axis (x = 0).
- Turning point: the lowest point (U-shape, minimum) or highest point (n-shape, maximum). Give it as (x, y).
- Line of symmetry: the vertical line x = (x-value of the turning point), halfway between the roots.
- A quadratic has 0, 1 or 2 roots.
- To find roots by algebra put y = 0: for (x + p)(x + q) = 0 each bracket in turn equals 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 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 (1, 0) and (3, 0).
- Write the roots. We only need the x-values (the roots are x-values, not coordinates).x = 1 or 3
Check. Substitute each root into the rule; y should be 0.
y = 12 − 4 × 1 + 3 = 1 − 4 + 3 = 0
y = 32 − 4 × 3 + 3 = 9 − 12 + 3 = 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). 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 as a coordinate pair. It is a maximum (highest point).(1, 9)
Check. Substitute x = 1: y = −(1)2 + 2 × 1 + 8 = −1 + 2 + 8 = 9. ✓
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. Put y = 0. When two brackets multiply to give 0, one of the brackets must be 0.
- Roots are where y = 0. The roots are the x-values where the curve meets the x-axis. On the x-axis y = 0, so put y = 0 in the equation.0 = (x − 3)(x + 2)
- Use the fact that a product is zero only if one factor is zero. If two things multiply to give 0, at least one of them must be 0. So at least one bracket must equal 0.(x − 3) = 0 or (x + 2) = 0
- Solve each little equation. Get x on its own by doing the inverse operation to both sides.x − 3 = 0 → x = 3
x + 2 = 0 → x = −2 - Write the roots. Both values make y = 0.x = 3 or −2
Check. Substitute: y = (−2)2 − (−2) − 6 = 4 + 2 − 6 = 0; y = 32 − 3 − 6 = 9 − 3 − 6 = 0 ✓
Watch out. The sign flips: the bracket (x + 3) gives the root x = −3, not +3.
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, −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 = 2
Check. The line of symmetry is halfway between the roots: (−1 + 5) ÷ 2 = 2. ✓
Watch out. A vertical line is written x = number (not y = number). y = number is a horizontal line.
Watch out
The roots are x-values (written x = 3). The y-intercept is a y-value. A turning point is a coordinate pair. Do not swap the order: the turning point of the curve in the diagram is (1, −4), not (−4, 1). And a vertical line is x = number, not y = number.Exam tip
The roots and the line of symmetry check each other: the line of symmetry is always halfway between the roots. Spot a mistake instantly by checking this.Exercise 13B Reading quadratic graphs
Non-calculatorKey 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 (−1, 0) and (3, 0).
- Write the roots. We only need the x-values (the roots are x-values, not coordinates).x = −1 or 3
Check. Substitute each root into the rule; y should be 0.
y = (−1)2 − 2 × (−1) − 3 = 1 + 2 − 3 = 0
y = 32 − 2 × 3 − 3 = 9 − 6 − 3 = 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 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, −3).
- Write the answer. The y-intercept is the y-value at that point.y-intercept = −3
Check. Substitute x = 0 into the rule: y = 02 − 2 × 0 − 3 = 0 + 0 − 3 = −3. ✓
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 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 (1, −4).
- Write the answer as a coordinate pair. It is a minimum (lowest point).(1, −4)
Check. Substitute x = 1: y = 12 − 2 × 1 − 3 = 1 − 2 − 3 = −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 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 = (1, −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 = 1
Check. The line of symmetry is halfway between the roots: (−1 + 3) ÷ 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 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 (1, 0) and (3, 0).
- Write the roots. We only need the x-values (the roots are x-values, not coordinates).x = 1 or 3
Check. Substitute each root into the rule; y should be 0.
y = 12 − 4 × 1 + 3 = 1 − 4 + 3 = 0
y = 32 − 4 × 3 + 3 = 9 − 12 + 3 = 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, −1).
- Write the answer as a coordinate pair. It is a minimum (lowest point).(2, −1)
Check. Substitute x = 2: y = 22 − 4 × 2 + 3 = 4 − 8 + 3 = −1. ✓
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. A U-shaped curve has a lowest point, so its turning point is a minimum.
- Look at the shape. The curve opens upwards: it is U-shaped.U-shaped
- Decide. The turning point of a U-shaped curve is its lowest point, so it is a minimum.Minimum (the lowest value of y is −1)
Check. The y-values either side of the turning point are higher: at x = 1 and x = 3, y = 0, which is above −1. ✓
Watch out. U = minimum, n = maximum.
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, 4).
- Write the answer. The y-intercept is the y-value at that point.y-intercept = 4
Check. Substitute x = 0 into the rule: y = −(0)2 + 4 = 0 + 4 = 4. ✓
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 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 (−2, 0) and (2, 0).
