Rule for g: g(x)=- x^3-4x^2+10 Transformation: Reflection in the y-axis and vertical translation up by 4 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)=f(- x)+4, 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
Reflections
In the x-axis
y=- f(x)
In the y-axis
y=f(- x)
Now, using the table, let's highlight the transformations of f(x).
g(x)=f(- x)+ 4
We can describe the transformations as a reflection in the y-axis and a vertical translation up by 4 units.
Finding the Rule for g(x)
Before finding the rule for g(x), let's first write the rule for f(- x). To do so, we will substitute - x for x in f(x).
f(x)=x^3-4x^2+6 ⇒ f(- x)= (- x)^3-4(- x)^2+6
Let's now simplify the above formula.
