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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

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)

1 answer

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

Vertex

The vertex of a parabola described by the equation \( y = ax^2 + bx + c \) can be found using the formula:

\[ x = -\frac{b}{2a} \]

In this case, \( a = 1 \) and \( b = -8 \):

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

To find the y-coordinate of the vertex, substitute \( x = 4 \) back into the function:

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

Thus, the vertex is at the point \( (4, -16) \).

X-Intercepts

The x-intercepts occur when \( y = 0 \):

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

Factoring gives:

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

Setting each factor to zero yields:

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

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

Y-Intercept

The y-intercept occurs when \( x = 0 \):

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

Thus, the y-intercept is at \( (0, 0) \).

Summary of Key Points

  • Vertex: \( (4, -16) \)
  • X-Intercepts: \( (0, 0) \) and \( (8, 0) \)
  • Y-Intercept: \( (0, 0) \)

Answer Selection

Based on the analysis:

  1. Vertex: \( (4, -16) \)
  2. X-Intercepts: \( (0, 0) \) and \( (8, 0) \)
  3. Y-Intercept: \( (0, 0) \)

From the provided responses, the closest match would be:

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

None of the options provided match this conclusion perfectly; however, the correct way to summarize is:

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