Rule for g: g(x)= 12x^3-2x^2 Transformation: Vertical shrink by a factor of 12 and vertical translation down by 3 units.
Practice makes perfect
We will describe the graph of g as a transformation of the graph of f. Then we will write a rule for g. Finally, we will graph the functions.
Describing the Transformation
To describe and graph the given transformation, g(x)= 12f(x)-3, let's look at the possible transformations. Then we can more clearly identify the ones being applied to the function f(x)=x^3-4x^2+6.
Transformations of f(x)
Vertical Translations
Translation up k units, k>0
y=f(x)+ k
Translation down k units, k>0
y=f(x)- k
Vertical Stretch or Shrink
Vertical stretch, a>1
y= af(x)
Vertical shrink, 0< a< 1
y= af(x)
Now, using the table, let's highlight the transformations of f(x).
g(x)= 1/2f(x)- 3
We can describe the transformations as a vertical shrink by a factor of 12 and a vertical translation down by 3 units.
Finding the Rule for g(x)
We will the rule for g(x) by substituting x^3-4x^2+6 for f(x) in g(x)= 12f(x)-3.
