For the topic of transforming functions, we need to understand the effect of translating, reflecting and stretching has on functions. You may be asked to describe the transformations, sketch graphs or find the coordinates of points that have been transformed. Whilst technology can be a big help understanding these transformations, questions often require you to answer them without your graphical calculator.
On this page, you should learn about
- transformations of graphs
- vertical translations: \(y=f(x)+b\)
- horizontal translations: \(y=f(x-a)\)
- reflection in x axis: \(y=-f(x)\)
- reflection in y axis: \(y=f(-x)\)
- vertical stretch: \(y=af(x)\)
- horizontal stretch: \(y=f(ax)\)
- Compositions of any of the above transformations
These graphs should help you understand the transformations of functions
Horizontal Stretch y = f(ax)
Click on the image to open an applet and explore this transformation:
Print from here
Here is a quiz that practises the skills from this page
START QUIZ!
The graph of y = f(x) is transformed using translations and/or stretches. Describe the transformation from y = f(x) in each case below by dragging the description to the correct place
translate 2 units up translate 2 units down translate 2 units right translate 2 units left vertical stretch with scale factor 2 horizontal stretch with scale factor 2 horizontal stretch with scale factor 0.5 vertical stretch with scale factor 0.5
| f(x) + 2 | 2f(x) |
| f(x - 2) | f(2x) |
| f(0.5x) | f(x + 2) |
f(x - a) is translation a units right
f(x) + a is a traslation a units up
af(x) is a vertical stretch with scale factor a
f(ax) is a horizontal stretch with scale factor \(\frac{1}{a}\)
The graph of y = f(x) is transformed using translations and/or stretches. Describe the transformation from y = f(x) in each case below by dragging the function to the correct place
f(-x) -f(x) f(4x) 4f(x) f(0.25x) f(4x) 0.25f(x) f(x + 4) f(x - 4) f(x) + 4 f(x) - 4
| Reflection in the y axis | vertical stretch with scale factor 4 |
| horizontal stretch with scale factor 4 | translation of \(\left( \begin{matrix} 4 \\ 0 \end{matrix} \right) \) |
| translation of \(\left( \begin{matrix} 0 \\ -4 \end{matrix} \right) \) | Reflection in the x axis |
f(x - a) is translation a units right
f(x) + a is a traslation a units up
af(x) is a vertical stretch with scale factor a
f(ax) is a horizontal stretch with scale factor \(\frac{1}{a}\)
The graph of y = f(x) is transformed. Give the equation of the new graph after these transformations.
Careful, no stretch is NOT a stretch factor of zero!
| Vertical stretch scale factor 3 | y = af(bx) | a = b = |
| Translation of \(\left( \begin{matrix} 1 \\ 2 \end{matrix} \right) \) | y = f(x - a) + b | a = b = |
Vertical stretch factor 2 and horizontal stretch scale factor 2 | y = af(bx) | a = b = |
Reflection in the y axis and vertical stretch scale factor 4 | y = af(bx) | a = b= |
Vertical stretch scale factor 0.5, Reflection in the x axis and Translation of \(\left( \begin{matrix} -1 \\ -3 \end{matrix} \right) \) | y = af(bx - c) + d | a = b = c = d = |
Reflection in the x axis, Horizontal stretch scale factor 0.2 and Translation of \(\left( \begin{matrix} 0 \\ 2 \end{matrix} \right) \) | y = af(bx - c) + d | a = b = c = d = |
f(x - a) is translation a units right
f(x) + a is a traslation a units up
af(x) is a vertical stretch with scale factor a
f(ax) is a horizontal stretch with scale factor \(\frac{1}{a}\)
The graph of the function y = f(x) is plotted in blue in each of the graphs below. Match up the functions with the other graph in red
f(x + 2) f(x - 2) -f(x - 2) -f(x) + 2 f(x) - 2 -f(x) - 2
y=
| y=
| y=
|
y=
| y=
| y=
|
f(-x) is a reflection in the y axis
-f(x) is a reflection in the x axis
The graph of the function y = f(x) is plotted in blue in each of the graphs below. Match up the functions with the other graph in red
-f(x) f(x) - 2 f(x - 2) f(-x) -f(-x) -f(x - 2)
y =
| y =
| y =
|
y =
| y =
| y =
|
f(-x) is a reflection in the y axis
-f(x) is a reflection in the x axis
The graph of the function y = f(x) is plotted in blue in each of the graphs below. Match up the functions with the other graph in red
f(x) - 1 f(x - 2) f(-x) -f(x) f(x + 2) f(x) + 1
y =
| y =
| y =
|
y =
| y =  | y =  |
The graph of y = f(x) with the coordinates of A, B and C is shown below
Give the coordinates of A', B' and C' after the transformation 2f(-x)
A'(xA , yA)
xA = yA =
B'(xB , yB)
xB = yB =
C'(xC , yC)
xC = yC =
The transformation is
- Reflection in the y axis
- Vertical stretch scale factor 2
The graph of y = f(x) with the coordinates of A, B and C is shown below
Give the coordinates of A', B' and C' after the transformation f(-2x) + 1
A'(xA , yA)
xA = yA =
B'(xB , yB)
xB = yB =
C'(xC , yC)
xC = yC =
The transformation is
- Reflection in the x axis
- Horizontal stretch scale factor 0.5
- Translation of \(\left( \begin{matrix} 0 \\ 1 \end{matrix} \right) \)
The graph of y = f(x) with the coordinates of A, B and C is shown below
Give the coordinates of A', B' and C' after the transformation -f(x + 1) - 2
A'(xA , yA)
xA = yA =
B'(xB , yB)
xB = yB =
C'(xC , yC)
xC = yC =
The transformation is
- Translation of \(\left( \begin{matrix} -1 \\ 0 \end{matrix} \right) \)
- Reflection in x axis
- Translation of \(\left( \begin{matrix} 0 \\ -2 \end{matrix} \right) \)
Let f and g be the functions such that g(x) = -3f(2x +1) +1
The point A (2,5) on the graph of f is mapped to the point A' on the graph of g
Find A'
A'(xA , yA)
xA = yA =
The transformation is
- Horizontal stretch scale factor 0.5
- Translation of \(\left( \begin{matrix} -1 \\ 0 \end{matrix} \right) \)
- Reflection in x axis
- Vertical stretch scale factor 3
- Translation of \(\left( \begin{matrix} 0 \\ 1 \end{matrix} \right) \)
a) The following diagram shows the graph of a function f
On the same set of axes, sketch the graph of f(-x) + 2
You can print this graph from here
b)
The following diagram shows the function af(x+b)
Write down the values of a and b
Hint
Full Solution
The graph of f(x) has a local maxima at \((1 - a , 2b)\) and a local minima at \((3a,b-3)\).
a) Find the coordinates of the local maxima of \(f(x+a)-2b\)
b) Find the coordinates of the local minima of \(2f(3x)\)
Hint
Full Solution
Consider the function f(x) = x3 - 4x² - x + 6 , \(x \in \mathbb{R}\)
The graph of f is translated two units to the left and 3 units up to form the function g(x). Express g(x) in the form ax3 + bx² + cx + d where \(a,b,c,d \in \mathbb{Z}\)
Hint
Full Solution
How much of Transforming Functions SL have you understood?
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