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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-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: (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-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)

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

1 answer

To analyze the quadratic function \( y = x^2 - 8x \), we first need to identify the key points, including 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 found using the vertex formula: \[ x = -\frac{b}{2a} \] For our equation, \( a = 1 \) and \( b = -8 \): \[ x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4 \] To find the corresponding \( y \)-coordinate, substitute \( x = 4 \) back into the equation: \[ y = (4)^2 - 8(4) = 16 - 32 = -16 \] Thus, the vertex is \( (4, -16) \).

2. X-Intercepts

X-intercepts occur where \( y = 0 \). So we set the equation equal to zero: \[ 0 = x^2 - 8x \] Factoring gives us: \[ 0 = x(x - 8) \] Setting each factor to zero gives us: \[ x = 0 \quad \text{or} \quad x = 8 \] Therefore, the x-intercepts are \( (0, 0) \) and \( (8, 0) \).

3. Y-Intercept

The y-intercept occurs when \( x = 0 \): \[ y = (0)^2 - 8(0) = 0 \] Thus, the y-intercept is \( (0, 0) \).

Summary

  • Vertex: \( (4, -16) \)
  • X-Intercepts: \( (0, 0) \) and \( (8, 0) \)
  • Y-Intercept: \( (0, 0) \)

Based on the analysis, the correct response would be:

  • x-intercepts: \( (0, 0) \) and \( (8, 0) \)
  • y-intercept: \( (0, 0) \)
  • vertex: \( (4, -16) \)

Thus, there are two possible answers:

  • Correct Option: \( x \)-intercepts: \( (0, 0) \) and \( (8, 0) \), y-intercept: \( (0, 0) \), vertex: \( (4, -16) \)