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Cambridge IGCSE0580

Sketching curves

Mathematics 0580 Chapter Notes

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Sketching curves
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Sketching curves notes

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In this chapter

  1. 1Straight lines
  2. 2Quadratic curves: shape, roots and symmetry
  3. 3Turning point of a quadratic
  4. 4Cubic curves: shape and intercepts
  5. 5Cubic curves: roots and repeated roots
  6. 6Reciprocal curves
  7. 7Exponential curves

1. Straight lines

A linear function gives a straight line. Write it as y = mx + c: m is the gradient and c is where the line crosses the y-axis.

To sketch a line, find the two points where it crosses the axes. Put x = 0 to find the y-intercept. Put y = 0 to find the x-intercept (the root). Join the points with a ruled line.

A line such as ax + by = c is rearranged to y = mx + c to read the gradient. A positive gradient rises to the right. A negative gradient falls.

y = mx + c

ax + by = c

Examiner insight

“Candidates would occasionally draw a line with a negative gradient, possibly y = 3 – x and there were some who drew a curve.” “Some lines were short and, in some cases, attempts to draw a freehand straight line from plotted points were seen but not always successfully.”

Worked example 13 marks

Do not use a calculator. Sketch the graph of 2x + y = 6. Mark where it crosses the axes.

  1. 1

    x = 0: y = 6, so the line crosses the y-axis at (0, 6) [1]

  2. 2

    y = 0: 2x = 6, so x = 3 and it crosses the x-axis at (3, 0) [1]

  3. 3

    y = -2x + 6 has a negative gradient: a ruled straight line down to the right through both points [1]

2. Quadratic curves: shape, roots and symmetry

A quadratic y = ax² + bx + c gives a smooth curve called a parabola. If a is positive it is U-shaped. If a is negative it is an upside-down U.

The roots are where the curve crosses the x-axis. Factorise and set each bracket to zero. The curve crosses the y-axis at c.

The curve is symmetrical. Its line of symmetry is halfway between the two roots. A sketch shows the shape and the intercepts. It does not need a table of values or a scale.

y = ax² + bx + c

x = (p + q)/2 for roots x = p, x = q

Examiner insight

“Some sketches had incorrect curvature at the extremes. Some candidates drew straight lines and others drew curves such as cubics and reciprocal graphs.” “Many candidates attempted to plot scales on the axes and plot points to draw their sketches. These should not be necessary.”

Worked example 14 marks

Do not use a calculator. Sketch the graph of y = x² - 2x - 3. Mark the roots, the y-intercept and the line of symmetry.

  1. 1

    y = (x - 3)(x + 1), so the roots are x = 3 and x = -1 [1]

  2. 2

    x = 0 gives y = -3: the y-intercept is (0, -3) [1]

  3. 3

    line of symmetry halfway between the roots: x = (3 + (-1))/2 = 1 [1]

  4. 4

    x² is positive, so the curve is U-shaped through all three marked points [1]

3. Turning point of a quadratic

The turning point of a quadratic is its lowest point (a minimum) or its highest point (a maximum). It lies on the line of symmetry.

Write the quadratic by completing the square. The turning point can then be read from the completed square form.

In y = (x - p)² + q the turning point is (p, q). Watch the sign of p: it is the opposite of the sign inside the bracket.

y = (x - p)² + q has turning point (p, q)

Examiner insight

“Many candidates identified the correct coordinates of the turning point. However, some failed to recognise that they could read the coordinates from the completed square form.” “Errors were often due to graphs passing through the origin or not intercepting with the y-axis.”

Worked example 13 marks

Do not use a calculator. Write y = x² - 6x + 5 in the form y = (x - p)² + q. Write down the coordinates of the turning point and say whether it is a maximum or a minimum.

  1. 1

    half of -6 is -3: (x - 3)² = x² - 6x + 9, so subtract 4 to give 5 [1]

  2. 2

    y = (x - 3)² - 4 [1]

  3. 3

    turning point (3, -4), a minimum because the x² term is positive [1]

4. Cubic curves: shape and intercepts

A cubic has an x³ term. If its coefficient is positive, the curve rises from bottom left to top right. If it is negative, the curve falls from top left to bottom right.

For y = ax³ + d the curve crosses the y-axis at d. Solve ax³ + d = 0 to find where it crosses the x-axis.

Draw a smooth curve with the right curvature at the two ends. Label the intercepts on the axes. Do not rely on a scale.

y = ax³ + d

Examiner insight

“Some curves were spoilt by incorrect curvature at the two extremes.” “Candidates would be well advised to clearly label the intercepts on their graph rather than rely on graph scales, for example scales going up in fives.”

Worked example 13 marks

Do not use a calculator. Sketch the graph of y = x³ - 8. Mark where it crosses the axes.

  1. 1

    the x³ term is positive: the curve rises from bottom left to top right [1]

  2. 2

    x = 0: y = -8, so it crosses the y-axis at (0, -8) [1]

  3. 3

    y = 0: x³ = 8, so x = 2 and it crosses the x-axis at (2, 0) [1]

5. Cubic curves: roots and repeated roots

A cubic in factorised form shows its roots. Set each bracket to zero. y = (x + 5)(x - 3)² has roots x = -5 and x = 3.

A squared bracket gives a repeated root. The curve touches the x-axis there and turns back. It does not cross. Here the touching point is a local minimum.

Put x = 0 to find the y-intercept. A cubic y = ax³ + bx² + cx has x as a factor, so it passes through the origin.

y = a(x - p)(x - q)²

y = ax³ + bx² + cx = x(ax² + bx + c)

Examiner insight

“Some candidates sketched a positive cubic with roots at –5, 3 and 5, or –5, –3 and 3.” “The other two marks were for the curve touched the x-axis at x = 2 and this being the only turning point on the x-axis.”

