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, $g (x) = - \frac{1}{2} f (x),$ let's look at the possible transformations. Then we can more clearly identify the ones being applied to the function $f (x) = 2 x^{3} - 2 x^{2} + 6 .$

Transformations of $f (x)$
Vertical Stretch or Shrink	$Vertical stretch, a > 1 y = a f (x)$
Vertical Stretch or Shrink	$Vertical shrink, 0 < a < 1 y = a f (x)$
Reflections	$In the x - axis y = - f (x)$
Reflections	$In the y - axis y = f (- x)$

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

f (x) .

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

We can describe the transformations as a reflection in the

x -

axis and a vertical shrink by a factor of

\frac{1}{2} .

Finding the Rule for $g (x)$

To obtain the rule for

g (x),

we will write the rule for

- \frac{1}{2} f (x) .

To do so, we will multiply the formula of

f (x)

by

- \frac{1}{2} .

f (x) = 2 x^{3} - 2 x^{2} + 6 ⇕ - \frac{1}{2} f (x) = - \frac{1}{2} (2 x^{3} - 2 x^{2} + 6)

Let's now simplify the above formula.

- \frac{1}{2} f (x) = - \frac{1}{2} (2 x^{3} - 2 x^{2} + 6)

Distr

Distribute $- \frac{1}{2}$

- \frac{1}{2} f (x) = - \frac{1}{2} (2 x^{3}) + \frac{1}{2} (2 x^{2}) - \frac{1}{2} (6)

MoveRightFacToNumOne

$\frac{1}{b} \cdot a = \frac{a}{b}$

- \frac{1}{2} f (x) = - \frac{2 x ^{3}}{2} + \frac{2 x ^{2}}{2} - \frac{6}{2}

SimpQuot

Simplify quotient

- \frac{1}{2} f (x) = - x^{3} + x^{2} - 3

Therefore, as

g (x) = - \frac{1}{2} f (x),

we got that

g (x) = - x^{3} + x^{2} - 3 .

Graphing the Functions

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

	$f (x)$		$g (x)$
$x$	$2 x^{3} - 2 x^{2} + 6$	$f (x) = 2 x^{3} - 2 x^{2} + 6$	$- x^{3} + x^{2} - 3$	$g (x) = - x^{3} + x^{2} - 3$
$- 2$	$2 (- 2)^{3} - 2 (- 2)^{2} + 6$	$- 18$	$- (- 2)^{3} + (- 2)^{2} - 3$	$9$
$- 1$	$2 (- 1)^{3} - 2 (- 1)^{2} + 6$	$2$	$- (- 1)^{3} + (- 1)^{2} - 3$	$- 1$
$0$	$2 (0)^{3} - 2 (0)^{2} + 6$	$6$	$- 0^{3} + 0^{2} - 3$	$- 3$
$1$	$2 (1)^{3} - 2 (1)^{2} + 6$	$6$	$- 1^{3} + 1^{2} - 3$	$- 3$
$2$	$2 (2)^{3} - 2 (2)^{2} + 6$	$14$	$- 2^{3} + 2^{2} - 3$	$- 7$

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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