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

y-intercept: (0,0)

vertex: (4,−16)

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

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

Answer 1

What point is symmetrical to (−1,−17) for the graph of y=−x2−16 ?(1 point)
Responses

(−1,17)
left parenthesis negative 1 comma 17 right parenthesis

(−17,−1)
left parenthesis negative 17 comma negative 1 right parenthesis

(1,17)
left parenthesis 1 comma 17 right parenthesis

(1,−17)

Answer 2

To analyze the function \( y = x^2 - 8x \), we can find its key points, including 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 for the x-coordinate of the vertex, \( x = -\frac{b}{2a} \).
- Here, \( a = 1 \) and \( b = -8 \).
- So, \( x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4 \).
- To find the y-coordinate of the vertex, substitute \( x = 4 \) back into the equation:
\[
y = (4)^2 - 8(4) = 16 - 32 = -16.
\]
- Therefore, the vertex is \( (4, -16) \).

2. **Finding the X-Intercepts**:
The x-intercepts occur where \( y = 0 \). So we set the equation to zero:
\[
0 = x^2 - 8x.
\]
Factoring this gives:
\[
x(x - 8) = 0.
\]
Thus, \( x = 0 \) or \( x = 8 \). The x-intercepts are:
- \( (0, 0) \) and \( (8, 0) \).

3. **Finding the Y-Intercept**:
The y-intercept occurs where \( x = 0 \):
\[
y = 0^2 - 8(0) = 0.
\]
Therefore, the y-intercept is \( (0, 0) \).

**Conclusion**:
- **Vertex**: \( (4, -16) \)
- **X-intercepts**: \( (0, 0) \) and \( (8, 0) \)
- **Y-intercept**: \( (0, 0) \)

Considering the options provided, the correct annotations would be:
- X-intercepts: \( (0, 0) \) and \( (8, 0) \)
- Vertex: \( (4, -16) \)
- Y-intercept: \( (0, 0) \)

Thus, a correct response would be:
**x-intercepts: (0,0) (8,0), vertex: (4,−16), y-intercept: (0,0)**.

Answer 3

To find the point symmetrical to \((-1, -17)\) for the graph of \(y = -x^2 - 16\), we first need to determine the vertex of the parabola since points symmetrical to a point on the graph will reflect across the vertex.

1. **Finding the Vertex**:
The equation \(y = -x^2 - 16\) is in standard form where:
- \(a = -1\)
- The vertex of a parabola in this form is found at the point \((h, k)\) with:
\[
h = -\frac{b}{2a} \quad \text{and in this case, } b = 0.
\]
So,
\[
h = 0 \quad \text{and } k = -16.
\]
Thus, the vertex is at \((0, -16)\).

2. **Finding the Symmetrical Point**:
The point \((-1, -17)\) is to the left of the vertex at \((0, -16)\).
- The distance from the vertex to the point \((-1, -17)\) can be calculated:
- This is 1 unit to the left horizontally and 1 unit down vertically.
- To find the symmetrical point, we move 1 unit to the right of the vertex while keeping the vertical position the same (since we only need to reflect across the horizontal axis at the y-coordinate of the vertex).
- The symmetrical point will be at:
\[
(1, -17).
\]

### Conclusion:
The point that is symmetrical to \((-1, -17)\) for the graph of \(y = -x^2 - 16\) is \( (1, -17) \).

**Final Response**:
\((1, -17)\)