2.5 Transformations of Graphs

The following diagram shows part of the graph of `f(x)`.

Consider the five graphs in the diagrams labelled A, B, C, D, E below.

(a) Which diagram is the graph of `f(x+2)`? 

(b) Which diagram is the graph of `-f(x)`? 

(c) Which diagram is the graph of `f(-x)`? 

(a) D

(b) C

(c) A

(a) Express `y=2x^2-12x+23` in the form of `y=2(x-c)^2+d` 

The graph of `y=x^2` is transformed into the graph of `y=2x^2-12x+23` by the transformations

• a vertical stretch with scale factor `k` followed by
• a horizontal translation of `p` units followed by
• a vertical translation of `q` units.

(b) Write down the value of

(i) `k`;

(ii) `p`;

(iii) `q`.

(a) For attempting to complete the square or expanding `y=2(x-c)^2+d`, or for showing the vertex is at (3, 5)

`y=2(x-3)^2+5` (accept `c=3,d=5`)

(b)

(i) `k=2`

(ii) `p=3`

(iii) `q=5`

The graph of `y=f(x)` is shown in the diagram.

(a) On each of the following diagrams draw the required graph. 

(i) `y=2f(x)`

(ii) `y=f(x-3)`

(b) The point `A`(3,`-`1) is on the graph of `f`. The point `A'` is the corresponding point on the graph of `y=-f(x)+1`. Find the coordinates of `A'`. 

(a)

(i)

(ii)

(b) `A'(3,2)` (accept `x=3,y=2`)

Let `g(x)=x^2+bx+11`. The point (−1,8) lies on the graph of `g`.

(a) Find the value of `b`. 

(b) The graph of `f(x)=x^2` is transformed to obtain the graph of `g`. 

Describe this transformation.

(a) valid attempt to substitute coordinates

e.g. `g`(`-`1) = 8

correct substitution

e.g. `(-1)^2+b(-1)+11=8, \ 1-b+11=8`

`b`= 4

(b) valid attempt to solve

e.g. `(x^2+4x+4)+7, h=-4/2, k=g(-2)`

correct working

e.g. `(x+2)^2+7, h=-2, k=7`

translation or shift (do not accept move) of vector `((-2),(7))` (accept left by 2 and up by 7)

The following diagram shows part of the graph of `f` with `x`-intercept (5,0) and `y`-intercept (0,8).

(a) Find the `y`-intercept of the graph of 

(i) `f(x)+3`

(ii) `f(4x)`

(b) Find the `x`-intercept of the graph of `f(2x)`. 

(c) Describe the transformation `f(x+1)`. 

(a)

(i) `y`-intercept is 11 (accept (0,11)) 

(ii) valid approach 

e.g. `f`(4 × 0) = `f`(0), recognizing stretch of `1/4` in `x`-direction 

`y`-intercept is 8 (accept (0,8))

(b) `x`-intercept is `5/2 \ (=2.5)` (accept `(5/2,0)` or `(2.5,0)`)

(c) correct name, correct magnitude and direction

e.g. name: translation, (horizontal) shift (do not accept move)

e.g. magnitude and direction: 1 unit to the left, `((-1),(0))`, horizontal by −1 

Consider `f(x)=4sin(x)+2.5` and `g(x)=4sin(x-{3pi}/2)+2.5+q`, where `x in RR` and `q > 0`. The graph of `g` is obtained by two transformations of the graph of `f`.

(a) Describe these two transformations.

(a) translation (shift) by `{3pi}/2` to the right/positive horizontal direction 

translation (shift) by `q` upwards/positive vertical direction

(b) minimum of `4sin(x-{3pi}/2)` is `-`4 (may be seen in sketch)

`-4+2.5+q >= 7`

`q >= 8.5` (accept `q=8.5`)

substituting `x` = 0 and their `q` (= 8.5) to find `r`

`(r=)\ 4sin({-3pi}/2)+2.5+8.5`

`4+2.5+8.5`

smallest value of `r` is 15

Let `f(x) = x^2 − 4x − 5`. The following diagram shows part of the graph of `f`.

