Question

graph $g(x) = -8|x| + 1$.

Explanation:

Step1: Recall 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 \( g(x)=-8|x| + 1 \), we have \( h = 0 \) and \( k = 1 \), so the vertex is \((0,1)\).

Step2: Analyze the slope (rate of change)

For \( x\geq0 \), the function becomes \( g(x)=-8x + 1 \), the slope \( m=-8 \). For \( x < 0 \), the function becomes \( g(x)=-8(-x)+1 = 8x + 1 \), the slope \( m = 8 \).

Step3: Find key points

• Vertex: \((0,1)\)
• For \( x = 1 \): \( g(1)=-8(1)+1=-7 \) (Wait, the given graph has a point at \((4,4)\) which is incorrect for \( g(x)=-8|x| + 1 \). Let's recalculate correctly. Wait, maybe the original graph is wrong, but we focus on the function \( g(x)=-8|x|+1 \). Let's find correct points:
• When \( x = 0 \), \( g(0)=-8(0)+1 = 1 \) (vertex).
• When \( x = 1 \), \( g(1)=-8(1)+1=-7 \)
• When \( x=-1 \), \( g(-1)=-8(|-1|)+1=-8 + 1=-7 \)
• The graph should be a V - shaped graph opening downward (since \( a=-8<0 \)) with vertex at \((0,1)\) and steeper slope (because \( |a| = 8 \) is large) compared to \( y = |x| \).

Answer:

To graph \( g(x)=-8|x| + 1 \):

1. Plot the vertex at \((0,1)\).
2. For \( x\geq0 \), use the slope \( -8 \): from \((0,1)\), move 1 unit right and 8 units down to get \((1, - 7)\), and draw a line with slope \(-8\) through these points.
3. For \( x < 0 \), use the slope \( 8 \): from \((0,1)\), move 1 unit left and 8 units down to get \((-1, - 7)\), and draw a line with slope \( 8 \) through these points. The graph is a downward - opening V - shape with vertex at \((0,1)\) and steep slopes.