Rule for g: g(x)=- x^3+7x^2-11x+5 Transformation: Reflection in the x-axis, horizontal translation to the right by 1 unit, and vertical translation up by 6 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-1)+6, 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
Horizontal Translations
Translation right h units, h>0
y=f(x- h)
Translation left h units, h>0
y=f(x+ h)
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- 1)+ 6
We can describe the transformations as a reflection in the x-axis, a horizontal translation to the right by 1 unit, and a vertical translation up by 6 units.
Finding the Rule for g(x)
Before finding the rule for g(x), let's first write the rule for f(x-1). To do so, we will substitute x-1 for x in f(x).
f(x)=x^3-4x^2+6
⇓
f( x-1)= ( x-1)^3-4( x-1)^2+6
Let's now simplify the above formula.
