Question

graph this function:
$y = |-4x| - 7$
click to plot the vertex first.

Explanation:

Step1: Simplify the absolute value function

The function is \( y = |-4x| - 7 \). Since \( |-4x|=|4x| \) (because the absolute value of a negative number is its positive counterpart, and \( |ab| = |a||b| \), so \( |-4x|=|-4||x| = 4|x| \)), the function simplifies to \( y = 4|x| - 7 \). The general form of an absolute value function is \( y = a|x - h| + k \), where \((h,k)\) is the vertex. In this case, \( h = 0 \) (because there is no horizontal shift, the expression inside the absolute value is \( x \), not \( x - h \) with \( h
eq0 \)) and \( k=-7 \). So the vertex of the absolute value function \( y = 4|x| - 7 \) is at \((0, -7)\).

Step2: Plot the vertex

On the given coordinate grid, the vertex is at the point where \( x = 0 \) and \( y=-7 \). So we locate the point \((0, -7)\) on the grid (the intersection of the \( y \)-axis and the horizontal line \( y = -7 \)).

Step3: Determine the slope of the lines (optional for plotting)

For the absolute value function \( y = a|x| + k \), when \( x\geq0 \), the function is \( y = 4x - 7 \) (slope \( m = 4 \)), and when \( x < 0 \), the function is \( y=-4x - 7 \) (slope \( m=-4 \)). To plot additional points, for \( x = 1 \), \( y = 4(1)-7=-3 \), so the point is \((1, -3)\). For \( x=-1 \), \( y=-4(-1)-7 = 4 - 7=-3 \), so the point is \((-1, -3)\). Connecting the vertex \((0, -7)\) with these points (and others) will form the V - shaped graph of the absolute value function. But the first step is to plot the vertex at \((0, -7)\).

Answer:

The vertex of the function \( y = |-4x| - 7 \) is at the point \((0, -7)\), so we plot the vertex at \((0, -7)\) on the coordinate grid.