Question

Create a graph; then answer the following question.

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

(1 point)
Responses

x-intercept: (4,−16)

y-intercept: (0,0)

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

vertex: (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-intercept: (0,8)

y-intercept: (0,0)

vertex: (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) (8,0)

y-intercept: (0,0)

vertex: (4,−16)

Answer 1

To analyze the function \( y = x^2 - 8x \), let's first determine the key points: the vertex, x-intercepts, and y-intercept.

### 1. Finding the Vertex
The vertex of a quadratic function in the form \( y = ax^2 + bx + c \) can be found using the formula:
\[
x = -\frac{b}{2a}
\]
For \( y = x^2 - 8x \):
- \( a = 1 \)
- \( b = -8 \)

Calculating the x-coordinate of the vertex:
\[
x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4
\]

Now, substituting \( x = 4 \) back into the function to find the y-coordinate:
\[
y = (4)^2 - 8(4) = 16 - 32 = -16
\]

So, the vertex is at:
\[
(4, -16)
\]

### 2. Finding the 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, we find:
\[
x = 0 \quad \text{or} \quad x = 8
\]

Thus, the x-intercepts are:
\[
(0, 0) \quad \text{and} \quad (8, 0)
\]

### 3. Finding the y-Intercept
To find the y-intercept, set \( x = 0 \):
\[
y = (0)^2 - 8(0) = 0
\]
Thus, the y-intercept is:
\[
(0, 0)
\]

### Summary of Key Points:
- **Vertex:** \( (4, -16) \)
- **x-Intercepts:** \( (0, 0) \) and \( (8, 0) \)
- **y-Intercept:** \( (0, 0) \)

### Correct Response:
Based on the analysis, the accurate interpretation of the graph yields:
- x-intercepts: (0, 0) and (8, 0)
- vertex: (4, -16)
- y-intercept: (0, 0)

Therefore, the best choice among the options given is:
- x-intercepts: (0, 0) (8, 0)
- vertex: (4, -16)
- y-intercept: (0, 0)