Question

find the vertex and sketch a graph of $f(x) = -0.5 |x + 2| + 3$
the vertex of the function are at

Explanation:

Step1: Identify vertex form of absolute value

The vertex form of an absolute value function is $f(x) = a|x - h| + k$, where $(h,k)$ is the vertex.

Step2: Rewrite given function to match form

Rewrite $f(x) = -0.5|x + 2| + 3$ as $f(x) = -0.5|x - (-2)| + 3$.

Step3: Extract vertex coordinates

From the rewritten form, $h=-2$ and $k=3$, so the vertex is $(-2, 3)$.

Step4: Identify points for graphing

• For $x=-2$: $f(-2)=-0.5|0|+3=3$ (vertex)
• For $x=0$: $f(0)=-0.5|2|+3=2$
• For $x=-4$: $f(-4)=-0.5|-2|+3=2$
• For $x=2$: $f(2)=-0.5|4|+3=1$
• For $x=-6$: $f(-6)=-0.5|-4|+3=1$

Answer:

The vertex of the function is at $(-2, 3)$

To sketch the graph:

1. Plot the vertex $(-2, 3)$
2. Plot the points $(0,2)$, $(-4,2)$, $(2,1)$, $(-6,1)$
3. Draw two straight lines from the vertex through these points, forming a downward-opening V-shape (since $a=-0.5<0$)