Create a graph; then answer the following question.

Question

What are the key points on the graph of y=x2−8x
y
=
x
2
−
8
x
? Name the vertex, x-intercept(s), and y-intercept.

(1 point)
Responses

x-intercept: (4,−16)
(
4
,
−
16
)

y-intercept: (0,0)
(
0
,
0
)

vertex: (8,0)
(
8
,
0
)

x -intercept:  left parenthesis 4 comma negative 16 right parenthesis y -intercept:  left parenthesis 0 comma 0 right parenthesis vertex:  left parenthesis 8 comma 0 right parenthesis

x-intercept: (0,8)
(
0
,
8
)

y-intercept: (0,0)
(
0
,
0
)

vertex: (4,−16)
(
4
,
−
16
)

x -intercept:  left parenthesis 0 comma 8 right parenthesis y -intercept:  left parenthesis 0 comma 0 right parenthesis vertex:  left parenthesis 4 comma negative 16 right parenthesis

x-intercepts: (0,0)
(
0
,
0
)
 (0,8)
(
0
,
8
)

vertex: (4,−16)
(
4
,
−
16
)

There is no y-intercept.

x -intercepts:  left parenthesis 0 comma 0 right parenthesis   left parenthesis 0 comma 8 right parenthesis   vertex:  left parenthesis 4 comma negative 16 right parenthesis There is no y -intercept.

x-intercepts: (0,0)
(
0
,
0
)
 (8,0)
(
8
,
0
)

y-intercept: (0,0)
(
0
,
0
)

vertex: (4,−16)

GPT-4o mini · Answer

To analyze the quadratic function $ y = x^2 - 8x $, we will find the vertex, x-intercepts, and y-intercept. ### 1. Vertex The vertex of a quadratic function in the form $ y = ax^2 + bx + c $ can be calculated using the formula: $$ x = -\frac{b}{2a} $$ In this case, $ a = 1 $ and $ b = -8 $: $$ x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4 $$ To find the y-coordinate of the vertex, substitute $ x = 4 $ into the function: $$ y = (4)^2 - 8(4) = 16 - 32 = -16 $$ Thus, the vertex is: $$ \text{Vertex: } (4, -16) $$ ### 2. X-Intercepts To find the x-intercepts, set $ y = 0 $: $$ 0 = x^2 - 8x $$ Factoring gives: $$ 0 = x(x - 8) $$ Setting each factor to zero: $$ x = 0 \quad \text{or} \quad x - 8 = 0 \rightarrow x = 8 $$ Thus, the x-intercepts are: $$ \text{X-Intercepts: } (0, 0) \text{ and } (8, 0) $$ ### 3. Y-Intercept To find the y-intercept, set $ x = 0 $: $$ y = (0)^2 - 8(0) = 0 $$ Thus, the y-intercept is: $$ \text{Y-Intercept: } (0, 0) $$ ### Summary of Key Points - **Vertex:** $ (4, -16) $ - **X-Intercepts:** $ (0, 0) $ and $ (8, 0) $ - **Y-Intercept:** $ (0, 0) $ ### Conclusion Now, looking at the provided options: - The responses indicating a vertex of (4, -16) are correct. - The x-intercepts should indeed be (0, 0) and (8, 0) with the y-intercept also being (0, 0). Correct selection reflecting this would be: - Vertex: (4, -16) - X-Intercepts: (0, 0) and (8, 0) - Y-Intercept: (0, 0)