GCSE Maths: Cubic and reciprocal graphs (Higher)
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)
Cubic and reciprocal graphs
Cubic graphs. A cubic rule has x³ (“x cubed”) as its highest power, for example y = x³ or y = x³ − 4x. You draw it in the same way as a quadratic: make a table of values, plot the points, join them with a smooth curve. Be careful with negatives: (−2)³ = (−2) × (−2) × (−2) = −8. An odd number of negative signs gives a negative answer, so a cube of a negative number is negative (unlike a square).
The graph of y = x³ is a smooth S-shaped curve. It goes up from the bottom left to the top right, passing through (0, 0). It is steeper the further you go from the origin. With a negative sign, y = −x³ is the same shape flipped so that it goes down from the top left to the bottom right. Many cubics also have a small “wiggle”, with one hump and one dip, as in y = x³ − 4x. A cubic can cross the x-axis once, twice or three times.
Reciprocal graphs. The reciprocal of a number is 1 divided by it. The graph of y = 1 ÷ x (written 1x) has a very different shape. You cannot divide by 0, so there is no point at x = 0. The graph has two separate branches: for positive x the values of y are positive and get smaller as x grows (1, 0.5, 0.25, …); for negative x the values of y are negative. Near x = 0 the curve shoots off steeply. The curve gets closer and closer to the axes but never touches them: the axes are called asymptotes.
The graph of y = k ÷ x (for example y = 6 ÷ x) has the same shape, further from the axes when k is bigger. If k is negative the two branches swap: they are in the top-left and bottom-right.
Key ideas
- A cubic has x³ as its highest power. A negative number cubed is negative: (−2)³ = −8.
- The graph of y = x³ is S-shaped, up from bottom-left to top-right, through (0, 0).
- y = −x³ goes down from top-left to bottom-right. Cubics can also have a hump and a dip.
- The graph of y = k ÷ x has two branches and no point at x = 0.
- Positive k: branches in top-right and bottom-left. Negative k: top-left and bottom-right.
- The curve gets close to the axes (asymptotes) but never touches them.
| x | −3 | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y |
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: −3, −2, −1, 0, 1, 2, 3 - Substitute the x-values one at a time. Do the same calculation every time, only the number changes. Use brackets round negative numbers.x = −3: y = (−3)3 = −27
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
x = 3: y = 33 = 27 - Write the y-values in the table. Each pair of numbers is a coordinate, ready to be plotted on a graph.y-values: −27, −8, −1, 0, 1, 8, 27
Coordinates: (−3, −27), (−2, −8), (−1, −1), (0, 0), (1, 1), (2, 8), (3, 27)
Check. First gaps: 19, 7, 1, 1, 7, 19. Gaps of the gaps: −12, −6, 0, 6, 12. Gaps of those: 6, 6, 6, 6, always the same, which is the pattern for a cubic. ✓
Watch out. A negative number cubed is negative: (−2)³ = −8.
| x | −3 | −2.5 | −2 | −1 | 0 | 1 | 2 | 2.5 | 3 |
|---|---|---|---|---|---|---|---|---|---|
| y |
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 = −3: y = (−3)3 − 4 × (−3) = −27 + 12 = −15
x = −2.5: y = (−2.5)3 − 4 × (−2.5) = −15.625 + 10 = −5.625
x = −2: y = (−2)3 − 4 × (−2) = −8 + 8 = 0
x = −1: y = (−1)3 − 4 × (−1) = −1 + 4 = 3
x = 0: y = 03 − 4 × 0 = 0 + 0 = 0
x = 1: y = 13 − 4 × 1 = 1 − 4 = −3
x = 2: y = 23 − 4 × 2 = 8 − 8 = 0
x = 2.5: y = 2.53 − 4 × 2.5 = 15.625 − 10 = 5.625
x = 3: y = 33 − 4 × 3 = 27 − 12 = 15 - List the coordinates. Each x with its y makes a point (x, y) to plot.(−3, −15), (−2.5, −5.625), (−2, 0), (−1, 3), (0, 0), (1, −3), (2, 0), (2.5, 5.625), (3, 15)
- Choose the scales for the axes. The x-values go from −3 to 3 and the y-values go from −15 to 15. The axes must cover all of these, and each square must always stand for the same amount.x-axis: −3 to 3. y-axis: −20 to 20.
