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Cambridge IGCSE0607

Transforming graphs of functions

International Mathematics 0607 Chapter Notes

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Transforming graphs of functions
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In this chapter

  1. 1Describing a translation
  2. 2Adding k outside the function: y = f(x) + k
  3. 3Adding k inside the function: y = f(x + k)
  4. 4Finding g(x) from a translation

1. Describing a translation

A 2-mark description needs two things: the name translation and a column vector beginpmatrixa bendpmatrix.

The top number a is the move to the right, parallel to the x-axis. The bottom number b is the move up, parallel to the y-axis. A negative number means left or down.

Only a single transformation is asked for. Two transformations score no marks.

translationbeginpmatrixa bendpmatrix: a along x, b along y

Examiner insight

“The word translation was required but some candidates used incorrect alternatives, such as move or translocation.” “The vector must not be given as a pair of coordinates.”

Worked example 12 marks

Describe fully the single transformation that maps the graph of y = g(x) onto the graph of y = g(x - 2).

  1. 1

    The name: translation [1]

  2. 2

    The vector: beginpmatrix2 0endpmatrix (2 to the right, 0 up) [1]

2. Adding k outside the function: y = f(x) + k

In y = f(x) + k the number k is added to every output. Every point moves up by k and the x-coordinates stay the same.

The graph of y = f(x) is mapped onto y = f(x) + k by a translation with vector beginpmatrix0 kendpmatrix. When k is negative the graph moves down.

y = f(x) + k Longrightarrow translation beginpmatrix0 kendpmatrix

Examiner insight

“Some tried to explain what the addition of the ‘+3’ meant but said that the movement was to the right instead of up.” “In this case the effect of the –2 in this position means the function moves down by 2 so this is a translation.”

Worked example 12 marks

Describe fully the single transformation that maps the graph of y = f(x) onto the graph of y = f(x) - 5.

  1. 1

    The name: translation [1]

  2. 2

    k = -5, so the graph moves down 5: vector beginpmatrix0 -5endpmatrix [1]

3. Adding k inside the function: y = f(x + k)

In y = f(x +k) the number k is added to x before the function acts. This moves the graph sideways, and the direction is the opposite of the sign.

The graph of y = f(x) is mapped onto y = f(x +k) by a translation with vector beginpmatrix-k 0endpmatrix. So f(x + 3) moves 3 to the left, and f(x - 2) moves 2 to the right.

y = f(x + k) Longrightarrow translation beginpmatrix-k 0endpmatrix

Examiner insight

“Some tried to explain what the ‘+3’ meant but said that the movement was to the right instead of left.” “Many candidates gave the correct answer, with the common mistake being giving the vector in the wrong direction.”

Worked example 12 marks

large f(x) = 5/x² + 1/x³, x ≠ 0.

large h(x) = 5/((x+1)²) + 1/((x+1)³), x ≠ -1.

Describe fully the single transformation that maps the graph of y = f(x) onto the graph of y = h(x).

  1. 1

    h(x) = f(x + 1), so the name is translation [1]

  2. 2

    x + 1 moves the graph 1 to the left: vector beginpmatrix-1 0endpmatrix [1]

4. Finding g(x) from a translation

A question can run the other way: it gives the vector and asks for g(x) in terms of x.

For a translation with vector beginpmatrixa 0endpmatrix, replace x by (x -a) wherever x appears. For a vector beginpmatrix0 bendpmatrix, add b to the whole function.

beginpmatrixa 0endpmatrix Longrightarrow g(x) = f(x - a) beginpmatrix0 bendpmatrix Longrightarrow g(x) = f(x) + b

Examiner insight

“In this part the transformation was given and candidates had to know about replacing x by (x – 2).” “The common error was to replace x by (x + 2).”

Worked example 14 marks

f(x) = x² - 3x. The graph of y = f(x) is mapped onto the graph of y = g(x) by a translation.

(a) The vector is beginpmatrix0 5endpmatrix. Find g(x) in terms of x.

(b) The vector is beginpmatrix4 0endpmatrix. Find g(x) in terms of x.

  1. 1

    (a) up 5, so add 5 to the function [1]

  2. 2

    g(x) = x² - 3x + 5 [1]

  3. 3

    (b) right 4, so replace x by x - 4 [1]

  4. 4

    g(x) = (x-4)² - 3(x-4) = x² - 11x + 28 [1]

Recap

  • A translation is described by its name and a column vector.
  • Say translation. Not move, shift or translocation.
  • f(x) + k is a translation up by k: vector beginpmatrix0 kendpmatrix.
  • f(x + k) is a translation left by k: vector beginpmatrix-k 0endpmatrix.
  • Describe one single transformation only.
  • From a vector beginpmatrixa 0endpmatrix: g(x) = f(x - a). From beginpmatrix0 bendpmatrix: g(x) = f(x) + b.

End-of-chapter exercise

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

  1. The graph of y = f(x) is mapped onto the graph of y = f(x) + 6. State the direction in which the graph moves.1 mark
  2. Describe fully the single transformation that maps the graph of y = f(x) onto the graph of y = f(x) - 7.2 marks
  3. Do not use a calculator. Describe fully the single transformation that maps the graph of y = g(x) onto the graph of y = g(x + 5).2 marks
  4. Describe fully the single transformation that maps the graph of y = g(x) onto the graph of y = g(x - 6).2 marks
  5. Do not use a calculator. f(x) = 3x + 2. The graph of y = f(x) is mapped onto the graph of y = g(x) by a translation with vector beginpmatrix0 -4endpmatrix. Find g(x) in terms of x.2 marks
  6. f(x) = x². The graph of y = f(x) is mapped onto the graph of y = g(x) by a translation with vector beginpmatrix-3 0endpmatrix. Find g(x) in terms of x.2 marks
  7. The point (2, 5) lies on the graph of y = f(x). Write down the coordinates of the image of this point on (a) the graph of y = f(x - 4) and (b) the graph of y = f(x) + 4.2 marks
  8. Do not use a calculator. g(x) = x² - 5x. The graph of y = g(x) is mapped onto the graph of y = h(x) by a translation with vector beginpmatrix-2 0endpmatrix. (a) Find h(x) in terms of x. (b) Work out h(2).4 marks
  9. Do not use a calculator. f(x) = x² - 6x + 4. The graph of y = f(x) is mapped onto the graph of y = g(x) by a translation with vector beginpmatrix-2 0endpmatrix. The graph of y = g(x) is then mapped onto the graph of y = h(x) by a translation with vector beginpmatrix0 kendpmatrix. (a) Find g(x) in the form x² + bx + c. (b) The point (4, 10) lies on the graph of y = h(x). Find the value of k.5 marks
  10. Do not use a calculator. f(x) = x² - 4x. The graph of y = f(x) is mapped onto the graph of y = g(x) by a translation with vector beginpmatrixk 0endpmatrix, where k is an integer. The point (1, 12) lies on the graph of y = g(x). (a) Show that k² + 2k - 15 = 0. (b) Find the two possible values of k. (c) For the positive value of k, describe fully the single transformation that maps the graph of y = g(x) onto the graph of y = f(x).7 marks

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