Functions and graphs: quadratic, cubic and reciprocal

Method. Replace x by the given number, then work out the value.

1. Write the function. The function is f = 2x + 3.
   f(4) = 2x + 3
2. Substitute. Replace x with 4 everywhere it appears.
   f(4) = 2(4) + 3
3. Work out the value. Use the order of operations.
   f(4) = 11
4. Write the answer.
   11

Check. Put x = 4 back into f(x) to check.

Watch out. Put brackets around a negative number when you substitute it, so that the signs are clear.

Method. Replace x by the given number, then work out the value.

1. Write the function. The function is f = 2x + 3.
   f(−2) = 2x + 3
2. Substitute. Replace x with −2 everywhere it appears.
   f(−2) = 2(−2) + 3
3. Work out the value. Use the order of operations.
   f(−2) = −1
4. Write the answer.
   −1

Check. Put x = −2 back into f(x) to check.

Watch out. Put brackets around a negative number when you substitute it, so that the signs are clear.

Method. Replace x by the given number, then work out the value.

1. Write the function. The function is f = 2x + 3.
   f(5) = 2x + 3
2. Substitute. Replace x with 5 everywhere it appears.
   f(5) = 2(5) + 3
3. Work out the value. Use the order of operations.
   f(5) = 13
4. Write the answer.
   13

Check. Put x = 5 back into f(x) to check.

Watch out. Put brackets around a negative number when you substitute it, so that the signs are clear.

Method. Replace x by the given number, then work out the value.

1. Write the function. The function is f = 2x + 3.
   f(0) = 2x + 3
2. Substitute. Replace x with 0 everywhere it appears.
   f(0) = 2(0) + 3
3. Work out the value. Use the order of operations.
   f(0) = 3
4. Write the answer.
   3

Check. Put x = 0 back into f(x) to check.

Watch out. Put brackets around a negative number when you substitute it, so that the signs are clear.

Method. Replace x by the given number, then work out the value.

1. Write the function. The function is g = x² − 3x.
   g(4) = x² − 3x
2. Substitute. Replace x with 4 everywhere it appears.
   g(4) = (4)² − 3(4)
3. Work out the value. Use the order of operations.
   g(4) = 4
4. Write the answer.
   4

Check. Put x = 4 back into g(x) to check.

Watch out. Put brackets around a negative number when you substitute it, so that the signs are clear.

Method. Replace x by the given number, then work out the value.

1. Write the function. The function is g = x² − 3x.
   g(−2) = x² − 3x
2. Substitute. Replace x with −2 everywhere it appears.
   g(−2) = (−2)² − 3(−2)
3. Work out the value. Use the order of operations.
   g(−2) = 10
4. Write the answer.
   10

Check. Put x = −2 back into g(x) to check.

Watch out. Put brackets around a negative number when you substitute it, so that the signs are clear.

Method. Replace x by the given number, then work out the value.

1. Write the function. The function is h = 10 − x².
   h(3) = 10 − x²
2. Substitute. Replace x with 3 everywhere it appears.
   h(3) = 10 − (3)²
3. Work out the value. Use the order of operations.
   h(3) = 1
4. Write the answer.
   1

Check. Put x = 3 back into h(x) to check.

Watch out. Put brackets around a negative number when you substitute it, so that the signs are clear.

Method. Set f(x) equal to 10, then solve the equation.

1. Write the equation. Set the function equal to 10.
   f(x) = 3x − 5 = 10
2. Undo the constant. Subtract −5 from both sides.
   3x = 15
3. Divide by the number in front of x. This leaves x on its own.
   x = 5
4. Write the answer.
   x = 5

Check. Substitute x = 5 into f(x): it gives 10.

Watch out. Solving f(x) = k is the reverse of finding f(x). Undo the operations in reverse order.

Method. Set f(x) equal to 18, then solve the equation.

1. Write the equation. Set the function equal to 18.
   f(x) = 5x − 2 = 18
2. Undo the constant. Subtract −2 from both sides.
   5x = 20
3. Divide by the number in front of x. This leaves x on its own.
   x = 4
4. Write the answer.
   x = 4

Check. Substitute x = 4 into f(x): it gives 18.

Watch out. Solving f(x) = k is the reverse of finding f(x). Undo the operations in reverse order.

Method. Set f(x) equal to −1, then solve the equation.

1. Write the equation. Set the function equal to −1.
   f(x) = 4x + 7 = −1
2. Undo the constant. Subtract 7 from both sides.
   4x = −8
3. Divide by the number in front of x. This leaves x on its own.
   x = −2
4. Write the answer.
   x = −2

Check. Substitute x = −2 into f(x): it gives −1.

