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

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

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

Answer

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.