Question

graph this function: y = |-4x| - 4 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. First, rewrite the given function \( y = |-4x| + 4 \) in the form \( y = |a(x - h)| + k \). We can factor out \(-4\) from the absolute value: \( y = |-4(x - 0)| + 4 \), so \( a=-4\), \( h = 0 \), \( k = 4 \). Thus, the vertex is \((0, 4)\).

Step2: Determine the slope of the lines

For the absolute value function \( y = |a(x - h)| + k \), the slope of the line to the right of the vertex (when \( x\geq h \)) is \( a \), and to the left (when \( x < h \)) is \(-a\). Here, \( a=-4 \), so for \( x\geq0 \), the slope is \(-4\) (since \( y = |-4x| + 4=4|x| + 4\), and for \( x\geq0 \), \( y = 4x + 4 \), wait, correction: \( |-4x|=4|x| \), so the function is \( y = 4|x| + 4 \). So for \( x\geq0 \), the equation is \( y = 4x + 4 \) (slope \( 4 \)), and for \( x < 0 \), the equation is \( y=-4x + 4 \) (slope \(-4\)). But first, plot the vertex \((0, 4)\). Then, to find another point on the right side ( \( x\geq0 \) ), let's take \( x = 1 \): \( y = 4(1)+4 = 8 \), so the point is \((1, 8)\). On the left side ( \( x < 0 \) ), take \( x=-1 \): \( y=-4(-1)+4 = 8 \), so the point is \((-1, 8)\). Now we can plot the vertex \((0, 4)\) first, then use the slopes to draw the two lines (or plot the other points and connect them).

Answer:

To graph the function \( y = |-4x| + 4 \):

1. Plot the vertex: The vertex is at \((0, 4)\) (since the function can be written as \( y = 4|x| + 4 \), with \( h = 0 \), \( k = 4 \)).
2. Plot additional points:

• For \( x = 1 \): \( y = 4(1)+4 = 8 \), so the point \((1, 8)\) is on the graph.
• For \( x=-1 \): \( y = 4|-1| + 4 = 8 \), so the point \((-1, 8)\) is on the graph.

1. Draw the lines: Connect the vertex \((0, 4)\) to \((1, 8)\) (right - hand line, slope \( 4 \)) and to \((-1, 8)\) (left - hand line, slope \(-4\)). The graph is a V - shaped graph with vertex at \((0, 4)\), opening upwards, with steeper slopes (since \( |a| = 4 \)) compared to \( y = |x| \).