To find the key points of the quadratic function \( y = x^2 - 16x + 48 \), we need to complete the square or use the quadratic formula to determine the vertex and intercepts.
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Vertex: A quadratic can be expressed in the standard form \( y = ax^2 + bx + c \). For this function:
- \( a = 1 \)
- \( b = -16 \)
- \( c = 48 \)
The x-coordinate of the vertex is given by the formula: \[ x = -\frac{b}{2a} = -\frac{-16}{2 \cdot 1} = \frac{16}{2} = 8 \]
Now, substitute \( x=8 \) back into the original equation to find the y-coordinate: \[ y = 8^2 - 16 \cdot 8 + 48 = 64 - 128 + 48 = -16 \]
So, the vertex is: \[ \text{Vertex: } (8, -16) \]
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X-Intercepts: To find the x-intercepts, set \( y = 0 \): \[ 0 = x^2 - 16x + 48 \] We can factor this equation: \[ 0 = (x - 12)(x - 4) \] Setting each factor to zero gives us the x-intercepts: \[ x - 12 = 0 \quad \Rightarrow \quad x = 12 \] \[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \]
Therefore, the ((smaller) and (larger)) x-intercepts are: \[ \text{(smaller) x-intercept: } (4, 0) \] \[ \text{(larger) x-intercept: } (12, 0) \]
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Y-Intercept: To find the y-intercept, set \( x = 0 \): \[ y = 0^2 - 16 \cdot 0 + 48 = 48 \]
Therefore, the y-intercept is: \[ \text{y-intercept: } (0, 48) \]
Summary of Key Points:
- Vertex: \( (8, -16) \)
- (smaller) x-intercept: \( (4, 0) \)
- (larger) x-intercept: \( (12, 0) \)
- y-intercept: \( (0, 48) \)
Fill in the values as follows:
- vertex: \( (8, -16) \)
- (smaller) x-intercept: \( (4, 0) \)
- (larger) x-intercept: \( (12, 0) \)
- y-intercept: \( (0, 48) \)