Question

move at least one of the 3 guide points below to complete the graph of $y = 4\sqrt{x - 1}$. 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 (graph with coordinate plane, points, and buttons as described in the image)

Explanation:

Step1: Identify the parent function

The parent function is \( y = \sqrt{x} \), which has a starting point (vertex) at \( (0, 0) \) and passes through \( (1, 1) \), \( (4, 2) \), etc.

Step2: Analyze the transformations

For the function \( y = 4\sqrt{x - 1} \), we have two transformations:

• Horizontal shift: The \( x - 1 \) shifts the graph 1 unit to the right. So the vertex (starting point) of the parent function \( (0, 0) \) moves to \( (1, 0) \).
• Vertical stretch: The coefficient 4 vertically stretches the graph by a factor of 4. So the points on the parent function will have their y - coordinates multiplied by 4.

Step3: Determine the key points

• Vertex: After the horizontal shift, the vertex is at \( (1, 0) \) (this is the blue point, so we should move the blue point from \( (0, 0) \) to \( (1, 0) \)).
• For the point \( (1, 1) \) on the parent function \( y=\sqrt{x} \), after the transformations: \( x = 1+ 0=1\) (wait, no, the horizontal shift is inside the square root. Let's take a general point \( (x,y) \) on \( y = \sqrt{x} \). For \( y = 4\sqrt{x - 1} \), we set \( x-1=t \), so \( x=t + 1 \) and \( y = 4\sqrt{t} \). When \( t = 0 \) (i.e., \( x=1 \)), \( y = 0 \) (vertex). When \( t = 1 \) (i.e., \( x=2 \)), \( y=4\times1 = 4 \). When \( t = 4 \) (i.e., \( x = 5 \)), \( y=4\times2=8 \).
• So the red points (which are the guide points for the stretched and shifted graph) should be adjusted. The first red point (near \( x = 1 \)): on the parent function at \( x = 1 \), \( y = 1 \), but after transformation, at \( x=2 \), \( y = 4 \). The second red point (near \( x = 4 \)): on the parent function at \( x = 4 \), \( y = 2 \), after transformation, at \( x=5 \), \( y = 8 \). And the blue point (vertex) should be at \( (1, 0) \).

To complete the graph:

1. Move the blue point (vertex) from \( (0, 0) \) to \( (1, 0) \) (this is the horizontal shift).
2. For the red points:

• The first red point (closer to the vertex) should be moved to \( (2, 4) \) (since when \( x = 2 \), \( y=4\sqrt{2 - 1}=4\sqrt{1}=4 \)).
• The second red point should be moved to \( (5, 8) \) (since when \( x = 5 \), \( y = 4\sqrt{5 - 1}=4\sqrt{4}=8 \)).

Answer:

Move the blue point to \( (1, 0) \), move the first red point to \( (2, 4) \) and the second red point to \( (5, 8) \) to complete the graph of \( y = 4\sqrt{x - 1} \).