Question

move at least one of the 5 guide points below to complete the graph of $y = (x + 6)^2$. 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 \( y=(x + 6)^2 \), which is in vertex form \( y=(x - h)^2+k \), where the vertex is \((h,k)\). Here, \( h=-6 \), \( k = 0 \), so the vertex (blue point) should be at \((-6,0)\).

Step2: Move the blue point

Currently, the blue point is at \((0,0)\). We need to move it to \((-6,0)\) to shift the parabola left by 6 units (since the original \( y = x^2 \) has vertex at \((0,0)\), and \( y=(x + 6)^2 \) is a horizontal shift left by 6).

Step3: Adjust red points (optional, but for accuracy)

After moving the vertex, the red points (which are symmetric around the vertex) will adjust their positions. For example, for \( y=(x + 6)^2 \), when \( x=-5 \), \( y=(-5 + 6)^2=1 \); when \( x=-7 \), \( y=(-7 + 6)^2 = 1 \); when \( x=-4 \), \( y=(-4+6)^2 = 4 \); when \( x=-8 \), \( y=(-8 + 6)^2=4 \). So the red points should be at \((-5,1)\), \((-7,1)\), \((-4,4)\), \((-8,4)\) (matching the vertex shift).

Answer:

Move the blue vertex point from \((0,0)\) to \((-6,0)\), and adjust the red points to be symmetric around \( x=-6 \) (e.g., \((-5,1)\), \((-7,1)\), \((-4,4)\), \((-8,4)\)) to complete the graph of \( y=(x + 6)^2 \).