To find the key points on the graph of the quadratic function \( y = x^2 - 16x + 48 \), we can complete the square or use the vertex formula and find the intercepts.
Step 1: Find the Vertex
The vertex of a parabola given by the equation \( y = ax^2 + bx + c \) can be found using the formula for the x-coordinate of the vertex:
\[ x = -\frac{b}{2a} \]
Here, \( a = 1 \) and \( b = -16 \):
\[ x = -\frac{-16}{2 \cdot 1} = \frac{16}{2} = 8 \]
To find the y-coordinate of the vertex, substitute \( x = 8 \) into the original equation:
\[ y = (8)^2 - 16(8) + 48 = 64 - 128 + 48 = -16 \]
Thus, the vertex is:
\[ \text{vertex: } (8, -16) \]
Step 2: Find the x-intercepts
The x-intercepts occur where \( y = 0 \). Set the equation to zero and solve for \( x \):
\[ 0 = x^2 - 16x + 48 \]
We can factor the quadratic:
\[ 0 = (x - 6)(x - 8) \]
Setting each factor to zero gives the x-intercepts:
\[ x - 6 = 0 \quad \Rightarrow \quad x = 6 \] \[ x - 8 = 0 \quad \Rightarrow \quad x = 8 \]
The x-intercepts are at:
\[ \text{(smaller) x-intercept: } (6, 0) \] \[ \text{(larger) x-intercept: } (8, 0) \]
Step 3: Find the y-intercept
The y-intercept occurs where \( x = 0 \). Substitute \( x = 0 \) into the original equation:
\[ y = (0)^2 - 16(0) + 48 = 48 \]
Thus, the y-intercept is:
\[ \text{y-intercept: } (0, 48) \]
Summary of Key Points
- Vertex: (8, -16)
- (smaller) x-intercept: (6, 0)
- (larger) x-intercept: (8, 0)
- y-intercept: (0, 48)
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