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Cambridge IGCSE0580

Graphs of functions

Mathematics 0580 Chapter Notes

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Graphs of functions
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In this chapter

  1. 1Straight-line graphs
  2. 2Quadratic curves
  3. 3Symmetry and the turning point
  4. 4Reciprocal graphs
  5. 5Solving equations from a graph
  6. 6Powers of x and sums of terms
  7. 7Rearranging to find the line
  8. 8Exponential growth and decay

1. Straight-line graphs

A table of values gives points to plot. Put each x into the equation, work out y, and plot the pairs.

Join the points with a ruled straight line. Use a ruler and extend the line across the whole grid.

In y = ax + b, the gradient is a. The line crosses the y-axis at b.

y = ax + b

Examiner insight

“Common errors included drawing inaccurate lines which were seen to be missing the coordinates that needed to intersect, or inaccuracies due to lack of use of a ruler.”

Worked example 13 marks

Do not use a calculator. Complete the table of values for y = 2x + 1. Then write down the gradient of the line.

x−2−1012
y−3…1…5
  1. 1

    x = -1: y = 2 × (-1) + 1 = -2 + 1 = -1 [1]

  2. 2

    x = 1: y = 2 × 1 + 1 = 3 [1]

  3. 3

    gradient = 2, the number multiplying x [1]

2. Quadratic curves

For y = x² + ax + b or y = -x² + ax + b, work out y for every x in the table. Put a negative x in brackets before you square it: (-3)² = 9.

Plot every point. Join them with one smooth curve. Never use straight lines between points.

A +x² term gives a U shape. A -x² term gives an upside-down U.

Examiner insight

“Common errors included ruled lines instead of curves and thick lines.” “Many candidates plotted the points but did not join with a smooth curve, either using straight lines or not joining them at all.”

Worked example 12 marks

Complete the table of values for y = x² + x - 3.

x−4−3−2−101234
y9…−1−3−3−1…9…
  1. 1

    x = -3: (-3)² + (-3) - 3 = 9 - 3 - 3 = 3 [1]

  2. 2

    x = 2: 4 + 2 - 3 = 3 and x = 4: 16 + 4 - 3 = 17 [1]

3. Symmetry and the turning point

A quadratic curve is symmetrical. Its line of symmetry is a vertical line, so its equation has the form x = a number.

The curve turns at the turning point, which lies on the line of symmetry. Two points with the same y-value sit the same distance either side of it. The line of symmetry is half-way between their x-values.

Read the turning point from the curve. The table may not list it.

Examiner insight

“The identification of the equation of this line proved more difficult for a number of candidates with x² – 3x – 1 = 1.5, y = 1.5 or 1.5 being common errors.” “Candidates did not use the values in the table to realise the line of symmetry of the graph was when x = 1.5, nor did they use the graph accurately to determine this.”

Worked example 13 marks

Do not use a calculator. Part of the table of values for y = x² - 6x + 5 is given. Write down the equation of the line of symmetry. Then find the coordinates of the lowest point of the curve.

x0123456
y50−3…−305
  1. 1

    y = -3 at both x = 2 and x = 4, so the turning point is half-way between them [1]

  2. 2

    line of symmetry: x = 3 [1]

  3. 3

    y = 3² - 6 × 3 + 5 = -4, so the lowest point is (3, -4) [1]

4. Reciprocal graphs

For y = a/x, the value x = 0 cannot be used. The graph has two separate branches. Never join them across the gap.

The curve gets closer and closer to the axes but never touches them. These lines are its asymptotes.

With a > 0, the branches lie where x and y are both positive and where both are negative. A negative x gives a negative y.

Examiner insight

“Quite often this approach was not successful as usually insufficient points were found to give the structure.” “Regardless of the shaded gap in the table, some joined the two sections of the curve.”

