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.
Key Concepts
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
Essentials
These graphs should help you understand the transformations of functions
Vertical Stretch y = af(x)
Drag the slider to see the effect
Horizontal Stretch y = f(ax)
Click on the image to open an applet and explore this transformation:
Translations y = f(x - a) + b
Click on the image to open an applet and explore this transformation:
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 uptranslate 2 units downtranslate 2 units righttranslate 2 units leftvertical stretch with scale factor 2horizontal stretch with scale factor 2horizontal stretch with scale factor 0.5vertical 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
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
It helps to sketch a graph of what the function might look like then mark on the local maxima and minima
a) Consider the 2 transformations for \(f(x+a)-2b\) and apply them to the point \((1 - a , 2b)\) b) Consider the 2 transformations for \(2f(3x)\)and apply them to the point \((3a,b-3)\)
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 \(a,b,c,d \in \mathbb{Z}\)