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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-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) (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

1 answer

To analyze the quadratic function \( y = x^2 - 8x \), we can rewrite it in vertex form by completing the square.

  1. Complete the square: \[ y = x^2 - 8x \] To complete the square: \[ y = (x^2 - 8x + 16) - 16 = (x - 4)^2 - 16 \]

    Thus, the vertex form of the equation is: \[ y = (x - 4)^2 - 16 \]

    From this form, we can identify key points:

    • Vertex: The vertex is located at \( (4, -16) \).

  2. Finding the x-intercepts: To find the x-intercepts, set \( y = 0 \): \[ 0 = (x - 4)^2 - 16 \] \[ (x - 4)^2 = 16 \] Taking the square root: \[ x - 4 = 4 \quad \text{or} \quad x - 4 = -4 \] Therefore: \[ x = 8 \quad \text{and} \quad x = 0 \] The x-intercepts are:

    • \( (0, 0) \)
    • \( (8, 0) \)

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

    • \( (0, 0) \)

Summary of Key Points:

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

Answer Selection:

Given the options provided, the correct response is:

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

So, the correct answer aligns with: x-intercepts: \( (0, 0) \) and \( (8, 0) \), y-intercept: \( (0, 0) \), vertex: \( (4, -16) \).