The function g(x) is equal to 3 times the function f(x). Note that f(x)=x^2+2.
B
b
We can find h(x) by inputting 3x into f(x) rather than just x.
C
c
Compare what happened in Part A and Part B.
A
a
g(x)=3x^2+6
The graph of g(x) is shifted up four units and is narrower than f(x).
B
b
h(x)=9x^2+2
The graph of h(x) is narrower than f(x).
C
c
See solution.
Practice makes perfect
a We will start by graphing f(x)=x^2+2. To draw a graph on a calculator, we first press the Y= button and type the function in one of the rows. After writing the function, we can push GRAPH to draw it.
Let's start by finding the equation for g(x). Since g(x)=3f(x), we can substitute in the equation of f(x) and simplify.
We have found that the equation for g(x) is g(x)=3x^2+6. Let's now graph this on our calculator and compare it with f(x).
We can see that the graph of g(x) is shifted up four units, and it is also narrower than f(x).
b
Let's find the equation of h(x). Since h(x)=f(3x), we can substitute 3x for the x-value in f(x).
f(x)=& x^2+2
⇓
h(x)=f(3x)=& (3x)^2+2
We can see that h(x)=(3x)^2+2. We further simplify this by the Power of a Product Property.
h(x)=9x^2+2
Let's now compare the graph of h(x) to f(x).
We can see that h(x) is narrower than f(x).
c
In Part A we multiplied a quadratic function by a number. In Part B we multiplied the x-value of a quadratic function by a number. Let's compare what happened. In Part A we saw that the function's width changed, and it also got shifted up or down. In Part B the function only had its width change.
This tells us that when we multiply a quadratic by a number, its width will change and it will be shifted up or down. When we multiply the x-value of a quadratic by a number, only the width will change.
