Question

move at least one of the 3 guide points below to complete the graph of $y = \frac{5}{2}\sqrt{x - 4}$. 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
answer
attempt 1 out of 2

Explanation:

Step1: Identify the parent function and transformations

The parent function is \( y = \sqrt{x} \), which has a vertex at \( (0,0) \) and passes through \( (1,1) \), \( (4,2) \), etc. The given function is \( y=\frac{5}{2}\sqrt{x - 4} \). This is a horizontal shift (right 4 units) and a vertical stretch by a factor of \( \frac{5}{2} \).

Step2: Determine the vertex (blue point)

For \( y=\frac{5}{2}\sqrt{x - 4} \), the argument of the square root is \( x - 4 \), so the horizontal shift is 4 units to the right. The vertex (blue point) of the parent \( y = \sqrt{x} \) is at \( (0,0) \), so after shifting right 4 units, the vertex of the given function should be at \( (4,0) \).

Step3: Determine the red points (using key points)

For the parent function \( y=\sqrt{x} \), when \( x = 1 \), \( y = 1 \); when \( x = 4 \), \( y = 2 \). For the given function \( y=\frac{5}{2}\sqrt{x - 4} \):

• When \( x - 4=1 \) (i.e., \( x = 5 \)), \( y=\frac{5}{2}\times1=\frac{5}{2}=2.5 \)
• When \( x - 4 = 4 \) (i.e., \( x = 8 \)), \( y=\frac{5}{2}\times2 = 5 \)

So we need to move the blue point (vertex) to \( (4,0) \), and the red points to \( (5, 2.5) \) and \( (8, 5) \) (or other corresponding points based on the stretch and shift).

Answer:

To complete the graph of \( y=\frac{5}{2}\sqrt{x - 4} \):

1. Move the blue vertex point from \( (0,0) \) to \( (4,0) \) (horizontal shift right 4 units).
2. Move the red points to reflect the vertical stretch (\( \frac{5}{2} \)) and horizontal shift: for example, one red point to \( (5, \frac{5}{2}) \) (or \( (5, 2.5) \)) and another to \( (8, 5) \) (using \( x - 4 = 1, 4 \) to find \( x,y \) values for the transformed function).

(Note: The exact coordinates of red points depend on the grid, but the key is shifting the vertex to \( (4,0) \) and adjusting red points for the vertical stretch \( \frac{5}{2} \) and horizontal shift right 4.)