We are given the graphs of two functions, namely, $f (x) = x^{2}$ and $g (x) .$

First, we see that the vertex of

g (x)

is

2

units above the vertex of

f (x) .

We will start by translating the graph of

f (x)

up

2

units.

h (x) = f (x) + 2

Now, let's plot

g (x)

and

h (x)

on the same coordinate plane and compare them.

We see that the graphs of $g (x)$ and $h (x)$ still do not match. However, we can see that the graph of $g (x)$ is wider, which means that we have to make a horizontal stretch. In other words, $g (x) = h (a x) .$

To figure out the scale factor, we will use the fact that

g (2) = 3 .

g (2) = 3 \Rightarrow h (a \cdot 2) = 3

Next, we substitute

h (x) = x^{2} + 2

and solve the equation for

a .

h (x) = x^{2} + 2

Substitute

$x = a \cdot 2$

h (a \cdot 2) = (a \cdot 2)^{2} + 2

▼

Solve for

a

Substitute

$h (a \cdot 2) = 3$

3 = (a \cdot 2)^{2} + 2

CalcPow

Calculate power

3 = 4 a^{2} + 2

SubEqn

$LHS - 2 = RHS - 2$

1 = 4 a^{2}

DivEqn

$LHS / 4 = RHS / 4$

\frac{1}{4} = a^{2}

RadicalEqn

$LHS = RHS$

\pm \frac{1}{4} = a

CalcRoot

Calculate root

\pm \frac{1}{2} = a

RearrangeEqn

Rearrange equation

a = \pm \frac{1}{2}

Since

a

must be positive, we will discard the negative option. Now, we are ready to write our function

g (x)

in terms of

f (x) .

g (x) = h (\frac{1}{2} x) = f (\frac{1}{2} x) + 2

In conclusion, to obtain

g (x)

we must translate the graph of

f (x)

2

units up and then stretch it horizontally by a factor of

\frac{1}{2} .

Exercise