Watch out. Solving f(x) = k is the reverse of finding f(x). Undo the operations in reverse order.

Method. Set the two outputs equal and solve.

1. Write the equation. Set f(a) equal to f(b).
   2a + 1 = 2b + 1
2. Subtract 1 from both sides. This removes the 1 on each side.
   2a = 2b
3. Divide by 2. This leaves a = b.
   a = b

Check. Each input gives a different output, so the function is one-to-one.

Method. Replace x by the given number, then work out the value.

1. Write the function. The function is f = 5 − 3x.
   f(−4) = 5 − 3x
2. Substitute. Replace x with −4 everywhere it appears.
   f(−4) = 5 − 3(−4)
3. Work out the value. Use the order of operations.
   f(−4) = 17
4. Write the answer.
   17

Check. Put x = −4 back into f(x) to check.

Watch out. Put brackets around a negative number when you substitute it, so that the signs are clear.

Method. Substitute each x value into y = x² − 2x.

1. Make a table. Use each x value in turn.

   x	−1	0	1	2	3
   y	3	0	−1	0	3

2. Write the y values. Read each y value from the table.
   3, 0, −1, 0, 3
3. Write the answer.
   3, 0, −1, 0, 3

Check. The lowest point, −1, is at x = 1.

Method. Substitute each x value into y = x² + 2.

1. Make a table. Square each x value and add 2.

   x	−2	−1	0	1	2
   y	6	3	2	3	6

2. Write the answer.
   6, 3, 2, 3, 6

Check. The graph is symmetrical about x = 0.

Method. Complete the square to find the turning point.

1. Complete the square. Write x² + bx as (x + b/2)² − (b/2)².
   y = (x − 2)² − 4 + 3
2. Find the turning point. The square is zero when x = −b/2, so the minimum value is the number outside the square.
   x = 2, y = −1
3. Write the answer. A positive x² term gives a minimum turning point.
   (2, −1)

Check. Substitute x = 2 into the equation: y = −1.

Watch out. The line of symmetry is x = −b/2, the same as the x-coordinate of the turning point.

Method. Complete the square to find the turning point.

1. Complete the square. Write x² + bx as (x + b/2)² − (b/2)².
   y = (x + 3)² − 9 + 5
2. Find the turning point. The square is zero when x = −b/2, so the minimum value is the number outside the square.
   x = −3, y = −4
3. Write the answer. A positive x² term gives a minimum turning point.
   (−3, −4)

Check. Substitute x = −3 into the equation: y = −4.

Watch out. The line of symmetry is x = −b/2, the same as the x-coordinate of the turning point.

Method. Complete the square to find the turning point.

1. Complete the square. Write x² + bx as (x + b/2)² − (b/2)².
   y = (x − 1)² − 1 − 8
2. Find the turning point. The square is zero when x = −b/2, so the minimum value is the number outside the square.
   x = 1, y = −9
3. Write the answer. A positive x² term gives a minimum turning point.
   (1, −9)

Check. Substitute x = 1 into the equation: y = −9.

Watch out. The line of symmetry is x = −b/2, the same as the x-coordinate of the turning point.

Method. Complete the square to find the turning point.

1. Complete the square. Write x² + bx as (x + b/2)² − (b/2)².
   y = (x + 2)² − 4 + 1
2. Find the turning point. The square is zero when x = −b/2, so the minimum value is the number outside the square.
   x = −2, y = −3
3. Write the answer. A positive x² term gives a minimum turning point.
   (−2, −3)

Check. Substitute x = −2 into the equation: y = −3.

Watch out. The line of symmetry is x = −b/2, the same as the x-coordinate of the turning point.

Method. Complete the square to find the turning point.

1. Complete the square. Write x² + bx as (x + b/2)² − (b/2)².
   y = (x − 3)² − 9 + 10
2. Find the turning point. The square is zero when x = −b/2, so the minimum value is the number outside the square.
   x = 3, y = 1
3. Write the answer. A positive x² term gives a minimum turning point.
   (3, 1)

Check. Substitute x = 3 into the equation: y = 1.

Watch out. The line of symmetry is x = −b/2, the same as the x-coordinate of the turning point.

Method. Complete the square to find the turning point.

1. Complete the square. Write x² + bx as (x + b/2)² − (b/2)².
   y = (x + 1)² − 1 − 3
2. Find the turning point. The square is zero when x = −b/2, so the minimum value is the number outside the square.
   x = −1, y = −4
3. Write the answer. A positive x² term gives a minimum turning point.
   (−1, −4)

Check. Substitute x = −1 into the equation: y = −4.

