Question

move at least one of the 5 guide points below to complete the graph of $y = \frac{1}{2}(x - 8)^2$. 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.
reset reflect over x-axis

Explanation:

Step1: Identify Vertex Form

The function \( y = \frac{1}{2}(x - 8)^2 \) is in vertex form \( y = a(x - h)^2 + k \), where the vertex is \( (h, k) \). Here, \( h = 8 \), \( k = 0 \), so the vertex (blue point) should be at \( (8, 0) \).

Step2: Move Blue Point

Currently, the blue point is at \( (0, 0) \). Move it to \( (8, 0) \) to shift the parabola right to the correct horizontal position.

Step3: Adjust Red Points (Optional for Stretch)

The coefficient \( a = \frac{1}{2} \) (vertical compression). For \( x = 8 + 2 = 10 \), \( y = \frac{1}{2}(10 - 8)^2 = \frac{1}{2}(4) = 2 \). For \( x = 8 + 4 = 12 \), \( y = \frac{1}{2}(12 - 8)^2 = \frac{1}{2}(16) = 8 \), but initial red points can be adjusted relative to the new vertex. However, the key first step is moving the blue point to \( (8, 0) \).

Answer:

Move the blue vertex point from \( (0, 0) \) to \( (8, 0) \), and adjust red points as needed (e.g., for \( x = 8 \pm 2 \), \( y = 2 \); for \( x = 8 \pm 4 \), \( y = 8 \)) to match \( y = \frac{1}{2}(x - 8)^2 \).