Question

move at least one of the 3 guide points below to complete the graph of $y = 8\sqrt{x + 6}$. 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 parent function and transformations

The parent function is \( y = \sqrt{x} \). The given function is \( y = 8\sqrt{x + 6} \), which has a horizontal shift left by 6 units and a vertical stretch by a factor of 8.

Step2: Find the vertex (blue point)

For the square root function \( y = \sqrt{x + h} + k \), the vertex is at \( (-h, k) \). Here, \( h = 6 \), \( k = 0 \), so the vertex should be at \( (-6, 0) \). The current blue point is at \( (0, 0) \), so we need to move the blue point to \( (-6, 0) \).

Step3: Find the red points (vertical stretch)

Take \( x = -5 \) (1 unit right of the vertex). Then \( y = 8\sqrt{-5 + 6} = 8\sqrt{1} = 8 \). Another point: \( x = 3 \), \( y = 8\sqrt{3 + 6} = 8\sqrt{9} = 24 \)? Wait, no, wait, the original red points are at \( x = 1 \) (y≈1) and \( x = 4 \) (y≈2). Let's recalculate. Wait, the parent function \( y = \sqrt{x} \) has points like (0,0), (1,1), (4,2), (9,3). For \( y = 8\sqrt{x + 6} \), when \( x + 6 = 0 \) (x=-6), y=0 (vertex). When \( x + 6 = 1 \) (x=-5), y=81=8. When \( x + 6 = 4 \) (x=-2), y=82=16. When \( x + 6 = 9 \) (x=3), y=8*3=24. So the red points should be adjusted to reflect the vertical stretch. The current red points are at (1,1) and (4,2), which are from \( y = \sqrt{x} \) (since at x=1, y=1; x=4, y=2). For \( y = 8\sqrt{x + 6} \), when x=-5 (x+6=1), y=8, so the point (-5, 8). When x=-2 (x+6=4), y=16, so (-2, 16). When x=3 (x+6=9), y=24, so (3, 24). So we need to move the red points up to these y-values and the blue point to (-6, 0).

Answer:

To complete the graph:

1. Move the blue (vertex) point to \( (-6, 0) \).
2. Move the red points to reflect the vertical stretch: for example, when \( x = -5 \), \( y = 8 \) (point \( (-5, 8) \)); when \( x = -2 \), \( y = 16 \) (point \( (-2, 16) \)); when \( x = 3 \), \( y = 24 \) (point \( (3, 24) \)).