Watch out. The line of symmetry is x = −b/2, the same as the x-coordinate of the turning point.

Method. The line of symmetry goes through the turning point.

1. Use x = −b/2. Here b = 6, so −b/2 = −3.
   x = −6/2 = −3
2. Write the answer.
   x = −3

Check. The turning point of this curve is at x = −3.

Method. Compare the two rules.

1. Compare the constants. The rule has −5 added on, so every y value is 5 less.
   y = x² − 5 = (y = x²) − 5
2. Write the answer. A subtraction on the outside moves the graph down.
   Translate it 5 units down.

Check. The turning point moves from (0, 0) to (0, −5).

Method. Adding a constant moves the whole graph up or down.

1. Start at (0, 0). The turning point of y = x² is (0, 0).
   (0, 0)
2. Add 4 to each y value. Each point moves up by 4.
   (0, 0) → (0, 4)
3. Write the answer.
   y = x² + 4

Check. The turning point is (0, 4), so the curve has moved up by 4.

Method. Complete the square to find the turning point.

1. Complete the square. Write x² + bx as (x + b/2)² − (b/2)².
   y = (x + 1)² − 1 − 8
2. Find the turning point. The square is zero when x = −b/2, so the minimum value is the number outside the square.
   x = −1, y = −9
3. Write the answer. A positive x² term gives a minimum turning point.
   (−1, −9)

Check. Substitute x = −1 into the equation: y = −9.

Watch out. The line of symmetry is x = −b/2, the same as the x-coordinate of the turning point.

Method. Complete the square to find the turning point.

1. Complete the square. Write x² + bx as (x + b/2)² − (b/2)².
   y = (x − 5)² − 25 + 21
2. Find the turning point. The square is zero when x = −b/2, so the minimum value is the number outside the square.
   x = 5, y = −4
3. Write the answer. A positive x² term gives a minimum turning point.
   (5, −4)

Check. Substitute x = 5 into the equation: y = −4.

Watch out. The line of symmetry is x = −b/2, the same as the x-coordinate of the turning point.

Method. Cube each x value.

1. Make a table. Cube each value of x.

   x	−2	−1	0	1	2
   y	−8	−1	0	1	8

2. Write the answer.
   −8, −1, 0, 1, 8

Check. Cubing a negative number gives a negative answer.

Method. Find the reciprocal of each x value.

1. Make a table. The y value is 1 divided by x.

   x	1	2	4	5	10
   y	1	1/2	1/4	1/5	1/10

2. Write the answer.
   1, 1/2, 1/4, 1/5, 1/10

Check. As x gets bigger, y gets smaller, but never reaches zero.

Method. Replace x by the given number, then work out the value.

1. Write the function. The function is y = x³.
   y(−3) = x³
2. Substitute. Replace x with −3 everywhere it appears.
   y(−3) = (−3)³
3. Work out the value. Use the order of operations.
   y(−3) = −27
4. Write the answer.
   −27

Check. Put x = −3 back into y(x) to check.

Watch out. Put brackets around a negative number when you substitute it, so that the signs are clear.

Method. Replace x by the given number, then work out the value.

1. Write the function. The function is y = x³ − 1.
   y(2) = x³ − 1
2. Substitute. Replace x with 2 everywhere it appears.
   y(2) = (2)³ − 1
3. Work out the value. Use the order of operations.
   y(2) = 7
4. Write the answer.
   7

Check. Put x = 2 back into y(x) to check.

Watch out. Put brackets around a negative number when you substitute it, so that the signs are clear.

Method. Use 0.5 = 1/2, then take the reciprocal.

1. Write 0.5 as a fraction. 0.5 = 1/2.
   x = 1/2
2. Take the reciprocal. The reciprocal of 1/2 is 2.
   1 ÷ (1/2) = 2

Check. 2 × 0.5 = 1.

Method. Find the number that cubed gives −27.

1. Write the equation. A negative number cubed is negative.
   x³ = −27
2. Find the cube root. (−3) × (−3) × (−3) = −27.
   x = −3

Check. (−3)³ = −27.

Method. Find the cube root of 64.

1. Write the equation. We need the number that, cubed, gives 64.
   x³ = 64
2. Find the number. 4 × 4 × 4 = 64.
   x = 4

Check. 4³ = 64.

Method. The asymptotes are the lines the graph gets close to.