- 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 (−3, −15), (−2.5, −5.625), (−2, 0), (−1, 3), (0, 0), (1, −3), (2, 0), (2.5, 5.625), (3, 15).
- 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 − 4x.
Check. Every plotted point should lie on the curve, and the curve should pass through (−3, −15) and (3, 15) at the ends. Check one point by substituting. ✓
Watch out. Join the points with a smooth curve, not with straight segments.
| x | −4 | −2 | −1 | −0.5 | 0.5 | 1 | 2 | 4 |
|---|---|---|---|---|---|---|---|---|
| y |
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 = −4: y = 1 ÷ (−4) = −0.25
x = −2: y = 1 ÷ (−2) = −0.5
x = −1: y = 1 ÷ (−1) = −1
x = −0.5: y = 1 ÷ (−0.5) = −2
x = 0.5: y = 1 ÷ 0.5 = 2
x = 1: y = 1 ÷ 1 = 1
x = 2: y = 1 ÷ 2 = 0.5
x = 4: y = 1 ÷ 4 = 0.25 - List the coordinates. Each x with its y makes a point (x, y) to plot.(−4, −0.25), (−2, −0.5), (−1, −1), (−0.5, −2), (0.5, 2), (1, 1), (2, 0.5), (4, 0.25)
- Choose the scales for the axes. The x-values go from −4 to 4 and the y-values go from −2 to 2. The axes must cover all of these, and each square must always stand for the same amount.x-axis: −4 to 4. y-axis: −3 to 3.
- 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 (−4, −0.25), (−2, −0.5), (−1, −1), (−0.5, −2), (0.5, 2), (1, 1), (2, 0.5), (4, 0.25).
- 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, in two separate branches, one on each side of x = 0.Label the curve y = 1x.
Check. Every plotted point should lie on the curve, and the curve should pass through (−4, −0.25) and (4, 0.25) at the ends. Check one point by substituting. ✓
Watch out. The two branches never join up: there is no point at x = 0.
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 = 6x. The equation we want to solve is the same expression set equal to 2.5.6x = 2.5 means y = 2.5 on this graph
- Draw the horizontal line y = 2.5. Every point on this line has a y-value of 2.5. Where the line meets the curve, the curve has y = 2.5, which is exactly when the equation is true.Start at 2.5 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 (2.4, 2.5).
- Read the x-values. The solutions of the equation are the x-values of the crossing points (the numbers along the bottom axis).x = 2.4
Check. Substitute back: y = 6 ÷ 2.4 = 2.5 ✓
Watch out. Read the x-values of the crossing points, not the y-values. The y-value is just the number you started with.
Watch out
A cube of a negative number is negative, but a square of a negative number is positive. (−2)³ = −8 but (−2)² = 4. And on a reciprocal graph never join the two branches, and never draw a point at x = 0.Exam tip
To check a reciprocal table, multiply x by y. For y = 6 ÷ x every pair must multiply to 6: 2 × 3 = 6, (−3) × (−2) = 6.Exercise 13D Cubic and reciprocal graphs
Non-calculator| x | −3 | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y | 1 | 3 |
| x | −3 | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y | −25 | −6 | 1 | 2 | 3 | 10 | 29 |
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 + 2
x-values: −3, −2, −1, 0, 1, 2, 3 - Substitute the x-values one at a time. Do the same calculation every time, only the number changes. Use brackets round negative numbers.x = −3: y = (−3)3 + 2 = −27 + 2 = −25
x = −2: y = (−2)3 + 2 = −8 + 2 = −6
x = −1: y = (−1)3 + 2 = −1 + 2 = 1
x = 0: y = 03 + 2 = 0 + 2 = 2
x = 1: y = 13 + 2 = 1 + 2 = 3
x = 2: y = 23 + 2 = 8 + 2 = 10
x = 3: y = 33 + 2 = 27 + 2 = 29 - Write the y-values in the table. Each pair of numbers is a coordinate, ready to be plotted on a graph.y-values: −25, −6, 1, 2, 3, 10, 29
Coordinates: (−3, −25), (−2, −6), (−1, 1), (0, 2), (1, 3), (2, 10), (3, 29)
Check. First gaps: 19, 7, 1, 1, 7, 19. Gaps of the gaps: −12, −6, 0, 6, 12. Gaps of those: 6, 6, 6, 6, always the same, which is the pattern for a cubic. ✓
Watch out. A negative number cubed is negative: (−2)³ = −8.