Worked example 14 marks

Sketch the graph of y = (x + 5)(x - 3)², indicating the values where the graph crosses the axes.

  1. 1

    the x³ term is positive: a positive cubic [1]

  2. 2

    root x = -5: the curve crosses the x-axis at -5 [1]

  3. 3

    (x - 3)² is a repeated root: the curve touches the x-axis at 3, a local minimum at (3, 0) [1]

  4. 4

    x = 0: y = 5 × (-3)² = 5 × 9 = 45, so it crosses the y-axis at 45 only [1]

6. Reciprocal curves

A reciprocal function y = m/x + n gives a curve in two separate branches. The two branches are mirror images of each other, so the curve is symmetrical.

The y-axis (x = 0) is a vertical asymptote. The line y = n is a horizontal asymptote. The branches get closer to these lines but never touch them.

Show both branches. Do not stop the curve at a plotted point, and do not let it curve back in.

y = m/x + n

x = 0 and y = n are the asymptotes

Examiner insight

“Many graphs started to curve back in rather than get closer to the axes.” “Candidates should understand that graph sketching involves demonstrating the shape of a curve or line and showing its main features such as where it crosses axes, turning points, asymptotes, etc. rather than plotting a series of points.”

Worked example 14 marks

Do not use a calculator. Sketch the graph of y = 1/x + 2. Mark the asymptotes and where the curve crosses the x-axis.

  1. 1

    vertical asymptote: x = 0 [1]

  2. 2

    as x gets large, 1/x gets close to 0, so the horizontal asymptote is y = 2 [1]

  3. 3

    y = 0: 1/x = -2, so the curve crosses the x-axis at x = -1/2 [1]

  4. 4

    branch for x > 0 lies above y = 2; branch for x < 0 lies below y = 2 and crosses the x-axis at x = -1/2 [1]

7. Exponential curves

An exponential function such as y = a^(x) has the variable in the index. Its graph is a smooth curve, not a straight line and not a parabola.

The curve passes through (0, 1) because a⁰ = 1. On one side it rises steeply. On the other side it gets closer and closer to a horizontal asymptote.

Adding n moves the curve up by n and moves the asymptote to y = n. Make sure the sketch shows the curve approaching the asymptote for negative x.

y = a^(x) + n

a⁰ = 1

Examiner insight

“Common incorrect responses were a straight line, a parabola and sometimes the reciprocal graph that should have been in part (a).” “This tended to be less successful for those that attempted the plotting points method as there usually weren’t enough points for x < 0 to clearly show it approaching the asymptote.”

Worked example 13 marks

Do not use a calculator. For the graph of y = 2^(x) + 3, find the y-intercept and the equation of the horizontal asymptote.

  1. 1

    x = 0: y = 2⁰ + 3 = 1 + 3 [1]

  2. 2

    the y-intercept is (0, 4) [1]

  3. 3

    for large negative x, 2^(x) gets close to 0, so the curve approaches the horizontal asymptote y = 3 [1]

Recap

  • A linear graph is a straight line. Find the two intercepts and join them.
  • A quadratic is U-shaped if a > 0 and an upside-down U if a < 0. Its line of symmetry is halfway between the roots.
  • Extended: complete the square. y = (x - p)² + q has its turning point at (p, q).
  • Extended: a positive cubic rises from bottom left to top right. A negative cubic does the opposite.
  • Extended: a squared bracket is a repeated root. The curve touches the axis there.
  • Extended: a reciprocal curve has two branches and two asymptotes, x = 0 and y = n.
  • Extended: y = a^(x) passes through (0, 1). y = a^(x) + n approaches the horizontal asymptote y = n.

End-of-chapter exercise

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

  1. Find the coordinates where the line y = 2x - 6 crosses the x-axis and the y-axis.2 marks
  2. Do not use a calculator. The line 3x + 2y = 12 crosses both axes at positive values. Find where it crosses each axis and state whether its gradient is positive or negative.3 marks
  3. Do not use a calculator. The graph of y = -x² + 4 is a parabola. State whether it is U-shaped or an upside-down U and find where it crosses the x-axis.3 marks
  4. The graph of y = x² - 5x + 6 is a parabola. Find where it crosses the x-axis and where it crosses the y-axis.3 marks
  5. Do not use a calculator. A quadratic graph crosses the x-axis at x = -3 and at x = 7. Write down the equation of its line of symmetry.2 marks
  6. A ball is hit from level ground. Its height y metres at a horizontal distance of x metres is given by y = x(10 - x). Find the horizontal distance at which it lands. Then write down the equation of the line of symmetry of its path.3 marks
  7. Do not use a calculator. Write y = x² + 8x + 7 in the form y = (x + a)² + b. Write down the coordinates of the turning point and say whether it is a maximum or a minimum.4 marks
  8. Do not use a calculator. A curve has the equation y = 2^(x) - 8. Find where it crosses the y-axis and where it crosses the x-axis. Write down the equation of its asymptote.4 marks
  9. Do not use a calculator. The graph of y = ax³ + bx² + cx crosses the x-axis at the origin and touches the x-axis at x = 3. It passes through the point (1, 8). Find a, b and c. State whether the point (3, 0) is a local maximum or a local minimum.5 marks
  10. Two taxi firms charge by the length of the journey. For a journey of x km, the average cost per km is y dollars. Firm A: y = 6/x + 2. Firm B: y = 8/x + 1.5. Write down the horizontal asymptote of the graph for Firm A and say what it means for a long journey. Find the journey length for which the two firms have the same average cost. State which firm has the lower average cost for very long journeys, giving a reason.5 marks

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