Question

move at least one of the 5 guide points below to complete the graph of $y = (x - 4)^2$. 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: Recall vertex form of parabola

The function \( y=(x - h)^2 + k \) has vertex at \((h,k)\). For \( y=(x - 4)^2 \), vertex is \((4,0)\).

Step2: Identify the blue point (vertex)

The blue point is the vertex. Currently, it's at \((0,0)\). Move the blue point to \((4,0)\) to shift the parabola right by 4 units (since \( h = 4 \) in \( y=(x - 4)^2 \)).

Step3: Adjust red points (if needed)

The red points are symmetric around the vertex. After moving the vertex to \((4,0)\), the red points should be symmetric about \( x = 4 \). For example, if a red point was at \( x = a \) before, it should now be at \( x = 8 - a \) (to be symmetric about \( x = 4 \)) with the same \( y \)-value (since the vertical stretch factor is 1, same as \( y = x^2 \) initially).

Answer:

Move the blue vertex point from \((0,0)\) to \((4,0)\), and adjust red points to be symmetric about \( x = 4 \) (e.g., if a red point was at \((-1,1)\), move it to \((9,1)\); if at \((1,1)\), move to \((7,1)\); the red points at \( y = 4 \) should move from \( x=-2 \) and \( x = 2 \) to \( x = 2 \) and \( x = 6 \) respectively, maintaining \( y = 4 \) since \((2 - 4)^2=4\) and \((6 - 4)^2 = 4\)).