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AS Pure Mathematics · Edexcel 8MA0 · free A-level maths worksheet

Sketching curves and transformations

Cubic, quartic and reciprocal graphs, intersections, translations and stretches f(x+a), f(x)+a, f(ax), af(x).

12 original questions, 63 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

Sketch the curve with equation y = (x + 2)(x − 1)(x − 4), showing the coordinates of the points where the curve meets the coordinate axes.

xy
3 marks
Show answer
positive cubic through (-2, 0), (1, 0), (4, 0) and (0, 8)
2

Sketch the curve with equation y = x2(3 − x), showing the coordinates of any points where the curve meets the coordinate axes.

xy
3 marks
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negative cubic touching the x-axis at (0, 0) and crossing at (3, 0)
3

Sketch the curve with equation y = 1x − 2 + 3. On your sketch show the equations of the asymptotes and the coordinates of the points where the curve crosses the coordinate axes.

xy
4 marks
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asymptotes x = 2 and y = 3; crosses at (0, 5/2) and (5/3, 0)
4

Sketch the curve with equation y = x2(x − 2)(x + 3), showing the coordinates of the points where the curve meets the coordinate axes.

xy
3 marks
Show answer
positive quartic touching at (0, 0), crossing at (-3, 0) and (2, 0)
5

The curve with equation y = f(x) has a maximum point at (3, 8). Write down the coordinates of the turning point on the curve with equation

  1. y = f(x − 2)
  2. y = f(x) + 3
  3. y = f(3x)
  4. y = −f(x)
  5. y = 2f(x + 1) − 4
5 marks
Show answer
a) (5, 8) b) (3, 11) c) (1, 8) d) (3, -8), a minimum e) (2, 12)
6

The diagram shows a sketch of the cubic curve with equation y = f(x). The curve crosses the x-axis at (−3, 0), touches it at (1, 0) and crosses the y-axis at (0, 3). On separate diagrams, sketch the curves with the following equations, showing the coordinates of the points where each curve meets the coordinate axes.

  1. y = f(x + 2)
  2. y = 2f(x)
  3. y = f(−x)
xy(-3, 0)(0, 3)(1, 0)y = f(x)xy(a)xy(b)xy(c)
7 marks
Show answer
a) (-5, 0), (-1, 0) and (0, 5) b) (-3, 0), (1, 0) and (0, 6) c) (3, 0), (-1, 0) and (0, 3)
7

The diagram shows a sketch of a cubic curve. The curve touches the x-axis at (−1, 0), crosses the x-axis at (3, 0) and crosses the y-axis at (0, −6). Find an equation for the curve.

xy(-1, 0)(3, 0)(0, -6)
3 marks
Show answer
y = 2(x + 1)2(x - 3)
8

The curve C has equation y = x2 − 6x + 13

  1. Write x2 − 6x + 13 in the form (x + a)2 + b, where a and b are integers.
  2. Hence describe fully the transformation that maps the curve with equation y = x2 onto C.
  3. C is translated k units down so that the new curve touches the x-axis. State the value of k.
  4. C is instead stretched parallel to the y-axis with scale factor 2. Write down the equation of the new curve.
6 marks
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a) (x - 3)2 + 4 b) translation by 3 units in the positive x direction and 4 units in the positive y direction c) k = 4 d) y = 2x2 - 12x + 26
9

f(x) = (x − 2)(x + 1)(x + 3)

  1. Sketch the curve with equation y = f(x), showing the coordinates of the points where it meets the coordinate axes.
  2. The curve with equation y = f(x + k), where k is a constant, passes through the origin. Find the three possible values of k.
  3. Write down the coordinates of the points where the curve with equation y = f(2x) meets the x-axis.
xy
6 marks
Show answer
a) positive cubic through (-3, 0), (-1, 0), (2, 0) and (0, -6) b) k = 2, -1, -3 c) (-3/2, 0), (-1/2, 0), (1, 0)
10

The curve C has equation y = x3 − 4x and the line l has equation y = 5x

  1. Factorise x3 − 4x completely.
  2. On the same axes, sketch C and l, showing the coordinates of the points where C meets the x-axis.
  3. Find the coordinates of all the points where C and l meet.
xy
7 marks
Show answer
a) x(x - 2)(x + 2) b) C crosses at (-2, 0), (0, 0), (2, 0); l through the origin with gradient 5 c) (-3, -15), (0, 0), (3, 15)
11

f(x) = 4x2, x ≠ 0

  1. Sketch the curve with equation y = f(x) − 1, stating the equations of the asymptotes and the coordinates of the points where the curve crosses the x-axis.
  2. The curve with equation y = f(x + a), where a is a constant, passes through the point (1, 1). Find the possible values of a.
  3. For each value of a, state the equation of the vertical asymptote of y = f(x + a).
xy
7 marks
Show answer
a) asymptotes x = 0, y = -1; crosses at (-2, 0) and (2, 0) b) a = 1 or a = -3 c) x = -1 or x = 3
12

The curve C has equation y = (x − 2)2(x + 1)

  1. Show that the equation of C can be written as y = x3 − 3x2 + 4
  2. Sketch C, showing the coordinates of the points where it meets the coordinate axes.
  3. Use differentiation to find the coordinates of the turning points of C.
  4. Hence find the set of values of k for which the equation (x − 2)2(x + 1) = k has three distinct real roots.
xy
9 marks
Show answer
a) shown b) touches at (2, 0), crosses at (-1, 0) and (0, 4) c) (0, 4) and (2, 0) d) 0 < k < 4

Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.

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