(a) Find the `x`-intercepts of the graph of `f`.

(b) Find the equation of the axis of symmetry of the graph of `f`.  

(c) The function can be written in the form `f(x) = (x − h)^2 + k`. 

(i) Write down the value of `h`. 

(ii) Find the value of `k`. 

The graph of a second function, `g`, is obtained by a reflection of the graph of `f` in the `y`-axis, followed by a translation of `((-3),(6))`.

(d) Find the coordinates of the vertex of the graph of `g`. 

(a) valid approach

e.g. `f(x)=0, x^2-4x-5=0`

valid attempt to solve quadratic equation

e.g. factorizing, formula, completing the square

evidence of correct working

e.g. `(x-5)(x+1), x={4+- sqrt{16-4(-5)}}/{2}`

`x=-1, x=5` (accept `(-1,0), (5,0)`)

(b) correct working 

e.g. `{-(-4)}/{2(1)},{-1+5}/2`

`x=2` (must be an equation with `x=`)

(c)

(i) `h=2`

(ii) valid approach, e.g. `f(2)` 

correct substitution, e.g. `2^2-4(2)-5`

`k=-9`

(d) vertex of `f` is at `(2,-9)` 

correct horizontal reflection

e.g. `x = −2, (−2, −9)`

valid approach for translation of their `x` or `y` value

e.g. `x-3,y+6, ((-2),(-9))+((-3),(6))`, one correct coordinate for vertex 

vertex of `g` is `(−5,−3)` (accept `x = −5,y = −3`)

The following diagram shows the graph of a function `f`, for `-4 <= x <= 2`.

(a) On the same axes, sketch the graph of `f(-x)`. 

(b) Another function, `g`, can be written in the form `g(x) = a × f(x + b)`. The following diagram shows the graph of `g`. 

Write down the value of `a` and of `b`.

(a)

(b) recognizing horizontal shift/translation of 1 unit

e.g. `b = 1`, moved 1 right

recognizing vertical stretch/dilation with scale factor 2

e.g. `a = 2, y × (−2)`

`a = −2, b = −1`

Let `f'(x)={6-2x}/{6x-x^2}`, for `0 < x < 6`.

(a) recognizing `f'(x)=0`

correct working

e.g. `6-2x=0`

`x=3`

(b) evidence of integration 

e.g. `int f', int {6-2x}/{6x-x^2}dx`

using substitution

e.g. `int 1/u du` where `u=6x-x^2`

correct integral

e.g. `ln(u) + c, ln(6x − x^2)`

substituting `(3, ln 27)` into their integrated expression (must have `c`)

e.g. `ln(6 × 3 − 3^2) + c = ln 27 , ln(18 − 9) + ln k = ln 27`

correct working

`c = ln 27 − ln 9 = ln 3`

attempt to substitute their value of `c` into `f(x)`

e.g. `f(x) = ln(6x − x^2) + ln 3 = ln(3(6x − x^2))`

(c) `a = 3` 

correct working

e.g. `{ln27}/{ln 3}`

correct use of log law

e.g. `{3ln3}/{ln 3},log_3 27`

`b=3`

Let `f(x)=sin(x+pi/4)+k`. The graph of 𝑓 passes through the point `(pi/4,6)`.

(a) Find the value of `k`. 

(b) Find the minimum value of `f(x)`. 

Let `g(x) = sin x`. The graph of `g` is translated to the graph of `f` by the vector `((p),(q))`.

(c) Write down the value of `p` and of `q`.  

(a) recognizing shift of `pi/4` left means maximum at 6 

recognizing `k` is difference of maximum and amplitude

e.g. `6-1`

`k=5`

(b) evidence of appropriate approach 

e.g. minimum value of `sin x` is `−1, −1 + k, f'(x) = 0, ({5pi}/4,4)` 

minimum value is 4

(c) `p={-pi}/4,q=5` (accept `(({-pi}/4),(5))`)