| x | −2 | −1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y |
| x | −2 | −1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y | −16 | −2 | 0 | 2 | 16 |
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 = 2x3
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 = −16
x = −1: y = 2 × (−1)3 = −2
x = 0: y = 2 × 03 = 0
x = 1: y = 2 × 13 = 2
x = 2: y = 2 × 23 = 16 - Write the y-values in the table. Each pair of numbers is a coordinate, ready to be plotted on a graph.y-values: −16, −2, 0, 2, 16
Coordinates: (−2, −16), (−1, −2), (0, 0), (1, 2), (2, 16)
Check. First gaps: 14, 2, 2, 14. Gaps of the gaps: −12, 0, 12. Gaps of those: 12, 12, always the same, which is the pattern for a cubic. ✓
Watch out. A negative number cubed is negative: (−2)³ = −8.
| 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 = −(0)3 = 0
x = 1: y = −(1)3 = −1
x = 2: y = −(2)3 = −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.
| x | −2 | −1.5 | −1 | −0.5 | 0 | 0.5 | 1 | 1.5 | 2 |
|---|---|---|---|---|---|---|---|---|---|
| y |
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 = −8
x = −1.5: y = (−1.5)3 = −3.375
x = −1: y = (−1)3 = −1
x = −0.5: y = (−0.5)3 = −0.125
x = 0: y = 03 = 0
x = 0.5: y = 0.53 = 0.125
x = 1: y = 13 = 1
x = 1.5: y = 1.53 = 3.375
x = 2: y = 23 = 8 - List the coordinates. Each x with its y makes a point (x, y) to plot.(−2, −8), (−1.5, −3.375), (−1, −1), (−0.5, −0.125), (0, 0), (0.5, 0.125), (1, 1), (1.5, 3.375), (2, 8)
- Choose the scales for the axes. The x-values go from −2 to 2 and the y-values go from −8 to 8. The axes must cover all of these, and each square must always stand for the same amount.x-axis: −2 to 2. y-axis: −10 to 10.
- 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, −8), (−1.5, −3.375), (−1, −1), (−0.5, −0.125), (0, 0), (0.5, 0.125), (1, 1), (1.5, 3.375), (2, 8).
- 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.
Check. Every plotted point should lie on the curve, and the curve should pass through (−2, −8) and (2, 8) at the ends. Check one point by substituting. ✓
Watch out. Join the points with a smooth curve, not with straight segments.
| x | −3 | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y |
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 = −3: y = (−3)3 − 4 × (−3) = −27 + 12 = −15
x = −2: y = (−2)3 − 4 × (−2) = −8 + 8 = 0
x = −1: y = (−1)3 − 4 × (−1) = −1 + 4 = 3
x = 0: y = 03 − 4 × 0 = 0 + 0 = 0
x = 1: y = 13 − 4 × 1 = 1 − 4 = −3
x = 2: y = 23 − 4 × 2 = 8 − 8 = 0
x = 3: y = 33 − 4 × 3 = 27 − 12 = 15 - List the coordinates. Each x with its y makes a point (x, y) to plot.(−3, −15), (−2, 0), (−1, 3), (0, 0), (1, −3), (2, 0), (3, 15)
- Choose the scales for the axes. The x-values go from −3 to 3 and the y-values go from −15 to 15. The axes must cover all of these, and each square must always stand for the same amount.x-axis: −3 to 3. y-axis: −20 to 20.
