Ask a New Question
Menu

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

1 answer

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

Step 1: Completing the Square

To complete the square for \( y = x^2 - 8x \):

  1. Take half of the coefficient of \( x \) (which is -8), square it, and add it inside the equation:

    • Half of -8 is -4, and squaring it gives 16.

  2. Rewrite the equation: \[ y = (x^2 - 8x + 16) - 16 \] \[ y = (x - 4)^2 - 16 \]

This shows that the vertex of the parabola is at \( (4, -16) \).

Step 2: Finding the Intercepts

Vertex:

  • The vertex is at \((4, -16)\).

x-intercepts:

To find the x-intercepts, set \( y = 0 \): \[ 0 = x^2 - 8x \] Factoring gives: \[ 0 = x(x - 8) \] This gives the solutions:

  • \( x = 0 \) and \( x = 8 \) So the x-intercepts are:
  • \((0, 0)\) and \((8, 0)\).

y-intercept:

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

  • \((0, 0)\).

Summary of Results

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

Answering the Original Question

Putting this all together, here are the key points again:

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

Therefore, the correct response includes:

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