Question

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

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 \( y = \frac{1}{4}\sqrt{x - 3} \). This involves a horizontal shift to the right by 3 units (because of \( x - 3 \)) and a vertical compression by a factor of \( \frac{1}{4} \) (because of the coefficient \( \frac{1}{4} \)).

Step2: Determine the vertex of the transformed function

For the function \( y = \frac{1}{4}\sqrt{x - 3} \), the vertex (the blue point, since it's the shift point) should be at \( (3, 0) \) (because the horizontal shift is 3 units right from the parent function's vertex \( (0, 0) \)). The original graph has the blue point at \( (0, 0) \), so we need to move the blue point to \( (3, 0) \).

Step3: Determine the red points (guide points for vertical stretch/compression)

For the parent function \( y = \sqrt{x} \), when \( x = 4 \), \( y = 2 \); when \( x = 16 \), \( y = 4 \), etc. For the transformed function \( y = \frac{1}{4}\sqrt{x - 3} \):

• When \( x = 3 \), \( y = \frac{1}{4}\sqrt{3 - 3} = 0 \) (vertex, already considered as the blue point).
• When \( x = 4 \), \( y = \frac{1}{4}\sqrt{4 - 3} = \frac{1}{4}(1) = \frac{1}{4} \).
• When \( x = 7 \), \( y = \frac{1}{4}\sqrt{7 - 3} = \frac{1}{4}(2) = \frac{1}{2} \).
• When \( x = 19 \), \( y = \frac{1}{4}\sqrt{19 - 3} = \frac{1}{4}(4) = 1 \).

So, the blue point (vertex) should be moved from \( (0, 0) \) to \( (3, 0) \). Then, the red points (which are the guide points for the curve) should be adjusted according to the vertical compression. For example, if one red point was at \( (1, 1) \) originally (from the parent function's transformation), it should now be at \( (4, \frac{1}{4}) \), another at \( (7, \frac{1}{2}) \), etc.

Step4: Move the points

1. Move the blue point (vertex) from \( (0, 0) \) to \( (3, 0) \) (this shifts the graph 3 units to the right).
2. For the red points, take a point that was at \( (x, y) \) on the original graph (which is \( y = \sqrt{x} \) scaled or shifted incorrectly) and adjust it. For example, a point that was at \( (1, 1) \) (from \( y = \sqrt{x} \)) should now be at \( (1 + 3, \frac{1}{4} \times 1) = (4, \frac{1}{4}) \) because of the horizontal shift and vertical compression. Similarly, a point at \( (4, 2) \) on the parent function becomes \( (4 + 3, \frac{1}{4} \times 2) = (7, \frac{1}{2}) \), and a point at \( (16, 4) \) becomes \( (16 + 3, \frac{1}{4} \times 4) = (19, 1) \).

By moving the blue point to \( (3, 0) \) and adjusting the red points to these new coordinates, the graph of \( y = \frac{1}{4}\sqrt{x - 3} \) will be correctly plotted.

Answer:

To complete the graph of \( y = \frac{1}{4}\sqrt{x - 3} \):

1. Move the blue (vertex) point from \( (0, 0) \) to \( (3, 0) \) (horizontal shift right by 3).
2. Move red guide points: For a point originally at \( (x, y) \) on \( y = \sqrt{x} \), adjust to \( (x + 3, \frac{1}{4}y) \). For example, \( (1, 1) \to (4, \frac{1}{4}) \), \( (4, 2) \to (7, \frac{1}{2}) \), \( (16, 4) \to (19, 1) \).

(Note: The exact coordinates of red points depend on the original guide points, but the key is shifting the vertex to \( (3, 0) \) and applying the vertical compression/horizontal shift to the other points.)