1. Look at the graph. The graph gets close to the y-axis when x is near 0.
   x = 0
2. Look at the other axis. It also gets close to the x-axis when y is near 0.
   y = 0
3. Write the answer.
   x = 0 and y = 0

Check. The graph never touches either line.

Method. Think about what happens at x = 0.

1. Try x = 0. The rule is y = 1 ÷ 0, which is not defined.
   y = 1 ÷ 0
2. Look at the graph. The curve is never at x = 0, so it cannot cross the y-axis.
   no point with x = 0
3. Write the answer.
   1 divided by 0 is undefined, so there is no point on the y-axis.

Check. The y-axis is the asymptote x = 0.

Method. Use the shape of y = x³.

1. Look at the shape. The cubic curve passes through the origin.
   (0, 0) is on the curve
2. Compare with a U-shape. A U-shape is a quadratic, not a cubic.
   y = x² is a U-shape
3. Write the answer.
   A curve through the origin that goes up on the right

Check. y = x³ has a negative part for negative x.

Method. Substitute x = −4 and find the reciprocal.

1. Substitute. y = 1 ÷ (−4).
   y = 1/(−4)
2. Simplify the sign. A positive divided by a negative is negative.
   = −1/4

Check. (−1/4) × (−4) = 1.

Method. Replace x by the given number, then work out the value.

1. Write the function. The function is y = 2^x.
   y(6) = 2^x
2. Substitute. Replace x with 6 everywhere it appears.
   y(6) = 2⁽⁶⁾
3. Work out the value. Use the order of operations.
   y(6) = 64
4. Write the answer.
   64

Check. Put x = 6 back into y(x) to check.

Watch out. Put brackets around a negative number when you substitute it, so that the signs are clear.

Method. Use the reciprocal of 2².

1. Write the reciprocal. 2⁽⁻²⁾ = 1 ÷ 2².
   1 ÷ 4
2. Write the answer.
   1/4

Check. 4 × (1/4) = 1.

Method. Double the value each time.

1. Make a table. Each step in x doubles y.

   x	0	1	2	3	4	5
   y	1	2	4	8	16	32

2. Write the answer.
   1, 2, 4, 8, 16, 32

Check. 2⁵ = 32.

Method. Multiply the starting number by 2 for each hour.

1. Write the rule. After x hours the number is 5 × 2^x.
   5 × 2⁴
2. Work out the power. 2⁴ = 16.
   5 × 16
3. Write the answer.
   80

Check. Doubling 5 four times: 10, 20, 40, 80.

Method. Work out both values.

1. Work out 2¹⁰. Ten 2s multiplied together.
   2¹⁰ = 1024
2. Work out 10². Two 10s multiplied together.
   10² = 100
3. Compare. A power of 2 grows faster than a power of 10 with the same index here.
   1024 > 100
4. Write the answer.
   2¹⁰ is bigger

Check. The base decides how fast a power grows, not just the index.

Method. Think about the sign of a power of 2.

1. Test a negative index. 2⁽⁻³⁾ = 1/8, which is positive.
   1/8 > 0
2. Test a large negative index. Each step down halves the value, but it never reaches zero.
   2⁽⁻¹⁰⁾ > 0
3. Write the answer.
   2^x is always positive, so it never reaches 0

Check. The x-axis is an asymptote.

Method. Solve the equation that the intersection gives, then round each root to 1 d.p.

1. Set the equations equal. At an intersection both equations give the same y, so x² = mx + k.
   x² = 2x + 1
2. Rearrange to 0. Move every term to one side.
   x² − (2x + 1) = 0
3. Use the formula. The quadratic formula gives the roots of ax² + bx + c = 0.
   x = (2 ± √(2² + 4 × 1))/2 = (2 ± √8)/2
4. Work out each root. Use a calculator or the graph to find the decimal values, then round to 1 d.p.
   x ≈ 2.4142 or x ≈ -0.4142
5. Write the answer. Round to 1 decimal place, as the question asks.
   x = −0.4 or x = 2.4

Check. Substitute each value into y = x² and the line: both give the same y, to 1 d.p.

Watch out. An intersection gives two points on a curve and a line, so there are usually two x-values.

Method. Solve the equation that the intersection gives, then round each root to 1 d.p.

1. Set the equations equal. At an intersection both equations give the same y, so x² = mx + k.
   x² = x + 1
2. Rearrange to 0. Move every term to one side.
   x² − (x + 1) = 0
3. Use the formula. The quadratic formula gives the roots of ax² + bx + c = 0.
   x = (1 ± √(1² + 4 × 1))/2 = (1 ± √5)/2
4. Work out each root. Use a calculator or the graph to find the decimal values, then round to 1 d.p.
   x ≈ 1.6180 or x ≈ -0.6180
5. Write the answer. Round to 1 decimal place, as the question asks.
   x = −0.6 or x = 1.6

Check. Substitute each value into y = x² and the line: both give the same y, to 1 d.p.