- 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 (−3, −15), (−2, 0), (−1, 3), (0, 0), (1, −3), (2, 0), (3, 15).
- 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 − 4x.
Check. Every plotted point should lie on the curve, and the curve should pass through (−3, −15) and (3, 15) at the ends. Check one point by substituting. ✓
Watch out. Join the points with a smooth curve, not with straight segments.
| x | −4 | −2 | −1 | −0.5 | 0.5 | 1 | 2 | 4 |
|---|---|---|---|---|---|---|---|---|
| y |
| x | −4 | −2 | −1 | −0.5 | 0.5 | 1 | 2 | 4 |
|---|---|---|---|---|---|---|---|---|
| y | −0.25 | −0.5 | −1 | −2 | 2 | 1 | 0.5 | 0.25 |
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 = 1x
x-values: −4, −2, −1, −0.5, 0.5, 1, 2, 4 - Substitute the x-values one at a time. Do the same calculation every time, only the number changes. Use brackets round negative numbers.x = −4: y = 1 ÷ (−4) = −0.25
x = −2: y = 1 ÷ (−2) = −0.5
x = −1: y = 1 ÷ (−1) = −1
x = −0.5: y = 1 ÷ (−0.5) = −2
x = 0.5: y = 1 ÷ 0.5 = 2
x = 1: y = 1 ÷ 1 = 1
x = 2: y = 1 ÷ 2 = 0.5
x = 4: y = 1 ÷ 4 = 0.25 - Write the y-values in the table. Each pair of numbers is a coordinate, ready to be plotted on a graph.y-values: −0.25, −0.5, −1, −2, 2, 1, 0.5, 0.25
Coordinates: (−4, −0.25), (−2, −0.5), (−1, −1), (−0.5, −2), (0.5, 2), (1, 1), (2, 0.5), (4, 0.25)
Check. For y = k ÷ x the y-value at −x is the negative of the y-value at x (opposite signs, same size). Both halves of the table follow this. ✓
Watch out. Take care with signs when x is negative.
| x | −8 | −4 | −2 | −1 | 1 | 2 | 4 | 8 |
|---|---|---|---|---|---|---|---|---|
| y |
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 = −8: y = 4 ÷ (−8) = −0.5
x = −4: y = 4 ÷ (−4) = −1
x = −2: y = 4 ÷ (−2) = −2
x = −1: y = 4 ÷ (−1) = −4
x = 1: y = 4 ÷ 1 = 4
x = 2: y = 4 ÷ 2 = 2
x = 4: y = 4 ÷ 4 = 1
x = 8: y = 4 ÷ 8 = 0.5 - List the coordinates. Each x with its y makes a point (x, y) to plot.(−8, −0.5), (−4, −1), (−2, −2), (−1, −4), (1, 4), (2, 2), (4, 1), (8, 0.5)
- Choose the scales for the axes. The x-values go from −8 to 8 and the y-values go from −4 to 4. The axes must cover all of these, and each square must always stand for the same amount.x-axis: −8 to 8. y-axis: −5 to 5.
- 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 (−8, −0.5), (−4, −1), (−2, −2), (−1, −4), (1, 4), (2, 2), (4, 1), (8, 0.5).
- 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, in two separate branches, one on each side of x = 0.Label the curve y = 4x.
