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Create a graph; then answer the following question.

What are the key points on the graph of y=x^2−8x
? Name the vertex, x-intercept(s), and y-intercept.

(1 point)
Responses

A) 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

B) 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.

C) 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

D) 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 find the key points: 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 formula \( x = -\frac{b}{2a} \). Here, \( a = 1 \) and \( b = -8 \).

    \[ x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4 \]

    Next, we find the y-coordinate of the vertex by substituting \( x = 4 \) back into the equation:

    \[ y = (4)^2 - 8(4) = 16 - 32 = -16 \]

    So, the vertex is \( (4, -16) \).

  2. Y-intercept: To find the y-intercept, we set \( x = 0 \):

    \[ y = (0)^2 - 8(0) = 0 \]

    So the y-intercept is \( (0, 0) \).

  3. X-intercepts: To find the x-intercepts, we set \( y = 0 \):

    \[ 0 = x^2 - 8x \]

    Factoring gives us:

    \[ x(x - 8) = 0 \]

    This gives us two solutions:

    \[ x = 0 \quad \text{or} \quad x = 8 \]

    Therefore, the x-intercepts are \( (0, 0) \) and \( (8, 0) \).

Now, we can summarize the key points:

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

From the responses you provided, the correct answer that matches these findings is:

D) x-intercepts: (0,0) (8,0)
y-intercept: (0,0)
vertex: (4,−16)