To analyze the graph of the quadratic function \( f(x) = x^2 - 8x + 15 \), we can follow these steps:
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Identify the vertex: The quadratic function is in the standard form \( f(x) = ax^2 + bx + c \), where:
- \( a = 1 \)
- \( b = -8 \)
- \( c = 15 \)
The x-coordinate of the vertex can be found using the formula \( x = -\frac{b}{2a} \): \[ x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4 \] Now, we can find the y-coordinate of the vertex by substituting \( x = 4 \) back into the function: \[ f(4) = 4^2 - 8 \cdot 4 + 15 = 16 - 32 + 15 = -1 \] Thus, the vertex is at the point \( (4, -1) \).
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Find the x-intercepts: To find the x-intercepts, set \( f(x) = 0 \): \[ x^2 - 8x + 15 = 0 \] We can factor this quadratic: \[ (x - 3)(x - 5) = 0 \] This gives us the solutions: \[ x = 3 \quad \text{and} \quad x = 5 \] So the x-intercepts are at the points \( (3, 0) \) and \( (5, 0) \).
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Find the y-intercept: To find the y-intercept, set \( x = 0 \): \[ f(0) = 0^2 - 8 \cdot 0 + 15 = 15 \] Thus, the y-intercept is at the point \( (0, 15) \).
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Direction of the parabola: Since the coefficient \( a = 1 \) is positive, the parabola opens upwards.
Now, summarizing:
- The vertex is at \( (4, -1) \).
- The x-intercepts are \( (3, 0) \) and \( (5, 0) \).
- The y-intercept is \( (0, 15) \).
- The parabola opens upwards.
To visualize the graph:
- The parabola's vertex is the lowest point.
- It crosses the x-axis at \( (3, 0) \) and \( (5, 0) \).
- It crosses the y-axis at \( (0, 15) \).
Using this information, you can sketch the graph of \( f(x) = x^2 - 8x + 15 \) showing all the key points mentioned.