Question

1. graph (h(x)=-2|x + 5|-3).
2. specify whether the vertex of (h(x)=-2|x + 5|-3) is a minimum or a maximum.

the vertex of (h(x)) is a minimum or maximum. (circle one)

1. describe the transformations that transform the function (f(x)=|x|) to (h(x)=-2|x + 5|-3).

complete each blank in the statement below by circling the correct response.
the function (f(x)) is shifted 5 units left or right and 3 units up or down. (circle one) (circle one)
reflected over the x - or y - axis, and vertically stretched or compressed by a factor 2. (circle one) (circle one)

Explanation:

Step1: Find the vertex of the absolute - value function

The general form of an absolute - value function is $y=a|x - h|+k$, and its vertex is $(h,k)$. For the function $h(x)=-2|x + 5|-3$, we have $h=-5$ and $k = - 3$. So the vertex is $(-5,-3)$.
To graph the function, we can find some other points. When $x=-5$, $h(-5)=-3$. When $x=-4$, $h(-4)=-2| - 4 + 5|-3=-2 - 3=-5$. When $x=-6$, $h(-6)=-2| - 6+5|-3=-2 - 3=-5$.

Step2: Determine if the vertex is a minimum or maximum

For an absolute - value function $y=a|x - h|+k$, if $a>0$, the graph opens up and the vertex is a minimum. If $a<0$, the graph opens down and the vertex is a maximum. Since $a=-2<0$ in $h(x)=-2|x + 5|-3$, the vertex of $h(x)$ is a maximum.

Step3: Describe the transformations

For the transformation from $f(x)=|x|$ to $h(x)=-2|x + 5|-3$:

• The $x$ value in $f(x)$ is replaced with $(x + 5)$, which means the function $f(x)$ is shifted 5 units to the left.
• The constant term $-3$ means the function is shifted 3 units down.
• The negative sign in front of the absolute - value and the coefficient $a=-2$ mean the function is reflected over the $x$ - axis and vertically stretched by a factor of 2.

Answer:

1. Plot the vertex at $(-5,-3)$ and additional points like $(-4,-5)$ and $(-6,-5)$ and draw the V - shaped graph opening down.
2. maximum
3. left; down; $x$; stretched