Question

move at least one of the 5 guide points below to complete the graph of $y=(x - 5)^2$. 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.

Explanation:

Step1: Identify the vertex form

The function is \( y=(x - 5)^2 \), which is a parabola in vertex form \( y = a(x - h)^2 + k \), where the vertex is \( (h,k)=(5,0) \).

Step2: Analyze the current graph

The current blue point (vertex) is at \( (0,0) \), but for \( y=(x - 5)^2 \), the vertex should be at \( (5,0) \). Also, the red points' positions depend on the vertex.

Step3: Adjust the points

• Move the blue point (vertex) from \( (0,0) \) to \( (5,0) \).
• For the red points, since the parabola \( y=(x - 5)^2 \) has a vertical stretch factor \( a = 1 \) (same as \( y = x^2 \) but shifted right), the red points should be adjusted relative to the new vertex. For example, when \( x = 5+1=6 \), \( y=(6 - 5)^2 = 1 \); when \( x = 5+2=7 \), \( y=(7 - 5)^2 = 4 \), etc. So move the red points to be symmetric around \( x = 5 \) with the correct \( y \)-values based on the function.

Answer:

To complete the graph of \( y=(x - 5)^2 \):

1. Move the blue vertex point from \( (0,0) \) to \( (5,0) \).
2. Adjust the red points to be symmetric about \( x = 5 \):

• For a point \( 1 \) unit right of \( x = 5 \) (i.e., \( x = 6 \)), \( y=(6 - 5)^2 = 1 \), so move the red point near \( x = 1 \) (originally) to \( (6,1) \).
• For a point \( 2 \) units right of \( x = 5 \) (i.e., \( x = 7 \)), \( y=(7 - 5)^2 = 4 \), so move the red point near \( x = 2 \) (originally) to \( (7,4) \).
• Similarly, mirror these on the left side of \( x = 5 \) (e.g., \( x = 4 \), \( y = 1 \); \( x = 3 \), \( y = 4 \)) by moving the left - side red points accordingly.