Check. Every plotted point should lie on the curve, and the curve should pass through (−8, −0.5) and (8, 0.5) at the ends. Check one point by substituting. ✓
Watch out. The two branches never join up: there is no point at x = 0.
| x | −4 | −2 | −1 | 1 | 2 | 4 |
|---|---|---|---|---|---|---|
| y |
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 = −4: y = −2 ÷ (−4) = 0.5
x = −2: y = −2 ÷ (−2) = 1
x = −1: y = −2 ÷ (−1) = 2
x = 1: y = −2 ÷ 1 = −2
x = 2: y = −2 ÷ 2 = −1
x = 4: y = −2 ÷ 4 = −0.5 - List the coordinates. Each x with its y makes a point (x, y) to plot.(−4, 0.5), (−2, 1), (−1, 2), (1, −2), (2, −1), (4, −0.5)
- Choose the scales for the axes. The x-values go from −4 to 4 and the y-values go from −2 to 2. The axes must cover all of these, and each square must always stand for the same amount.x-axis: −4 to 4. y-axis: −3 to 3.
- 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 (−4, 0.5), (−2, 1), (−1, 2), (1, −2), (2, −1), (4, −0.5).
- 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, in two separate branches, one on each side of x = 0.Label the curve y = −2x.
Check. Every plotted point should lie on the curve, and the curve should pass through (−4, 0.5) and (4, −0.5) at the ends. Check one point by substituting. ✓
Watch out. The two branches never join up: there is no point at x = 0.
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. Try the calculation and think about what division means.
- Write the calculation. Put x = 0 into the rule.y = 1 ÷ 0
- Explain. Dividing by 0 has no answer: no number multiplied by 0 gives 1.There is no value of y
- Say what it means for the graph. There is no point on the graph at x = 0, so the curve never meets the y-axis. The graph has two separate branches.No point at x = 0; the y-axis is an asymptote.
Check. Try x = 0.001: y = 1000, and x = 0.0001: y = 10 000. The curve rises forever as x gets close to 0, never reaching a value at x = 0. ✓
Watch out. Never write a y-value for x = 0 on a reciprocal graph.
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 6x.y = 6x
- Replace x by 2. Substituting means putting the number in place of the letter. Put a negative number in brackets: (−2)² means (−2) × (−2) = 4.y = 6 ÷ 2
- Work it out in the right order. Powers first, then multiplying and dividing, then adding and subtracting (BIDMAS).y = 3
- Write the answer. This tells us the curve passes through the point (2, 3).When x = 2, y = 3
Check. Type it into a calculator using brackets for negatives. The display shows 3. ✓
Watch out. You cannot divide by zero, so x = 0 has no y-value on a reciprocal graph.
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 6x.y = 6x
- 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 = 6 ÷ (−3)
- Work it out in the right order. Powers first, then multiplying and dividing, then adding and subtracting (BIDMAS).y = −2
- Write the answer. This tells us the curve passes through the point (−3, −2).When x = −3, y = −2
Check. Type it into a calculator using brackets for negatives. The display shows −2. ✓
Watch out. You cannot divide by zero, so x = 0 has no y-value on a reciprocal graph.
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 = 6x. The equation we want to solve is the same expression set equal to 1.5.6x = 1.5 means y = 1.5 on this graph
- Draw the horizontal line y = 1.5. Every point on this line has a y-value of 1.5. Where the line meets the curve, the curve has y = 1.5, which is exactly when the equation is true.Start at 1.5 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 (4, 1.5).
- Read the x-values. The solutions of the equation are the x-values of the crossing points (the numbers along the bottom axis).x = 4
Check. Substitute back: y = 6 ÷ 4 = 1.5 ✓
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. 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 = 6x. The equation we want to solve is the same expression set equal to −2.6x = −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 (−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 = −3
Check. Substitute back: y = 6 ÷ (−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.
| x | −12 | −6 | −4 | −3 | −2 | −1 | 1 | 2 | 3 | 4 | 6 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| y |
(c) x = 2.4
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. Make the table, plot and join each branch, then read across at y = 5.