Watch out. An intersection gives two points on a curve and a line, so there are usually two x-values.

Method. Solve the equation that the intersection gives, then round each root to 1 d.p.

1. Set the equations equal. At an intersection both equations give the same y, so x² = mx + k.
   x² = 2
2. Rearrange to 0. Move every term to one side.
   x² − (2) = 0
3. Use the formula. The quadratic formula gives the roots of ax² + bx + c = 0.
   x = (0 ± √(0² + 4 × 2))/2 = (0 ± √8)/2
4. Work out each root. Use a calculator or the graph to find the decimal values, then round to 1 d.p.
   x ≈ 1.4142 or x ≈ -1.4142
5. Write the answer. Round to 1 decimal place, as the question asks.
   x = −1.4 or x = 1.4

Check. Substitute each value into y = x² and the line: both give the same y, to 1 d.p.

Watch out. An intersection gives two points on a curve and a line, so there are usually two x-values.

Method. Solve the equation that the intersection gives, then round each root to 1 d.p.

1. Set the equations equal. At an intersection both equations give the same y, so x² = mx + k.
   x² = −x + 3
2. Rearrange to 0. Move every term to one side.
   x² − (−x + 3) = 0
3. Use the formula. The quadratic formula gives the roots of ax² + bx + c = 0.
   x = (-1 ± √(-1² + 4 × 3))/2 = (-1 ± √13)/2
4. Work out each root. Use a calculator or the graph to find the decimal values, then round to 1 d.p.
   x ≈ 1.3028 or x ≈ -2.3028
5. Write the answer. Round to 1 decimal place, as the question asks.
   x = −2.3 or x = 1.3

Check. Substitute each value into y = x² and the line: both give the same y, to 1 d.p.

Watch out. An intersection gives two points on a curve and a line, so there are usually two x-values.

Method. Solve the equation that the intersection gives, then round each root to 1 d.p.

1. Set the equations equal. At an intersection both equations give the same y, so x² = mx + k.
   x² = 3x − 1
2. Rearrange to 0. Move every term to one side.
   x² − (3x − 1) = 0
3. Use the formula. The quadratic formula gives the roots of ax² + bx + c = 0.
   x = (3 ± √(3² + 4 × -1))/2 = (3 ± √5)/2
4. Work out each root. Use a calculator or the graph to find the decimal values, then round to 1 d.p.
   x ≈ 2.6180 or x ≈ 0.3820
5. Write the answer. Round to 1 decimal place, as the question asks.
   x = 0.4 or x = 2.6

Check. Substitute each value into y = x² and the line: both give the same y, to 1 d.p.

Watch out. An intersection gives two points on a curve and a line, so there are usually two x-values.

Method. Solve the equation that the intersection gives, then round each root to 1 d.p.

1. Set the equations equal. At an intersection both equations give the same y, so x² = mx + k.
   x² = 4x − 2
2. Rearrange to 0. Move every term to one side.
   x² − (4x − 2) = 0
3. Use the formula. The quadratic formula gives the roots of ax² + bx + c = 0.
   x = (4 ± √(4² + 4 × -2))/2 = (4 ± √8)/2
4. Work out each root. Use a calculator or the graph to find the decimal values, then round to 1 d.p.
   x ≈ 3.4142 or x ≈ 0.5858
5. Write the answer. Round to 1 decimal place, as the question asks.
   x = 0.6 or x = 3.4

Check. Substitute each value into y = x² and the line: both give the same y, to 1 d.p.

Watch out. An intersection gives two points on a curve and a line, so there are usually two x-values.

Method. Set the two rules equal and factorise.

1. Set the rules equal. The graphs meet where x³ = 2x.
   x³ − 2x = 0
2. Take out the common factor. Both terms have an x.
   x(x² − 2) = 0
3. Solve each factor. x = 0, or x² = 2, so x = ±√2.
   x = 0, x = √2 or x = −√2
4. Write the answer.
   x = 0, x = −√2 or x = √2

Check. The three points of intersection are (−√2, −2√2), (0, 0) and (√2, 2√2).

Method. Think about how each graph behaves in the first quadrant.

