Question

move at least one of the 5 guide points below to complete the graph of $y = (x - 7)^2 - 8$. 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 \( y=(x - 7)^2-8 \) is in vertex form \( y = a(x - h)^2 + k \), where the vertex is \( (h,k)=(7,-8) \). The blue point (vertex) in the graph is currently at \( (0,0) \), so we need to move it.

Step2: Move the Blue Point

Move the blue vertex point from \( (0,0) \) to \( (7,-8) \). For the red points (which depend on the vertex), since the parent function \( y = x^2 \) has points like \( (1,1) \), \( (-1,1) \), \( (2,4) \), \( (-2,4) \) relative to the vertex \( (0,0) \), relative to the new vertex \( (7,-8) \), the red points should be at \( (7 + 1,-8+1)=(8,-7) \), \( (7 - 1,-8+1)=(6,-7) \), \( (7 + 2,-8+4)=(9,-4) \), \( (7 - 2,-8+4)=(5,-4) \). So we move the blue point to \( (7,-8) \) and adjust the red points accordingly (by moving them to the positions relative to the new vertex).

Answer:

Move the blue vertex point to \( (7, -8) \) and adjust the red points to their respective positions relative to the new vertex (e.g., red points at \( (6, -7) \), \( (8, -7) \), \( (5, -4) \), \( (9, -4) \)) to complete the graph of \( y=(x - 7)^2-8 \).