- Write the roots. We only need the x-values (the roots are x-values, not coordinates).x = −2 or 2
Check. Substitute each root into the rule; y should be 0.
y = −(−2)2 + 4 = −4 + 4 = 0
y = −(2)2 + 4 = −4 + 4 = 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 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 (0, 4).
- Write the answer. The maximum (highest point) value of y is the y-coordinate of the turning point.maximum value of y = 4
Check. Substitute x = 0: y = −(0)2 + 4 = 0 + 4 = 4. ✓
Watch out. The minimum value is the y-value (height), not the x-value.
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 (−4, 0) and (2, 0).
- Write the roots. We only need the x-values (the roots are x-values, not coordinates).x = −4 or 2
Check. Substitute each root into the rule; y should be 0.
y = (−4)2 + 2 × (−4) − 8 = 16 − 8 − 8 = 0
y = 22 + 2 × 2 − 8 = 4 + 4 − 8 = 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 (−1, −9).
- Write the answer as a coordinate pair. It is a minimum (lowest point).(−1, −9)
Check. Substitute x = −1: y = (−1)2 + 2 × (−1) − 8 = 1 − 2 − 8 = −9. ✓
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 minimum 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). 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 (−1, −9).
- Write the answer. The minimum (lowest point) value of y is the y-coordinate of the turning point.minimum 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. 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 = (−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: (−4 + 2) ÷ 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. Roots are where the curve meets the x-axis (y = 0).
- Remember what a root is. A root is an x-value where the curve meets the x-axis, where y = 0.
- Look for crossing points. The curve just touches the x-axis at its turning point and turns back, without crossing.It touches the x-axis at (3, 0).
- Write the answer. There is only one place, so there is one (repeated) root.x = 3
Check. Substitute x = 3: y = 32 − 6 × 3 + 9 = 9 − 18 + 9 = 0. ✓
Watch out. A curve that only touches the x-axis has one root, not two.
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. Roots are where the curve meets the x-axis. Check whether it does.
- Remember what a root is. A root is an x-value where the curve meets the x-axis, where y = 0.
- Look for crossing points. Follow the curve and see whether it ever touches or crosses the x-axis.The curve is above the x-axis everywhere, so it never meets it.
- Write the answer. No crossing points means there are no roots.No roots. (there is no real x-value with y = 0)
Check. The lowest point is y = 2, which is above 0, so the curve can never reach y = 0. ✓
Watch out. "No roots" is a proper answer. Do not invent crossing points.
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 exactly halfway between the roots, so use the midpoint.
- Add the roots. The midpoint is the average of the two numbers.1 + 5 = 6
- Halve it. Divide by 2.6 ÷ 2 = 3
- Write the vertical line. A vertical line is x = number.x = 3
Check. 3 is 2 away from 1 and 2 away from 5. ✓
Watch out. Vertical lines are x = number.
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 exactly halfway between the roots, so use the midpoint.
- Add the roots. Remember to add a negative.−4 + 2 = −2
- Halve it. Divide by 2.−2 ÷ 2 = −1
- Write the vertical line. A vertical line is x = number.x = −1
Check. −1 is 3 away from −4 and 3 away from 2. ✓
Watch out. Take care adding a negative number.
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 exactly halfway between the roots, so use the midpoint.
- Add the roots. Both are negative.−7 + (−1) = −8
- Halve it. Divide by 2.−8 ÷ 2 = −4
- Write the vertical line. A vertical line is x = number.x = −4
Check. −4 is 3 away from −7 and 3 away from −1. ✓
Watch out. Vertical lines are x = number.
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. Put y = 0. When two brackets multiply to give 0, one of the brackets must be 0.
- Roots are where y = 0. The roots are the x-values where the curve meets the x-axis. On the x-axis y = 0, so put y = 0 in the equation.0 = (x − 3)(x + 2)
- Use the fact that a product is zero only if one factor is zero. If two things multiply to give 0, at least one of them must be 0. So at least one bracket must equal 0.(x − 3) = 0 or (x + 2) = 0
- Solve each little equation. Get x on its own by doing the inverse operation to both sides.x − 3 = 0 → x = 3
x + 2 = 0 → x = −2 - Write the roots. Both values make y = 0.x = 3 or −2
Check. Substitute: y = (−2)2 − (−2) − 6 = 4 + 2 − 6 = 0; y = 32 − 3 − 6 = 9 − 3 − 6 = 0 ✓
Watch out. The sign flips: the bracket (x + 3) gives the root x = −3, not +3.
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. Put y = 0. When two brackets multiply to give 0, one of the brackets must be 0.
