To determine the key points of the quadratic function \( y = x^2 - 8x \), we can follow these steps:
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Vertex: The vertex of a quadratic function in the form \( y = ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \). Here, \( a = 1 \) and \( b = -8 \).
\[ x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4 \]
Now, substitute \( x = 4 \) back into the original equation to find the \( y \)-coordinate of the vertex:
\[ y = (4)^2 - 8(4) = 16 - 32 = -16 \]
Therefore, the vertex is at the point \( (4, -16) \).
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Y-intercept: The y-intercept occurs when \( x = 0 \).
\[ y = (0)^2 - 8(0) = 0 \]
Thus, the y-intercept is at the point \( (0, 0) \).
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X-intercepts: The x-intercepts occur when \( y = 0 \). We set the equation to zero and solve for \( x \):
\[ 0 = x^2 - 8x \]
Factoring gives:
\[ 0 = x(x - 8) \]
Setting each factor to zero results in:
\[ x = 0 \quad \text{or} \quad x = 8 \]
Thus, the x-intercepts are at the points \( (0, 0) \) and \( (8, 0) \).
Summary of Key Points:
- Vertex: \( (4, -16) \)
- Y-intercept: \( (0, 0) \)
- X-intercepts: \( (0, 0) \) and \( (8, 0) \)
Keep in mind that the y-intercept and one of the x-intercepts coincide at \( (0, 0) \).