Question

question
move at least one of the 5 guide points below to complete the graph of $y = (x - 7)^2$. 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.
(there is a graph with coordinate axes, a parabola - like graph with red and blue points, and buttons labeled reset and reflect over x axis)

Explanation:

Step1: Identify the vertex of \( y=(x - 7)^2 \)

The vertex form of a parabola is \( y=a(x - h)^2 + k \), where \((h,k)\) is the vertex. For \( y=(x - 7)^2 \), \( h = 7 \), \( k = 0 \), so the vertex is \((7,0)\).

Step2: Move the blue vertex point

The current blue vertex is at \((0,0)\). We need to move it to \((7,0)\) by shifting it 7 units to the right (since \( h = 7 \) and we start at \( x = 0 \)).

Step3: Adjust red points (optional for stretch/compression, but first move vertex)

After moving the vertex to \((7,0)\), the red points will adjust according to the parabola \( y=(x - 7)^2 \). For example, when \( x = 8 \), \( y=(8 - 7)^2=1 \); when \( x = 9 \), \( y=(9 - 7)^2 = 4 \), etc. So we move the red points to their corresponding positions on \( y=(x - 7)^2 \).

Answer:

Move the blue vertex point from \((0,0)\) to \((7,0)\) (shift 7 units right), and adjust red points to lie on \( y=(x - 7)^2 \) (e.g., for \( x = 7 + 1 = 8 \), \( y = 1 \); \( x = 7 + 2 = 9 \), \( y = 4 \), etc.).