Create a graph; then answer the following question.

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

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 2

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 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: \((1, -17)\)