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, $g (x) = 3 f (x),$ let's look at the possible transformations. Then we can more clearly identify the ones being applied to the function $f (x) = x^{5} - 2 x + 3 .$

Vertical Stretch or Shrink
$Vertical stretch, a > 1 y = a f (x)$	$Vertical shrink, 0 < a < 1 y = a f (x)$

Now, using the table, let's highlight the transformations of

f (x) .

g (x) = 3 f (x)

We can describe the transformation as a vertical stretch by a factor of

3 .

Finding the Rule for $g (x)$

To obtain the rule for

g (x),

we will write the rule for

3 f (x) .

To do so, we will multiply the formula of

f (x)

by

3 .

f (x) = x^{5} - 2 x + 3 \Rightarrow 3 f (x) = 3 (x^{5} - 2 x + 3)

Let's now simplify the above formula.

3 f (x) = 3 (x^{5} - 2 x + 3)

Distr

Distribute $3$

3 f (x) = 3 x^{5} + 3 \cdot (- 2 x) + 3 \cdot 3

Multiply

3 f (x) = 3 x^{5} - 6 x + 9

Therefore, as

g (x) = 3 f (x),

we got that

g (x) = 3 x^{5} - 6 x + 9 .

Graphing the Functions

Let's make a table of values for both functions.

Input	$f (x)$		$g (x)$
$x$	$x^{5} - 2 x + 3$	$f (x) = x^{5} - 2 x + 3$	$3 x^{5} - 6 x + 9$	$g (x) = 3 x^{5} - 6 x + 9$
$- 1.5$	$(- 1.5)^{5} - 2 (- 1.5) + 3$	$- 1.59375$	$3 (- 1.5)^{5} - 6 (- 1.5) + 9$	$- 4.78125$
$- 1$	$(- 1)^{5} - 2 (- 1) + 3$	$4$	$3 (- 1)^{5} - 6 (- 1) + 9$	$12$
$0$	$0^{5} - 2 \cdot 0 + 3$	$3$	$3 \cdot 0^{5} - 6 \cdot 0 + 9$	$9$
$1$	$1^{5} - 2 \cdot 1 + 3$	$2$	$3 \cdot 1^{5} - 6 \cdot 1 + 9$	$6$
$1.5$	$1.5^{5} - 2 \cdot 1.5 + 3$	$7.59375$	$3 \cdot 1.5^{5} - 6 \cdot 1.5 + 9$	$22.78125$

Finally, we plot the points of each function and connect them with smooth curves.

Describing the Transformation

Finding the Rule for g(x)

Graphing the Functions

Finding the Rule for $g (x)$

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