1. Look at the reciprocal. y = 1/x falls from very large values towards zero.
   falls
2. Look at the line. y = x + 1 rises from 1 upwards.
   rises
3. Write the answer. A falling and a rising graph cross once.
   One point of intersection

Check. The point is about x = 0.6, where the graphs cross.

Method. Solve the equation that the intersection gives, then round each root to 1 d.p.

1. Set the equations equal. At an intersection both equations give the same y, so x² = mx + k.
   x² = 5
2. Rearrange to 0. Move every term to one side.
   x² − (5) = 0
3. Use the formula. The quadratic formula gives the roots of ax² + bx + c = 0.
   x = (0 ± √(0² + 4 × 5))/2 = (0 ± √20)/2
4. Work out each root. Use a calculator or the graph to find the decimal values, then round to 1 d.p.
   x ≈ 2.2361 or x ≈ -2.2361
5. Write the answer. Round to 1 decimal place, as the question asks.
   x = −2.2 or x = 2.2

Check. Substitute each value into y = x² and the line: both give the same y, to 1 d.p.

Watch out. An intersection gives two points on a curve and a line, so there are usually two x-values.

Method. Use the straight-line rule from the graph to convert the value.

1. Read the rule for the graph. The graph is a straight line through (0, 0), so the rule is a multiplication.
   euros = 1.2 × pounds
2. Find the value. Put in 40 pounds.
   40 × 1.2 = 48
3. Write the answer.
   48

Check. Check: 48 divided by 1.2 is 40, the original value.

Watch out. Read the rule from the graph first, then use it for every value.

Method. Use the straight-line rule from the graph to convert the value.

1. Read the rule for the graph. The graph is a straight line through (0, 0), so the rule is a multiplication.
   euros = 1.2 × pounds
2. Find the value. Put in 25 pounds.
   25 × 1.2 = 30
3. Write the answer.
   30

Check. Check: 30 divided by 1.2 is 25, the original value.

Watch out. Read the rule from the graph first, then use it for every value.

Method. The gradient of a distance-time graph is the speed.

1. Choose two points on the straight line. Read the coordinates of two points on the line.
   (10, 2) and (30, 6)
2. Find the gradient. Gradient = change in distance divided by change in time.
   (6 − 2) ÷ (30 − 10) = 4 ÷ 20
3. Write the speed with units. Speed is the gradient of a distance-time graph.
   0.2

Check. Check: in 20 time units the distance changes by 4.

Watch out. A flat part of a distance-time graph means the object is not moving: the speed is zero.

Method. The gradient of a distance-time graph is the speed.

1. Choose two points on the straight line. Read the coordinates of two points on the line.
   (0, 0) and (3, 180)
2. Find the gradient. Gradient = change in distance divided by change in time.
   (180 − 0) ÷ (3 − 0) = 180 ÷ 3
3. Write the speed with units. Speed is the gradient of a distance-time graph.
   60

Check. Check: in 3 time units the distance changes by 180.

Watch out. A flat part of a distance-time graph means the object is not moving: the speed is zero.

Method. Find the fixed charge and the charge per mile.

1. Find the fixed charge. The £3 is paid once, whatever the distance.
   fixed charge = 3
2. Find the charge per mile. Each mile adds £2.
   2 × m
3. Write the rule. Add the two parts.
   C = 2m + 3

Check. For 4 miles, C = 2 × 4 + 3 = 11.

Method. A horizontal line has gradient zero.

1. Look at the line. The distance does not change, so the object is stopped.
   change in distance = 0
2. Find the speed. Speed = change in distance ÷ change in time = 0 ÷ 10 = 0.
   0 km/min
3. Write the answer.
   0 km/min

Check. A stopped object has speed zero.

Method. Divide the number of dollars by 1.5.

1. Write the rule. dollars = 1.5 × pounds, so pounds = dollars ÷ 1.5.
   pounds = 30 ÷ 1.5
2. Work out the value. 30 ÷ 1.5 = 20.
   20
3. Write the answer.
   20 pounds

Check. 20 × 1.5 = 30.

Method. Compare the gradients of the two parts.

1. Look at the steepness. A steeper straight line means a bigger change in distance for each unit of time.
   bigger gradient
2. Write the answer. A steeper part is faster.
   The steeper part is faster

Check. The gradient is the speed.

Method. The gradient of a distance-time graph is the speed.

1. Choose two points on the straight line. Read the coordinates of two points on the line.
   (0, 0) and (5, 15)
2. Find the gradient. Gradient = change in distance divided by change in time.
   (15 − 0) ÷ (5 − 0) = 15 ÷ 5
3. Write the speed with units. Speed is the gradient of a distance-time graph.
   3

Check. Check: in 5 time units the distance changes by 15.

