To find the key points of the quadratic equation \( y = x^2 - 8x \), we can analyze the equation step by step.
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Vertex: The vertex form of a parabolic equation can be found by completing the square or using the vertex formula \( x = -\frac{b}{2a} \) for \( y = ax^2 + bx + c \). Here, \( a = 1 \) and \( b = -8 \).
\[ x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4 \]
Now, substitute \( x = 4 \) back into the equation to find the \( y \)-coordinate of the vertex:
\[ y = (4)^2 - 8(4) = 16 - 32 = -16 \]
Therefore, the vertex is at \( (4, -16) \).
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X-intercepts: To find the x-intercepts, set \( y \) to 0 and solve for \( x \):
\[ 0 = x^2 - 8x \] Factor the equation:
\[ 0 = x(x - 8) \]
This gives us the solutions:
\[ x = 0 \quad \text{and} \quad x = 8 \]
Thus, the x-intercepts are \( (0, 0) \) and \( (8, 0) \).
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Y-intercept: To find the y-intercept, set \( x = 0 \) in the equation:
\[ y = (0)^2 - 8(0) = 0 \]
Therefore, the y-intercept is \( (0, 0) \).
Summary of Key Points:
- Vertex: \( (4, -16) \)
- X-intercepts: \( (0, 0) \) and \( (8, 0) \)
- Y-intercept: \( (0, 0) \)
Note: The x-intercept and the y-intercept are the same point at \( (0, 0) \).