To analyze the function \( y = x^2 - 8x \), we can follow these steps:
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Find the vertex: The vertex form of a quadratic equation is \( y = a(x-h)^2 + k \), where \((h,k)\) is the vertex. To find the vertex of the quadratic in standard form \( y = ax^2 + bx + c \), we can use the formula for the x-coordinate of the vertex: \[ x = -\frac{b}{2a} \] In our case:
- \( a = 1 \)
- \( b = -8 \)
Plugging in the values: \[ x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4 \]
To find the y-coordinate of the vertex, substitute \( x = 4 \) back into the equation: \[ y = (4)^2 - 8(4) = 16 - 32 = -16 \] So, the vertex is \( (4, -16) \).
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Find the y-intercept: The y-intercept occurs when \( x = 0 \): \[ y = (0)^2 - 8(0) = 0 \] Therefore, the y-intercept is \( (0, 0) \).
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Find the x-intercepts: The x-intercepts occur when \( y = 0 \): \[ 0 = x^2 - 8x \] Factoring out \( x \): \[ 0 = x(x - 8) \] This gives us two solutions: \[ x = 0 \quad \text{and} \quad x = 8 \] Thus, the x-intercepts are \( (0, 0) \) and \( (8, 0) \).
Now, summarizing the key points:
- Vertex: \( (4, -16) \)
- Y-intercept: \( (0, 0) \)
- X-intercepts: \( (0, 0) \) and \( (8, 0) \)
With this information, we can now respond directly to your question.
The correct key points are:
- Vertex: \( (4, -16) \)
- X-intercept(s): \( (0, 0) \) and \( (8, 0) \)
- Y-intercept: \( (0, 0) \)
Therefore, the statement "x-intercept: (0,8) y-intercept: (0,0) vertex: (4,−16)" is incorrect for the x-intercept and correct for the vertex and y-intercept based on our calculations.
The accurate response should be:
- X-intercepts: \( (0, 0) \) and \( (8, 0) \)
- Y-intercept: \( (0, 0) \)
- Vertex: \( (4, -16) \)