Question

graph the equation.
$y = -6|x| + 3$

Explanation:

Step1: Find the vertex of the graph

The vertex form of an absolute value function is $y=a|x-h|+k$, where $(h,k)$ is the vertex. For $y=-6|x|+3$, $h=0$, $k=3$. So vertex is $(0, 3)$.

Step2: Find x=1 point

Substitute $x=1$ into the equation:
$y=-6|1|+3 = -6+3=-3$
Point: $(1, -3)$

Step3: Find x=-1 point

Substitute $x=-1$ into the equation:
$y=-6|-1|+3 = -6+3=-3$
Point: $(-1, -3)$

Step4: Find x=0.5 point

Substitute $x=0.5$ into the equation:
$y=-6|0.5|+3 = -3+3=0$
Point: $(0.5, 0)$

Step5: Find x=-0.5 point

Substitute $x=-0.5$ into the equation:
$y=-6|-0.5|+3 = -3+3=0$
Point: $(-0.5, 0)$

Step6: Plot and connect points

Plot the vertex $(0,3)$, $(1,-3)$, $(-1,-3)$, $(0.5,0)$, $(-0.5,0)$, then draw two straight lines from the vertex through the respective points to form a downward-opening V-shape.

Answer:

The graph is a downward-opening V with vertex at $(0, 3)$, passing through points $(-1, -3)$, $(1, -3)$, $(-0.5, 0)$, and $(0.5, 0)$, forming two linear segments:

1. For $x\geq0$: line through $(0,3)$ and $(1,-3)$ (slope = -6)
2. For $x\leq0$: line through $(0,3)$ and $(-1,-3)$ (slope = 6)