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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) (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

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

1 answer

To analyze the quadratic function \( y = x^2 - 8x \), let's follow these steps:

  1. Find the vertex: The vertex of a parabola in the form \( y = ax^2 + bx + c \) can be found using the formula: \[ x = -\frac{b}{2a} \] Here, \( a = 1 \) and \( b = -8 \). \[ x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4 \] Now, substitute \( x = 4 \) back into the equation to find the \( y \)-coordinate of the vertex: \[ y = (4)^2 - 8(4) = 16 - 32 = -16 \] So, the vertex is at \( (4, -16) \).

  2. Find the x-intercepts: Set \( y = 0 \) to find the x-intercepts: \[ 0 = x^2 - 8x \] Factor the equation: \[ 0 = x(x - 8) \] This gives us the solutions: \[ x = 0 \quad \text{and} \quad x = 8 \] Therefore, the x-intercepts are \( (0, 0) \) and \( (8, 0) \).

  3. Find the y-intercept: Set \( x = 0 \) to find the y-intercept: \[ y = (0)^2 - 8(0) = 0 \] This gives us the y-intercept \( (0, 0) \).

Summary of Results

  • Vertex: \( (4, -16) \)
  • X-intercepts: \( (0, 0) \), \( (8, 0) \)
  • Y-intercept: \( (0, 0) \)

Correct Response

Considering the options you provided, the correct observations are:

  • X-intercepts: \( (0, 0) \), \( (8, 0) \)
  • Y-intercept: \( (0, 0) \)
  • Vertex: \( (4, -16) \)

Therefore, the correct choice from your list is:

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