Watch out. A flat part of a distance-time graph means the object is not moving: the speed is zero.

Method. Use the straight-line rule from the graph to convert the value.

1. Read the rule for the graph. The graph is a straight line through (0, 0), so the rule is a multiplication.
   km = 0.6 × miles
2. Find the value. Put in 50 miles.
   50 × 0.6 = 30
3. Write the answer.
   30

Check. Check: 30 divided by 0.6 is 50, the original value.

Watch out. Read the rule from the graph first, then use it for every value.

Method. Replace x by the given number, then work out the value.

1. Write the function. The function is f = 4x − 3.
   f(5) = 4x − 3
2. Substitute. Replace x with 5 everywhere it appears.
   f(5) = 4(5) − 3
3. Work out the value. Use the order of operations.
   f(5) = 17
4. Write the answer.
   17

Check. Put x = 5 back into f(x) to check.

Watch out. Put brackets around a negative number when you substitute it, so that the signs are clear.

Method. Replace x by the given number, then work out the value.

1. Write the function. The function is g = x² + 2x.
   g(−3) = x² + 2x
2. Substitute. Replace x with −3 everywhere it appears.
   g(−3) = (−3)² + 2(−3)
3. Work out the value. Use the order of operations.
   g(−3) = 3
4. Write the answer.
   3

Check. Put x = −3 back into g(x) to check.

Watch out. Put brackets around a negative number when you substitute it, so that the signs are clear.

Method. Set f(x) equal to 1, then solve the equation.

1. Write the equation. Set the function equal to 1.
   f(x) = 2x + 9 = 1
2. Undo the constant. Subtract 9 from both sides.
   2x = −8
3. Divide by the number in front of x. This leaves x on its own.
   x = −4
4. Write the answer.
   x = −4

Check. Substitute x = −4 into f(x): it gives 1.

Watch out. Solving f(x) = k is the reverse of finding f(x). Undo the operations in reverse order.

Method. Complete the square to find the turning point.

1. Complete the square. Write x² + bx as (x + b/2)² − (b/2)².
   y = (x − 3)² − 9 + 2
2. Find the turning point. The square is zero when x = −b/2, so the minimum value is the number outside the square.
   x = 3, y = −7
3. Write the answer. A positive x² term gives a minimum turning point.
   (3, −7)

Check. Substitute x = 3 into the equation: y = −7.

Watch out. The line of symmetry is x = −b/2, the same as the x-coordinate of the turning point.

Method. Use x = −b/2.

1. Find b. Here b = −8.
   b = −8
2. Use x = −b/2. −(−8)/2 = 4.
   x = 4
3. Write the answer.
   x = 4

Check. The turning point is at x = 4.

Method. Take the reciprocal of 2⁴.

1. Write the reciprocal. 2⁽⁻⁴⁾ = 1 ÷ 2⁴.
   1 ÷ 16
2. Write the answer.
   1/16

Check. 16 × (1/16) = 1.

Method. Substitute and find the reciprocal.

1. Substitute. y = 1 ÷ (−2).
   y = 1/(−2)
2. Write the answer.
   −1/2

Check. (−1/2) × (−2) = 1.

Method. Find the cube root of 125.

1. Write the equation. We need the number that, cubed, gives 125.
   x³ = 125
2. Find the number. 5 × 5 × 5 = 125.
   x = 5

Check. 5³ = 125.

Method. Use the straight-line rule from the graph to convert the value.

1. Read the rule for the graph. The graph is a straight line through (0, 0), so the rule is a multiplication.
   dollars = 1.25 × pounds
2. Find the value. Put in 60 pounds.
   60 × 1.25 = 75
3. Write the answer.
   75

Check. Check: 75 divided by 1.25 is 60, the original value.

Watch out. Read the rule from the graph first, then use it for every value.

Method. Solve the equation that the intersection gives, then round each root to 1 d.p.

1. Set the equations equal. At an intersection both equations give the same y, so x² = mx + k.
   x² = x + 3
2. Rearrange to 0. Move every term to one side.
   x² − (x + 3) = 0
3. Use the formula. The quadratic formula gives the roots of ax² + bx + c = 0.
   x = (1 ± √(1² + 4 × 3))/2 = (1 ± √13)/2
4. Work out each root. Use a calculator or the graph to find the decimal values, then round to 1 d.p.
   x ≈ 2.3028 or x ≈ -1.3028
5. Write the answer. Round to 1 decimal place, as the question asks.
   x = −1.3 or x = 2.3

Check. Substitute each value into y = x² and the line: both give the same y, to 1 d.p.

