Questionample 4: Write the equation $y=|-2 x-8|-3$ in the form $y=a|x-p|+q$ , then sketch the graph using transformations. $\begin{aligned} \text { vertex } & =(p, q) \\ & =(-4,3) \end{aligned}$

Studdy Solution

STEP 1

1. We are given the equation $y = |-2x - 8| - 3$ .
2. We need to rewrite it in the form $y = a|x - p| + q$ .
3. We will use transformations to sketch the graph.

STEP 2

1. Rewrite the equation in the desired form.
2. Identify the transformations from the standard absolute value function.
3. Determine the vertex of the graph.
4. Sketch the graph using the transformations.

STEP 3

Rewrite the equation $y = |-2x - 8| - 3$ in the form $y = a|x - p| + q$ .
First, factor out $-2$ from the expression inside the absolute value:
$|-2x - 8| = |-2(x + 4)|$
Thus, the equation becomes:
$y = |-2(x + 4)| - 3$
Rewrite it as:
$y = 2|x + 4| - 3$
This is now in the form $y = a|x - p| + q$ with $a = 2$ , $p = -4$ , and $q = -3$ .

STEP 4

Identify the transformations from the standard absolute value function $y = |x|$ .
1. The factor $a = 2$ indicates a vertical stretch by a factor of 2.
2. The term $x + 4$ indicates a horizontal shift to the left by 4 units.
3. The term $-3$ indicates a vertical shift downward by 3 units.

STEP 5

Determine the vertex of the graph.
The vertex of the transformed graph is at $(p, q) = (-4, -3)$ .

STEP 6

Sketch the graph using the transformations.
1. Start with the basic graph of $y = |x|$ .
2. Apply a vertical stretch by a factor of 2.
3. Shift the graph 4 units to the left.
4. Shift the graph 3 units downward.
The vertex of the graph is at $(-4, -3)$ , and the graph opens upwards with a slope of 2 on either side of the vertex.

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