Question

move at least one of the 5 guide points below to complete the graph of $y = (x - 6)^2 - 8$. moving the red points changes the vertical stretch or compression. moving the blue point shifts the function left/right/up/down. click the buttons below to start over or reflect over the x-axis.
reset reflect over x-axis

Explanation:

Step1: Identify Vertex Form

The function \( y=(x - 6)^2 - 8 \) is in vertex form \( y = a(x - h)^2 + k \), where \((h,k)\) is the vertex. Here, \( h = 6 \), \( k = -8 \), so the vertex (blue point) should be at \((6, -8)\).

Step2: Move Blue Point

Currently, the blue point is at \((0,0)\). Move it to \((6, -8)\) to shift the function right 6 units and down 8 units.

Step3: Adjust Red Points (Optional)

Since the coefficient of \((x - 6)^2\) is 1 (no vertical stretch/compression), red points can be adjusted relative to the new vertex. For example, for \( x = 6 + 1 = 7 \), \( y=(7 - 6)^2 - 8 = 1 - 8 = -7 \); for \( x = 6 - 1 = 5 \), \( y=(5 - 6)^2 - 8 = 1 - 8 = -7 \); for \( x = 6 + 2 = 8 \), \( y=(8 - 6)^2 - 8 = 4 - 8 = -4 \); for \( x = 6 - 2 = 4 \), \( y=(4 - 6)^2 - 8 = 4 - 8 = -4 \). Move red points to these coordinates to match the parabola.

Answer:

Move the blue vertex point to \((6, -8)\) and adjust red points as needed (e.g., \((5, -7)\), \((7, -7)\), \((4, -4)\), \((8, -4)\)) to complete the graph of \( y=(x - 6)^2 - 8 \).