Watch out. An intersection gives two points on a curve and a line, so there are usually two x-values.

Method. Replace x by the given number, then work out the value.

1. Write the function. The function is f = 3x² − 1.
   f(−2) = 3x² − 1
2. Substitute. Replace x with −2 everywhere it appears.
   f(−2) = 3(−2)² − 1
3. Work out the value. Use the order of operations.
   f(−2) = 11
4. Write the answer.
   11

Check. Put x = −2 back into f(x) to check.

Watch out. Put brackets around a negative number when you substitute it, so that the signs are clear.

Method. Set f(x) equal to −6, then solve the equation.

1. Write the equation. Set the function equal to −6.
   f(x) = 5x + 4 = −6
2. Undo the constant. Subtract 4 from both sides.
   5x = −10
3. Divide by the number in front of x. This leaves x on its own.
   x = −2
4. Write the answer.
   x = −2

Check. Substitute x = −2 into f(x): it gives −6.

Watch out. Solving f(x) = k is the reverse of finding f(x). Undo the operations in reverse order.

Method. Complete the square to find the turning point.

1. Complete the square. Write x² + bx as (x + b/2)² − (b/2)².
   y = (x + 2)² − 4 + 9
2. Find the turning point. The square is zero when x = −b/2, so the minimum value is the number outside the square.
   x = −2, y = 5
3. Write the answer. A positive x² term gives a minimum turning point.
   (−2, 5)

Check. Substitute x = −2 into the equation: y = 5.

Watch out. The line of symmetry is x = −b/2, the same as the x-coordinate of the turning point.

Method. Use x = −b/2.

1. Find b. Here b = 10.
   b = 10
2. Use x = −b/2. −10/2 = −5.
   x = −5
3. Write the answer.
   x = −5

Check. The turning point is at x = −5.

Method. Take the reciprocal of 2³.

1. Write the reciprocal. 2⁽⁻³⁾ = 1 ÷ 8.
   1 ÷ 8
2. Write the answer.
   1/8

Check. 8 × (1/8) = 1.

Method. Think about x = 0.

1. Look at x = 0. The rule 1 ÷ 0 is not defined.
   1 ÷ 0 is undefined
2. Write the answer. The graph is split by the y-axis, which is an asymptote.
   Two separate branches

Check. One branch is in the first quadrant and one in the third.

Method. Solve the equation that the intersection gives, then round each root to 1 d.p.

1. Set the equations equal. At an intersection both equations give the same y, so x² = mx + k.
   x² = 2x + 2
2. Rearrange to 0. Move every term to one side.
   x² − (2x + 2) = 0
3. Use the formula. The quadratic formula gives the roots of ax² + bx + c = 0.
   x = (2 ± √(2² + 4 × 2))/2 = (2 ± √12)/2
4. Work out each root. Use a calculator or the graph to find the decimal values, then round to 1 d.p.
   x ≈ 2.7321 or x ≈ -0.7321
5. Write the answer. Round to 1 decimal place, as the question asks.
   x = −0.7 or x = 2.7

Check. Substitute each value into y = x² and the line: both give the same y, to 1 d.p.

Watch out. An intersection gives two points on a curve and a line, so there are usually two x-values.

Method. Multiply the starting number by 2 for each hour.

1. Write the rule. After x hours the number is 3 × 2^x.
   3 × 2⁵
2. Work out. 2⁵ = 32, and 3 × 32 = 96.
   96
3. Write the answer.
   96

Check. Doubling 3 five times: 6, 12, 24, 48, 96.

Method. The gradient of a distance-time graph is the speed.

1. Choose two points on the straight line. Read the coordinates of two points on the line.
   (0, 0) and (4, 20)
2. Find the gradient. Gradient = change in distance divided by change in time.
   (20 − 0) ÷ (4 − 0) = 20 ÷ 4
3. Write the speed with units. Speed is the gradient of a distance-time graph.
   5

Check. Check: in 4 time units the distance changes by 20.

Watch out. A flat part of a distance-time graph means the object is not moving: the speed is zero.

Method. Use the straight-line rule from the graph to convert the value.

1. Read the rule for the graph. The graph is a straight line through (0, 0), so the rule is a multiplication.
   pounds = 0.8 × euros
2. Find the value. Put in 25 euros.
   25 × 0.8 = 20
3. Write the answer.
   20

Check. Check: 20 divided by 0.8 is 25, the original value.

Watch out. Read the rule from the graph first, then use it for every value.