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, $g (x) = - f (x),$ let's look at the possible transformations. Then we can more clearly identify the ones being applied to the function $f (x) = x^{5} - 4 x + 6 .$

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)

We can describe the transformations as a reflection in the

x -

axis.

Finding the Rule for $g (x)$

To find the rule for

g (x),

let's write the rule for

- f (x) .

f (x) = x^{5} - 4 x + 6 \Rightarrow - f (x) = - (x^{5} - 4 x + 6)

Let's now simplify the above formula.

- f (x) = - (x^{5} - 4 x + 6)

Distr

Distribute $- 1$

- f (x) = - x^{5} + 4 x - 6

We know that

g (x) = - f (x) .

Therefore, the rule for

g

is

g (x) = - x^{5} + 4 x - 6 .

Graphing the Functions

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

Input	$f (x)$		$g (x)$
$x$	$x^{5} - 4 x + 6$	$f (x) = x^{5} - 4 x + 6$	$- x^{5} + 4 x - 6$	$g (x) = - x^{5} + 4 x - 6$
$- 2$	$(- 2)^{5} - 4 (- 2) + 6$	$- 18$	$- (- 2)^{5} + 4 (- 2) - 6$	$18$
$- 1$	$(- 1)^{5} - 4 (- 1) + 6$	$9$	$- (- 1)^{5} + 4 (- 1) - 6$	$- 9$
$0$	$0^{5} - 4 (0) + 6$	$6$	$- 0^{4} + 4 (0) - 6$	$- 6$
$1$	$1^{5} - 4 (1) + 6$	$3$	$- 1^{5} + 4 (1) - 6$	$- 3$
$1.5$	$1.5^{5} - 4 (1.5) + 6$	$7.594$	$- 1.5^{5} + 4 (1.5) - 6$	$- 7.594$

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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