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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-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: (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)

1 answer

To analyze the function \( y = x^2 - 8x \), let's first determine the key points: the vertex, x-intercepts, and y-intercept.

1. Finding the Vertex

The vertex of a quadratic function in the form \( y = ax^2 + bx + c \) can be found using the formula: \[ x = -\frac{b}{2a} \] For \( y = x^2 - 8x \):

  • \( a = 1 \)
  • \( b = -8 \)

Calculating the x-coordinate of the vertex: \[ x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4 \]

Now, substituting \( x = 4 \) back into the function to find the y-coordinate: \[ y = (4)^2 - 8(4) = 16 - 32 = -16 \]

So, the vertex is at: \[ (4, -16) \]

2. Finding the x-Intercepts

To find the x-intercepts, set \( y = 0 \): \[ 0 = x^2 - 8x \] Factoring gives: \[ 0 = x(x - 8) \] Setting each factor to zero, we find: \[ x = 0 \quad \text{or} \quad x = 8 \]

Thus, the x-intercepts are: \[ (0, 0) \quad \text{and} \quad (8, 0) \]

3. Finding the y-Intercept

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

Summary of Key Points:

  • Vertex: \( (4, -16) \)
  • x-Intercepts: \( (0, 0) \) and \( (8, 0) \)
  • y-Intercept: \( (0, 0) \)

Correct Response:

Based on the analysis, the accurate interpretation of the graph yields:

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

Therefore, the best choice among the options given is:

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