Question

sketch a graph of $f(x) = -0.5|x - 1| + 1$. 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 vertex (max point)

The vertex of $f(x)=a|x-h|+k$ is $(h,k)$. Here $h=1, k=1$, so vertex is $(1,1)$. This is a maximum since $a=-0.5<0$.

Step2: Find left side point

Set $x=0$:
$f(0)=-0.5|0-1|+1 = -0.5(1)+1 = 0.5$
Point: $(0, 0.5)$

Step3: Find right side point

Set $x=2$:
$f(2)=-0.5|2-1|+1 = -0.5(1)+1 = 0.5$
Point: $(2, 0.5)$

Step4: Find additional left point

Set $x=-1$:
$f(-1)=-0.5|-1-1|+1 = -0.5(2)+1 = 0$
Point: $(-1, 0)$

Step5: Find additional right point

Set $x=3$:
$f(3)=-0.5|3-1|+1 = -0.5(2)+1 = 0$
Point: $(3, 0)$

Answer:

1. Plot the vertex/maximum point at $(1, 1)$.
2. Plot the points $(0, 0.5)$, $(2, 0.5)$, $(-1, 0)$, $(3, 0)$.
3. Draw two straight lines: one connecting $(-1,0)$ to $(1,1)$ to $(3,0)$, extending outward in both directions to form a downward-opening V-shape.