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Cambridge IGCSE0607

Graphs of functions

International Mathematics 0607 Chapter Notes

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Graphs of functions
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In this chapter

  1. 1Straight lines and parabolas
  2. 2Cubic, reciprocal and exponential graphs
  3. 3Trigonometric graphs: period and amplitude
  4. 4Reading values from a graph with the calculator
  5. 5Finding a, b, c and d from a graph

1. Straight lines and parabolas

A linear function f(x) = ax + b has a straight line as its graph. The number a is the gradient and b is where the line crosses the y-axis. So f(x) = 3x - 2 is the straight line that rises and crosses the y-axis at -2.

A quadratic function f(x) = ax² + bx + c has a smooth curve as its graph. It is a U-shape when a > 0 and an upside-down U when a < 0.

The curve has one turning point, a local minimum or a local maximum, on its line of symmetry. The zeros are the values of x where it crosses the x-axis.

f(x) = ax + b

f(x) = ax² + bx + c

Examiner insight

“However, candidates should have been aware that what they were drawing in this part was a straight line, and there were a few solutions where the candidates drew a distinctly curved line.” “A few candidates did not use a ruler to draw their line.”

Worked example 13 marks

Write down the zeros of y = x² - 4x + 3.

Write down the equation of the line of symmetry of y = x² - 4x + 3.

Write down the coordinates of the local minimum.

  1. 1

    enter y = x² - 4x + 3 on the calculator and use the zero tool where the curve crosses the x-axis: x = 1 and x = 3 [1]

  2. 2

    the line of symmetry is halfway between the zeros: x = 2 [1]

  3. 3

    at x = 2, y = 2² - 4(2) + 3 = -1, so the local minimum is (2, -1) [1]

2. Cubic, reciprocal and exponential graphs

A cubic function f(x) = ax³ + bx² + cx + d has a smooth S-shaped curve. Do not slide it sideways: it still crosses the y-axis at d.

A reciprocal function f(x) = a/x has two separate branches. Each branch gets close to the two axes without touching them. A line the curve approaches but never meets is an asymptote: here x = 0 and y = 0.

An exponential function f(x) = a^(x) rises quickly when a > 1 and falls towards the x-axis when 0 < a < 1. Both curves pass through (0, 1).

f(x) = ax³ + bx² + cx + d

f(x) = a/x

f(x) = a^(x)

Examiner insight

“The majority of candidates knew the basic shape of a cubic graph, but some candidates translated the graph in the x-direction.” “Some candidates only drew one branch.”

Worked example 13 marks

Use your graph to solve this equation.

x³ + x² - 5x + 2 = x² + 2

  1. 1

    draw y = x³ + x² - 5x + 2 and y = x² + 2 on the calculator and find where they cross

  2. 2

    one crossing is at x = 0 [1]

  3. 3

    the others are at x = -2.24 and x = 2.24 [2]

  4. 4

    check: the equation simplifies to x³ - 5x = 0, so x = 0 or x = ±√5

3. Trigonometric graphs: period and amplitude

The graphs of f(x) = asin(bx) and f(x) = acos(bx) are waves that repeat. The amplitude is the height of the wave above its middle line. It is |a|.

The period is the length of one full repeat of the wave. With x in degrees it is 360° ÷ b. Without b, sin x and cos x repeat every 360°.

The graph of f(x) = tan x is not a wave of this kind. It repeats every 180° and has vertical asymptotes.

amplitude = |a|

period = (360^(°))/b

Examiner insight

“Some candidates thought this was a graph transformation question.” “The majority of candidates were able to write down the amplitude but were unsure as to how to find the period.”

Worked example 12 marks

State the amplitude and the period of y = 5sin(3x), where x is in degrees.

  1. 1

    amplitude = 5 [1]

  2. 2

    period = 360° ÷ 3 = 120° [1]

4. Reading values from a graph with the calculator

Put the function into the graphic display calculator and choose a window that shows the whole shape. Then read the zeros, the y-intercept, a local maximum or local minimum, or an intersection with the calculator's own tools.

Do not use the trace key: it gives a near value, not the exact one. Give answers to 3 significant figures unless they are exact.

A local maximum is a turning point, not the highest value on the screen. Write a point as coordinates.

Examiner insight

“Inexact answers suggested that some candidates had used the trace feature on the calculator rather than opt for finding the intersection with the x-axis.” “A few candidates tried to work out the answer algebraically.”

