Question

graph this function:
y = 2|x| - 6
click to plot the vertex first.

Explanation:

Step1: Identify the vertex form of absolute value function

The general form of an absolute value function is \( y = a|x - h| + k \), where \((h, k)\) is the vertex. For the function \( y = 2|x| - 6 \), we can rewrite it as \( y = 2|x - 0| - 6 \). So, \( h = 0 \) and \( k = -6 \).

Step2: Plot the vertex

The vertex is at the point \((0, -6)\). On the given coordinate plane, we find the point where \( x = 0 \) and \( y = -6 \) and plot it.

Step3: Determine the slope for each side

For \( x \geq 0 \), the function becomes \( y = 2x - 6 \), so the slope \( m = 2 \). For \( x < 0 \), the function becomes \( y = -2x - 6 \), so the slope \( m = -2 \).

Step4: Plot additional points

For \( x = 1 \) (using the positive slope equation): \( y = 2(1) - 6 = -4 \), so the point is \((1, -4)\). For \( x = -1 \) (using the negative slope equation): \( y = -2(-1) - 6 = -4 \), so the point is \((-1, -4)\). We can plot these points and draw the two lines (one with positive slope for \( x \geq 0 \) and one with negative slope for \( x < 0 \)) to complete the graph.

Answer:

The vertex is at \((0, -6)\), and the graph is a V - shaped graph with the vertex at \((0, -6)\), passing through points like \((1, -4)\) and \((-1, -4)\) (and other points following the slopes for \( x \geq 0 \) and \( x < 0 \)).