Question

sketch a graph of $f(x) = -\frac{1}{2}|x|$\
before sketching the graph, determine where the function has its minimum or maximum value so you can\
place your first point there. think in terms of transformations of the parent function when sketching the\
graph.

Explanation:

Step1: Identify parent function

Parent function: $g(x)=|x|$

Step2: Analyze transformations

Vertical compression by $\frac{1}{2}$, reflection over x-axis: $f(x)=-\frac{1}{2}|x|$

Step3: Find extremum point

For $f(x)=-\frac{1}{2}|x|$, $|x|\geq0$, so $f(x)\leq0$. Maximum at $x=0$: $f(0)=0$

Step4: Plot key points

When $x=2$, $f(2)=-\frac{1}{2}|2|=-1$; when $x=-2$, $f(-2)=-\frac{1}{2}|-2|=-1$

Step5: Sketch the graph

Connect $(0,0)$, $(2,-1)$, $(-2,-1)$ with two straight lines forming a downward-opening V-shape.

Answer:

The graph is a downward-opening V with vertex at $(0, 0)$, passing through points $(2, -1)$ and $(-2, -1)$, with straight lines extending from the vertex through these points.