Question

move at least one of the 5 guide points below to complete the graph of $y = 3(x - 8)^2$. 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 the vertex form

The function is \( y = 3(x - 8)^2 \), which 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 the blue point

Currently, the blue point is at \( (0, 0) \). We need to move it to \( (8, 0) \) along the x - axis (since \( k = 0 \), no vertical movement).

Step3: Adjust red points (optional for stretch, but let's check the vertical stretch)

The coefficient \( a = 3 \). For a standard parabola \( y = x^2 \), when \( x = 1 \), \( y = 1 \); for \( y = 3(x - 8)^2 \), when \( x = 9 \) (since \( x-8 = 1 \)), \( y = 3(1)^2=3 \); when \( x = 7 \) (since \( x - 8=- 1 \)), \( y = 3(-1)^2 = 3 \). When \( x=10 \) ( \( x - 8 = 2 \) ), \( y=3(2)^2 = 12 \); when \( x = 6 \) ( \( x - 8=-2 \) ), \( y = 3(-2)^2=12 \). But first, moving the vertex is essential. So the main action is moving the blue point to \( (8, 0) \) and then adjusting red points according to the function's values.

Answer:

Move the blue vertex point from \( (0, 0) \) to \( (8, 0) \). Then, for red points, for example, when \( x = 9 \) (1 unit right of vertex), \( y=3(1)^2 = 3 \); when \( x = 7 \) (1 unit left of vertex), \( y = 3 \); when \( x = 10 \) (2 units right of vertex), \( y=3(2)^2=12 \); when \( x = 6 \) (2 units left of vertex), \( y = 12 \). So adjust the red points to match these coordinates relative to the new vertex \( (8, 0) \). (The key first step is moving the blue point to \( (8, 0) \))