Question

graph the following function by moving the green and blue dots (if necessary).

$y = (x - 5)^3 + 4$

graph with grid, axes, a curve, blue dots, a green dot, and a \start over\ button

answer
attempt 1 out of 2

Explanation:

Step1: Recall the parent function

The parent function of \( y=(x - 5)^{3}+4 \) is \( y = x^{3} \), which has a point at \( (0,0) \) (the green dot for the parent function).

Step2: Apply horizontal shift

For the function \( y=(x - h)^{3}+k \), the horizontal shift is \( h \) units. Here \( h = 5 \), so we shift the parent function's key point (the green dot) 5 units to the right. So the new x - coordinate of the green dot (the vertex - like point for the cubic function) is \( 0+5 = 5 \), and the y - coordinate is \( 0 + 4=4 \) (from the vertical shift \( k = 4 \)).

Step3: Apply vertical shift

The vertical shift \( k = 4 \), so we shift the graph of \( y=(x - 5)^{3} \) (which has its key point at \( (5,0) \)) 4 units up. So the green dot (the main point of the cubic function \( y=(x - 5)^{3}+4 \)) should be at \( (5,4) \).

For the blue dots, we can find some points by plugging in x - values. For example, if \( x=4 \), then \( y=(4 - 5)^{3}+4=(- 1)^{3}+4=-1 + 4 = 3 \), so the point \( (4,3) \); if \( x = 6 \), then \( y=(6 - 5)^{3}+4=1^{3}+4=1 + 4 = 5 \), so the point \( (6,5) \); if \( x=3 \), then \( y=(3 - 5)^{3}+4=(-2)^{3}+4=-8 + 4=-4 \), so the point \( (3,-4) \); if \( x = 7 \), then \( y=(7 - 5)^{3}+4=2^{3}+4=8 + 4 = 12 \) (but since the graph in the given grid has y - axis up to 10, we can choose appropriate x - values within the grid range). The main transformation is shifting the parent cubic function \( y = x^{3} \) 5 units to the right and 4 units up.

Answer:

To graph \( y=(x - 5)^{3}+4 \):

1. Move the green dot (the key point) from \( (0,0) \) (parent function's key point) to \( (5,4) \) (5 units right, 4 units up).
2. Adjust the blue dots by using the equation \( y=(x - 5)^{3}+4 \) to find points. For example, when \( x = 4 \), \( y = 3 \) (point \( (4,3) \)); when \( x=6 \), \( y = 5 \) (point \( (6,5) \)) etc. The green dot should be at \( (5,4) \) and the blue dots should be placed according to the function's values.