Question

move at least one of the 5 guide points below to complete the graph of $y = (x - 6)^2 + 3$. 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 \( y=(x - 6)^2+3 \) is in vertex form \( y = a(x - h)^2 + k \), where \((h,k)\) is the vertex. Here, \( h = 6 \), \( k = 3 \), so the vertex (blue point) should be at \((6, 3)\).

Step2: Move Blue Point

The current blue point is at \((0,0)\). Shift it 6 units right (to \( x = 6 \)) and 3 units up (to \( y = 3 \)) to \((6, 3)\).

Step3: Adjust Red Points (Optional, but for stretch/compression, since \( a = 1 \), red points can stay with same relative distances. For example, a red point at \( x = 6 + 1 = 7 \) should have \( y=(7 - 6)^2+3=4 \), so move a red point to \((7, 4)\) (similar for \( x = 5 \), \( y = 4 \)), and others accordingly to match the parabola shape.

Answer:

Move the blue vertex point to \((6, 3)\) and adjust red points to match \( y=(x - 6)^2+3 \) (e.g., red points at \( (5, 4) \), \( (7, 4) \), \( (4, 7) \), \( (8, 7) \) etc. based on the parabola's shape).