Worked example 13 marks

Do not use a calculator. Complete the table of values for y = 8/x. Explain why the graph has no point at x = 0.

x−4−2−1124
y…−4…8…2
  1. 1

    x = -4: y = 8 ÷ (-4) = -2 [1]

  2. 2

    x = -1: y = -8 and x = 2: y = 4 [1]

  3. 3

    8 ÷ 0 cannot be worked out, so there is no point at x = 0 and the curve has two branches [1]

5. Solving equations from a graph

A root of an equation is a value of x where the curve meets the x-axis, because y = 0 there. To solve f(x) = 0, read off those x-values.

To solve f(x) = k, draw the line y = k on the graph of y = f(x). Read the x-values at the intersection.

The answers are x-values. They are not the y-value of the line.

Examiner insight

“Successful candidates often drew the line y = 7 and then read off where the line and the curve intersected.” “The most common wrong answers came from candidates writing the x coordinates of the intersection of the curve with the x axis.”

Worked example 13 marks

Do not use a calculator. Some points on the curve y = x² - 2x - 3 are given.(a) Write down the roots of x² - 2x - 3 = 0.(b) State the line to draw to solve x² - 2x - 3 = 5, and write down the solutions.

x−2−101234
y50−3−4−305
  1. 1

    (a) roots are where y = 0: x = -1 and x = 3 [1]

  2. 2

    (b) draw the line y = 5 [1]

  3. 3

    it meets the curve at x = -2 and x = 4 [1]

6. Powers of x and sums of terms

Extended functions use axⁿ with n = -2, -1, -1/2, 0, 1/2, 1, 2, 3. Examples are y = x³ + x - 4 and y = 2x + 3/x². Sums have no more than three terms.

Work out each term separately, then add them. Draw one smooth curve. A negative power cannot be worked out at x = 0, so the curve breaks there.

A cubic with a positive x³ term rises from bottom-left to top-right.

Examiner insight

“A minority of candidates did not have a single curve, but feathered or very thick curves or ruled segments; none of these were accepted for full marks.” “Others misread their table, for example, plotting (–3, 2.5) instead of (–3, –2.5).”

Worked example 13 marks

The table shows some values, correct to 1 decimal place, for y = x³ - 2x² - 2. Complete the table.

x−1−0.500.511.522.53
y…−2.6−2−2.4…−3.1−2…7
  1. 1

    x = -1: (-1)³ - 2(-1)² - 2 = -1 - 2 - 2 = -5 [1]

  2. 2

    x = 1: 1 - 2 - 2 = -3 [1]

  3. 3

    x = 2.5: 15.625 - 12.5 - 2 = 1.125, which is 1.1 to 1 decimal place [1]

7. Rearranging to find the line

Sometimes the equation is not the one that was plotted. Rearrange it so that one side is the expression for the plotted curve. The other side is the straight line to draw.

For the curve y = x³ - 2x² - 2, the equation x³ - 2x² - x + 1 = 0 becomes x³ - 2x² - 2 = x - 3. The line is y = x - 3.

Draw the line with a ruler. Read every x-value where it meets the curve. A cubic can give three.

Examiner insight

“Common errors seen included drawing the line y = 10, reading off the x-intercepts from their graph in part (b), or trying to solve the equation either by trying to use the quadratic formula or rearranging the equation.” “If a candidate did manage to draw the line y = 2x successfully they nearly always gained full marks.”

Worked example 13 marks

Do not use a calculator. The graph of y = x² - 2x - 3 is drawn. State the equation of the straight line to draw on it to solve x² - 3x - 1 = 0. Say what you then read off.

  1. 1

    add x and subtract 2 on both sides: x² - 2x - 3 = x - 2 [1]

  2. 2

    the line to draw is y = x - 2 [1]

  3. 3

    the solutions are the x-values where this ruled line meets the curve [1]

8. Exponential growth and decay

In y = a × b^(x) + c, the variable x is the power. Substitute into the table as usual: 2⁻² = 1/4.

In growth the curve climbs more and more steeply. In decay it falls and flattens out. At the flat end the curve gets close to a level line but never reaches it. That line is the asymptote.