- Roots are where y = 0. The roots are the x-values where the curve meets the x-axis. On the x-axis y = 0, so put y = 0 in the equation.0 = (x + 1)(x − 5)
- Use the fact that a product is zero only if one factor is zero. If two things multiply to give 0, at least one of them must be 0. So at least one bracket must equal 0.(x + 1) = 0 or (x − 5) = 0
- Solve each little equation. Get x on its own by doing the inverse operation to both sides.x + 1 = 0 → x = −1
x − 5 = 0 → x = 5 - Write the roots. Both values make y = 0.x = −1 or 5
Check. Substitute: y = (−1)2 − 4 × (−1) − 5 = 1 + 4 − 5 = 0; y = 52 − 4 × 5 − 5 = 25 − 20 − 5 = 0 ✓
Watch out. The sign flips: the bracket (x + 3) gives the root x = −3, not +3.
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. Put y = 0. When two brackets multiply to give 0, one of the brackets must be 0.
- Roots are where y = 0. The roots are the x-values where the curve meets the x-axis. On the x-axis y = 0, so put y = 0 in the equation.0 = x(x − 4)
- Use the fact that a product is zero only if one factor is zero. If two things multiply to give 0, at least one of them must be 0. So at least one bracket must equal 0.x = 0 or (x − 4) = 0
- Solve each little equation. Get x on its own by doing the inverse operation to both sides.x = 0
x − 4 = 0 → x = 4 - Write the roots. Both values make y = 0.x = 0 or 4
Check. Substitute: y = 02 − 4 × 0 = 0 + 0 = 0; y = 42 − 4 × 4 = 16 − 16 = 0 ✓
Watch out. The sign flips: the bracket (x + 3) gives the root x = −3, not +3.
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. Put y = 0, get x² on its own, then take the square root (positive and negative).
- Roots are where y = 0. Put y = 0 in the equation.0 = x2 − 25
- Get x² on its own. Add 25 to both sides to undo the −25.x2 = 25
- Square root both sides. We want the numbers that, when squared, make 25. There are two: a positive and a negative, because (−5)² = 25 as well as 5² = 25.x = 5 or x = −5
- Write the roots. The roots are the same distance either side of 0, as the parabola is symmetrical about the y-axis.x = ±5
Check. Substitute x = 5: y = 52 − 25 = 25 − 25 = 0. And x = −5: y = (−5)2 − 25 = 25 − 25 = 0. ✓
Watch out. Remember the negative root as well. A quadratic like this has two roots, not one.
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. Put y = 0, get x² on its own, then take the square root (positive and negative).
- Roots are where y = 0. Put y = 0 in the equation.0 = x2 − 49
- Get x² on its own. Add 49 to both sides to undo the −49.x2 = 49
- Square root both sides. We want the numbers that, when squared, make 49. There are two: a positive and a negative, because (−7)² = 49 as well as 7² = 49.x = 7 or x = −7
- Write the roots. The roots are the same distance either side of 0, as the parabola is symmetrical about the y-axis.x = ±7
Check. Substitute x = 7: y = 72 − 49 = 49 − 49 = 0. And x = −7: y = (−7)2 − 49 = 49 − 49 = 0. ✓
Watch out. Remember the negative root as well. A quadratic like this has two roots, not one.
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. Put y = 0. When two brackets multiply to give 0, one of the brackets must be 0.
- Roots are where y = 0. The roots are the x-values where the curve meets the x-axis. On the x-axis y = 0, so put y = 0 in the equation.0 = (x − 6)(x − 6)
- Use the fact that a product is zero only if one factor is zero. If two things multiply to give 0, at least one of them must be 0. So at least one bracket must equal 0.(x − 6) = 0 or (x − 6) = 0
- Solve each little equation. Get x on its own by doing the inverse operation to both sides.x − 6 = 0 → x = 6
x − 6 = 0 → x = 6 - Write the roots. Both values make y = 0.x = 6
Check. Substitute: y = 62 − 12 × 6 + 36 = 36 − 72 + 36 = 0 ✓
Watch out. The sign flips: the bracket (x + 3) gives the root x = −3, not +3.
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 (1, 0) and (5, 0).
- Write the roots. We only need the x-values (the roots are x-values, not coordinates).x = 1 or 5
Check. Substitute each root into the rule; y should be 0.
y = −(1)2 + 6 × 1 − 5 = −1 + 6 − 5 = 0
y = −(5)2 + 6 × 5 − 5 = −25 + 30 − 5 = 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). 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 (3, 4).
- Write the answer as a coordinate pair. It is a maximum (highest point).(3, 4)
Check. Substitute x = 3: y = −(3)2 + 6 × 3 − 5 = −9 + 18 − 5 = 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 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 (3, 4).
