Question

graph this function:
y = 6|x| - 7
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 \( y = 6|x| - 7 \), we can rewrite it as \( y = 6|x - 0| - 7 \). So, the vertex \((h, k)\) is \((0, -7)\).

Step2: Plot the vertex

On the given coordinate plane, find the point where \( x = 0 \) and \( y = -7 \). This is the vertex of the absolute - value graph.

Step3: Determine the slope of the two branches

For the absolute - value function \( y = a|x - h|+k \), when \( x\geq h \), the function is \( y = a(x - h)+k \), and the slope is \( a \); when \( x < h \), the function is \( y=-a(x - h)+k \), and the slope is \(-a\). Here, \( a = 6 \), \( h = 0 \), and \( k=-7 \).

• When \( x\geq0 \), the function is \( y = 6x-7 \), and the slope \( m = 6 \). To find another point on this branch, we can use \( x = 1 \). Substitute \( x = 1 \) into \( y = 6x-7 \), we get \( y=6(1)-7=-1 \). So, the point \((1, - 1)\) is on this branch.
• When \( x < 0 \), the function is \( y=-6x - 7 \), and the slope \( m=-6 \). To find another point on this branch, we can use \( x=-1 \). Substitute \( x = - 1 \) into \( y=-6x - 7 \), we get \( y=-6(-1)-7 = 6 - 7=-1 \). So, the point \((-1, - 1)\) is on this branch.
• We can then plot these points and draw the two branches of the absolute - value graph. The left branch (for \( x < 0 \)) has a slope of \(-6\) and the right branch (for \( x\geq0 \)) has a slope of \( 6 \), both passing through the vertex \((0, - 7)\).

Answer:

First, plot the vertex at the point \((0, - 7)\) on the coordinate plane. Then, use the slope of \(6\) (for \(x\geq0\)) and \(- 6\) (for \(x < 0\)) to plot additional points and draw the two branches of the V - shaped graph of \(y = 6|x|-7\). The vertex is at \((0, - 7)\), and the graph has a steeper slope (since \(|a| = 6>1\)) compared to the parent function \(y = |x|\), opening upwards.