For y = a × b^(x) + c the asymptote is y = c. Give it as an equation, not just a number.

Examiner insight

“Many gave numerical values only rather than an equation.”

Worked example 13 marks

The table shows some values for y = 3 × 2^(x) - 5, correct to 1 decimal place. Complete the table.

x−2−1.5−1−0.500.511.52
y…−3.9…−2.9…−0.813.57
  1. 1

    x = -2: 3 × 2⁻² - 5 = 0.75 - 5 = -4.25, which is -4.3 to 1 decimal place [1]

  2. 2

    x = -1: 3 × 0.5 - 5 = -3.5 [1]

  3. 3

    x = 0: 3 × 1 - 5 = -2 [1]

Recap

  • A straight line y = ax + b has gradient a. Draw it with a ruler.
  • Quadratic curve: work out every y, plot, draw one smooth curve. Its line of symmetry is x = a number.
  • Reciprocal graph y = a/x: two branches, no point at x = 0.
  • Solve f(x) = k: draw y = k and read the x-values at the intersection.
  • Extended: powers of x, rearranging to a line, and the asymptote y = c of y = a × b^(x) + c.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. Complete the table of values for y = 3x - 2. | x | −1 | 0 | 2 | 3 | | --- | --- | --- | --- | --- | | y | … | … | … | 7 |3 marks
  2. Do not use a calculator. Write down the gradient of the line y = -2x + 5. Write down the coordinates of the point where it crosses the y-axis.2 marks
  3. For the curve y = -x² + 4x + 1: (a) Find y when x = -1. (b) Find y when x = 5. (c) Use your answers to write down the equation of the line of symmetry.4 marks
  4. Do not use a calculator. (a) Find y for x = -6, x = -4 and x = 3 on the curve y = 12/x. (b) Does the curve cross the y-axis? Explain.4 marks
  5. Do not use a calculator. Some points on the curve y = x² - 4x - 1 are given. Solve x² - 4x - 1 = 4 by stating the line to draw and reading the x-values. | x | −1 | 0 | 1 | 2 | 3 | 4 | 5 | | --- | --- | --- | --- | --- | --- | --- | --- | | y | 4 | −1 | −4 | −5 | −4 | −1 | 4 |3 marks
  6. Some points on the curve y = x² - 2x - 8 are given. (a) Write down the roots of x² - 2x - 8 = 0. (b) Write down the equation of the line of symmetry. (c) Write down the coordinates of the lowest point. | x | −2 | −1 | 0 | 1 | 2 | 3 | 4 | | --- | --- | --- | --- | --- | --- | --- | --- | | y | 0 | −5 | −8 | −9 | −8 | −5 | 0 |4 marks
  7. Do not use a calculator. Find y for y = x³ + x - 4 when x = -2, x = 1 and x = 2.3 marks
  8. Use a calculator. The function is y = 2x + 3/x². (a) Find y when x = -2, x = 1 and x = 0.5. (b) Explain why y cannot be found when x = 0.4 marks
  9. Do not use a calculator. The table gives points on the curve y = x³ - 4x. The equation x³ - 6x - 4 = 0 is solved by drawing a straight line on this graph. (a) Find the equation of the line. (b) Work out the y-values of the line at x = -2, -1, 0, 2 and 3. (c) Write down the solution that is an integer. (d) The other two solutions are not integers. For each one, write down the two consecutive integers it lies between. | x | −3 | −2 | −1 | 0 | 1 | 2 | 3 | | --- | --- | --- | --- | --- | --- | --- | --- | | y | −15 | 0 | 3 | 0 | −3 | 0 | 15 |6 marks
  10. Use a calculator. A hot drink cools in a room. Its temperature, T °C, after t hours is T = 64 × 2^(-t) + 18. (a) Find T when t = 0 and when t = 2. (b) The graph of T against t is drawn. State the equation of the line to draw on it to solve 64 × 2^(-t) = 4. Then find the solution. (c) Write down the equation of the asymptote of the graph. Say what it tells you about the drink.6 marks

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