Worked example 13 marks

Find the coordinates of the point where the graph of y = 6/(4 -x) crosses the y-axis.

Find the coordinates of the points of intersection of y = 6/(4 -x) and y = 5 - x.

  1. 1

    the graph crosses the y-axis where x = 0: the calculator gives y = 1.5, so the point is (0, 1.5) [1]

  2. 2

    enter both functions on the calculator and use its intersection tool: (2, 3) [1]

  3. 3

    and (7, -2) [1]

5. Finding a, b, c and d from a graph

Some of a, b, c or d may be 0, so check each one. Two readings do most of the work.

The y-intercept is the point where x = 0. It gives b in a line, c in a quadratic and d in a cubic.

Then put the coordinates of another point on the graph into the function and solve for what is left. For a quadratic, a local minimum means a > 0 and a local maximum means a < 0.

Examiner insight

“Some incorrect answers were (0, 0), (0, 2) and (0, 6).”

Worked example 13 marks

The graph shows the curve y = x² + bx + c. Use the graph to find b and c.

  1. 1

    the curve crosses the y-axis at (0, 5), so c = 5 [1]

  2. 2

    the curve passes through (1, 2), so 2 = 1² + b(1) + 5 [1]

  3. 3

    2 = 6 + b, so b = -4 (check with (2, 1): 4 - 8 + 5 = 1) [1]

Recap

  • Linear f(x) = ax + b is a straight line. Quadratic f(x) = ax² + bx + c is U-shaped for a > 0 and upside-down U for a < 0.
  • Cubic is S-shaped. Reciprocal a/x has two branches and the asymptotes x = 0 and y = 0. Exponential a^(x) passes through (0, 1).
  • asin(bx) and acos(bx): amplitude |a|, period 360° ÷ b.
  • The y-intercept is f(0). It gives b, c or d. Substitute another point for the rest.
  • Use the calculator's zero, minimum, maximum and intersection tools. Never the trace key.
  • Write asymptotes as equations and points as coordinates, to 3 significant figures unless exact.
  • Draw straight lines with a ruler and draw asymptotes first.

End-of-chapter exercise

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

  1. Do not use a calculator. The graph of y = -x² + 4 has one turning point. State whether it is a local maximum or a local minimum.2 marks
  2. Do not use a calculator. The straight line y = 3x - 2 is of the form y = ax + b. Write down the gradient and the coordinates of the point where it crosses the y-axis.2 marks
  3. Do not use a calculator. The straight line y = ax + b passes through (0, 5) and (2, 11). Find a and b.3 marks
  4. Use a graphic display calculator for y = x² - 6x + 4. (a) Find the coordinates of the local minimum. (b) Write down the equation of the line of symmetry.3 marks
  5. Do not use a calculator. (a) Name the type of function f(x) = 4/x. (b) Write down the equations of its two asymptotes.3 marks
  6. Do not use a calculator. State the amplitude and the period of y = 4sin(2x), where x is in degrees.2 marks
  7. Do not use a calculator. The graph of y = a^(x) passes through (3, 8). Find a. Then find the value of y when x = -1.3 marks
  8. Use a graphic display calculator for f(x) = x³ - 4x + 1. (a) Find the x-coordinates of the points where the graph crosses the x-axis. (b) Find the coordinates of the local maximum.5 marks
  9. A weight bounces on the end of a spring. Its height above its rest position is d cm, t seconds after it passes upwards through the rest position. d is negative when the weight is below its rest position. The model is d = asin(bt), where bt is in degrees and a and b are positive. The greatest height above the rest position is 6 cm. The weight is next at its rest position when t = 4. (a) Write down the value of a. (b) Find the period and the value of b. (c) Use a graphic display calculator to find the first two times when the weight is 4 cm below its rest position. (d) Find how long the weight spends more than 4 cm below its rest position in one period.6 marks
  10. Use a graphic display calculator. A company sells x thousand items and makes a profit of P thousand dollars. The model is P = x³ - 6x² + 9x - 1 for 0 leqslant x leqslant 4.5. (a) Find the coordinates of the local maximum. (b) Find the three values of x for which the profit is zero. (c) The equation P = k has three solutions for 0 leqslant x leqslant 4.5. Find the range of values of k. (d) Find the greatest profit in the model. Give your answer in dollars.9 marks

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