- Write the answer. The maximum (highest point) value of y is the y-coordinate of the turning point.maximum value of y = 4
Check. Substitute x = 3: y = −(3)2 + 6 × 3 − 5 = −9 + 18 − 5 = 4. ✓
Watch out. The minimum value is the y-value (height), not the x-value.
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 = (3, 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 = 3
Check. The line of symmetry is halfway between the roots: (1 + 5) ÷ 2 = 3. ✓
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 on the line of symmetry, which is halfway between the roots. Find its x-value first, then substitute to find its y-value.
- Find the x-value of the turning point. It is halfway between the roots 1 and 5.(1 + 5) ÷ 2 = 3
- Substitute x = 3 into the rule to find y. This gives the height of the curve at the turning point.x = 3: y = 32 − 6 × 3 + 5 = 9 − 18 + 5 = −4
- Write the coordinates. x first, then y.(3, −4)
Check. The curve is U-shaped (positive x²), so the turning point is a minimum, and −4 is below the roots at y = 0. ✓
Watch out. Do not stop at the line of symmetry: the turning point needs a y-value too.
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 halfway between the roots.
- Add the roots. −3 + 7 = 4
- Halve. This is the average of the two roots.4 ÷ 2 = 2
- Write the line. A vertical line.x = 2
Check. 2 is 5 away from −3 and 5 away from 7. ✓
Watch out. Vertical lines are x = number.
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 on the line of symmetry, and its y-value is the maximum value given.
- Use the line of symmetry for x. The turning point has the same x-value as the line of symmetry.x = 2
- Use the maximum for y. The highest point has y = 25.y = 25
- Write the coordinates. x first.(2, 25)
Check. The turning point lies on the mirror line x = 2. ✓
Watch out. Write coordinates as (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 line of symmetry is halfway between the roots. Work it out and compare with Sam's answer.
- Find the midpoint of 2 and 6. Add them and halve.(2 + 6) ÷ 2 = 4
- Compare. Sam's 8 is the sum of the roots (2 + 6 = 8), but he forgot to halve it.Correct line: x = 4; Sam said x = 8
- Write the conclusion. Sam is wrong. The line of symmetry is x = 4.
Check. x = 4 is 2 away from 2 and 2 away from 6, so it is halfway. x = 8 is not between the roots at all. ✓
Watch out. Halfway means add and then halve.
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. Think about how a U-shaped or n-shaped curve can meet a horizontal line.
- Picture the shape. A parabola goes in one direction to its turning point and then comes back the other way, once.
- Count the crossings. A U-shaped curve with its lowest point above the axis never meets the axis (no roots). If the lowest point is on the axis it touches once (one root). If the lowest point is below the axis, the curve comes down through the axis and goes back up through it: two crossings.
- Conclude. The curve turns only once, so it can cross the axis at most twice.No roots, one root or two roots; never three.
Check. Sketch a U and slide a horizontal line up and down. It meets the curve 0, 1 or 2 times. ✓
Watch out. This is the key idea behind "how many solutions".
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. Check her answer against the graph and the rule.
- Substitute her x-value. If (−4, 1) were on the curve, then x = −4 would give y = 1.x = −4: y = (−4)2 − 2 × (−4) − 3 = 16 + 8 − 3 = 21
- Compare. We get 21, not 1, so (−4, 1) is not even on the curve. She has written the coordinates the wrong way round.The correct point has x = 1 and y = −4.
- Write the correct coordinates. The lowest point is where x = 1: y = 1 − 2 − 3 = −4.(1, −4)
Check. x = 1: y = 12 − 2 × 1 − 3 = 1 − 2 − 3 = −4. ✓
Watch out. Coordinates are (x, y): across first, then up or down.
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 curve crosses the y-axis where x = 0.
- Put x = 0 in the rule. Every point on the y-axis has x = 0.x = 0: y = 02 − 2 × 0 − 3 = 0 + 0 − 3 = −3
- Write the coordinates. (0, −3)
Check. The rule has −3 on its own, which is the y-intercept. ✓
Watch out. x = 0 on the y-axis.
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 x = 1 (through the turning point). The mirror image of (0, −3) is the same distance on the other side.
- Find the distance from the line of symmetry. x = 0 is 1 to the left of x = 1.1 − 0 = 1
- Go the same distance to the right. The mirror image is 1 to the right of x = 1.1 + 1 = 2
- Write the point. The y-value stays the same.(2, −3)
Check. x = 2: y = 22 − 2 × 2 − 3 = 4 − 4 − 3 = −3 ✓
Watch out. The y-value does not change.
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. Say how symmetry gives the second point.
- State the mirror line. The line of symmetry is x = 1.
- Reflect the point. A point and its mirror image are the same distance from the line and at the same height.(0, −3) reflects to (2, −3)
Check. Both points give y = −3 in the rule. ✓
Watch out. Give a reason in words.