Create a graph; then answer the following question.

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-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

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
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Answer 1

To analyze the function \( y = x^2 - 8x \), we can rewrite it in vertex form to identify key points such as the vertex, x-intercepts, and y-intercept.

1. Vertex: To find the vertex, we can complete the square or use the vertex formula. 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 \( b = -8 \): \[ x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4 \] Substitute \( x = 4 \) back into the function to find the y-coordinate: \[ y = (4)^2 - 8(4) = 16 - 32 = -16 \] Thus, the vertex is \( (4, -16) \).

2. X-intercepts: To find the x-intercepts, set \( y = 0 \): \[ 0 = x^2 - 8x \] Factoring gives: \[ 0 = x(x - 8) \] This yields: \[ x = 0 \quad \text{or} \quad x = 8 \] Therefore, the x-intercepts are \( (0, 0) \) and \( (8, 0) \).

3. 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: \( (0, 0) \)

Among the provided answers, the correct correspondence is:

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