To find the key features of the parabola given by the equation \( y = x^2 - 16x + 28 \), we can follow these steps:
Step 1: Find the vertex
The vertex of a parabola in the standard form \( y = ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \).
For the equation \( y = x^2 - 16x + 28 \):
- \( a = 1 \)
- \( b = -16 \)
- \( c = 28 \)
Calculate x-coordinate of the vertex: \[ x = -\frac{-16}{2 \cdot 1} = \frac{16}{2} = 8 \]
Calculate y-coordinate of the vertex: \[ y = (8)^2 - 16(8) + 28 = 64 - 128 + 28 = -36 \]
Vertex: \[ (8, -36) \]
Step 2: Find x-intercepts
To find the x-intercepts, we set \( y = 0 \): \[ 0 = x^2 - 16x + 28 \] We can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Plugging \( a = 1, b = -16, c = 28 \): \[ x = \frac{16 \pm \sqrt{(-16)^2 - 4 \cdot 1 \cdot 28}}{2 \cdot 1} \] \[ x = \frac{16 \pm \sqrt{256 - 112}}{2} \] \[ x = \frac{16 \pm \sqrt{144}}{2} \] \[ x = \frac{16 \pm 12}{2} \]
This gives us two solutions:
- \( x = \frac{28}{2} = 14 \) (larger x-intercept)
- \( x = \frac{4}{2} = 2 \) (smaller x-intercept)
X-intercepts:
- Smaller x-intercept: \( (2, 0) \)
- Larger x-intercept: \( (14, 0) \)
Step 3: Find y-intercept
To find the y-intercept, set \( x = 0 \): \[ y = 0^2 - 16(0) + 28 = 28 \] Y-intercept: \[ (0, 28) \]
Step 4: Axis of symmetry
The axis of symmetry is the x-coordinate of the vertex: \[ x = 8 \]
Summary of the Key Features:
- Vertex: \( (8, -36) \)
- Smaller x-intercept: \( (2, 0) \)
- Larger x-intercept: \( (14, 0) \)
- Y-intercept: \( (0, 28) \)
- Axis of symmetry: \( x = 8 \)
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