- Make the table. Each y-value is 12 divided by x. Use brackets for negatives.x = −12: y = 12 ÷ (−12) = −1
x = −6: y = 12 ÷ (−6) = −2
x = −4: y = 12 ÷ (−4) = −3
x = −3: y = 12 ÷ (−3) = −4
x = −2: y = 12 ÷ (−2) = −6
x = −1: y = 12 ÷ (−1) = −12
x = 1: y = 12 ÷ 1 = 12
x = 2: y = 12 ÷ 2 = 6
x = 3: y = 12 ÷ 3 = 4
x = 4: y = 12 ÷ 4 = 3
x = 6: y = 12 ÷ 6 = 2
x = 12: y = 12 ÷ 12 = 1 - Plot and join. Plot every pair, then join the positive branch and the negative branch separately with smooth curves.Two branches; no point at x = 0.
- Read x when y = 5. Draw a horizontal line at y = 5 to the positive branch, then read down.x = 2.4 (between x = 2 and x = 3, so about 2.4)
- Check. x × y = 2.4 × 5 = 12 ✓
Check. 2.4 × 5 = 12. ✓
Watch out. Check every pair multiplies to 12.
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. Substitute x = 0.
- Substitute. The origin is (0, 0), so x = 0.y = 0³ = 0
- Conclude. The point (0, 0) is on the graph.True
Check. The table for x³ has y = 0 at x = 0. ✓
Watch out. Substitute to test a point.
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. A point on the y-axis has x = 0.
- Try x = 0. The y-axis is where x = 0.y = 1 ÷ 0, which cannot be worked out
- Conclude. There is no point with x = 0, so the graph never meets the y-axis.False
Check. The branches get closer to the y-axis without touching it. ✓
Watch out. Asymptotes are never crossed.
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. Think about the shape and the table.
- Look at the table. The y-values for x = −2, −1, 0, 1, 2 are −8, −1, 0, 1, 8.They keep increasing
- Conclude. A U-shape comes back down; x³ goes up all the way from bottom-left to top-right.False: it is S-shaped
Check. There is no lowest point. ✓
Watch out. Only squares give U-shapes.
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. Cube a negative number.
- Try an example. Take x = −2.(−2) × (−2) × (−2) = 4 × (−2) = −8
- Explain. Three negative factors: the signs multiply to negative.True
Check. x = −1: (−1)³ = −1 ✓
Watch out. Odd powers keep the sign.
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. Roots are where the curve crosses the x-axis.
- Find the crossings. Read the x-values at the three places where the curve meets the x-axis.x = −2, x = 0, x = 2
- Write them. x = −2, 0 or 2
Check. x³ − 4x = x(x² − 4) = x(x − 2)(x + 2), which is 0 at these values. ✓
Watch out. A cubic can have up to three roots.
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 = x3 − 4x. The equation we want to solve is the same expression set equal to 1.x3 − 4x = 1 means y = 1 on this graph
- Draw the horizontal line y = 1. Every point on this line has a y-value of 1. Where the line meets the curve, the curve has y = 1, which is exactly when the equation is true.Start at 1 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 (−1.9, 1), (−0.3, 1) and (2.1, 1).
- Read the x-values. The solutions of the equation are the x-values of the crossing points (the numbers along the bottom axis).x ≈ −1.9, −0.3 or 2.1 (accept answers within 0.1)
Check. Substitute back: at x = −1.9 the rule gives 1.00; at x = −0.3 the rule gives 1.00; at x = 2.1 the rule gives 1.00 ✓
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. Read the coordinates where the two graphs meet.
- Find the crossings. The curve and the line meet in the top-right and the bottom-left.Top-right: (1, 1). Bottom-left: (−1, −1).
- Write the answer. Give both points.(1, 1) and (−1, −1)
Check. Both points have y = x, and 1 ÷ x = x there. ✓
Watch out. Give both coordinates.
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. Substitute each x-value into both sides.
- Try x = 1. Left side 1 ÷ x, right side x.1 ÷ 1 = 1 and the right side is 1. Equal.
- Try x = −1. Take care with the negative.1 ÷ (−1) = −1 and the right side is −1. Equal.
- Conclude. Both x-values make the two sides equal.So x = 1 and x = −1 solve 1 ÷ x = x. ✓
Check. Rearranging gives x² = 1, which has solutions x = 1 and x = −1. ✓
Watch out. Show the substitution for both sides.