To analyze the function \( f(x) = x^2 - 8x + 5 \), let's determine the true statements one by one.
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Vertex Form: The vertex form of a quadratic is given by \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex. First, we can complete the square to rewrite the function in vertex form.
\[ f(x) = x^2 - 8x + 5 \] To complete the square: \[ f(x) = (x^2 - 8x + 16) - 16 + 5 = (x - 4)^2 - 11 \] Thus, the function in vertex form is \( f(x) = (x - 4)^2 - 11 \). This statement is true.
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Vertex of the Function: From the vertex form we found, the vertex is \((h, k) = (4, -11)\). Therefore, the vertex is not \((-8, 5)\). This statement is false.
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Axis of Symmetry: The axis of symmetry can be found from the vertex, which is \(x = h\). From the vertex \((4, -11)\), the axis of symmetry is \(x = 4\). This statement is false since it is not \(x = 5\).
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Y-Intercept: The y-intercept occurs when \(x = 0\). We can find the y-intercept by calculating \(f(0)\): \[ f(0) = (0)^2 - 8(0) + 5 = 5 \] Thus, the y-intercept is \((0, 5)\). This statement is true.
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X-Intercepts: The function crosses the x-axis where \(f(x) = 0\). We can find the roots with the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{8 \pm \sqrt{(-8)^2 - 4(1)(5)}}{2(1)} = \frac{8 \pm \sqrt{64 - 20}}{2} = \frac{8 \pm \sqrt{44}}{2} = \frac{8 \pm 2\sqrt{11}}{2} = 4 \pm \sqrt{11} \] Since there are two distinct real solutions, the function does indeed cross the x-axis twice. This statement is true.
In summary, the true statements about the function \( f(x) = x^2 - 8x + 5 \) are:
- The function in vertex form is \( f(x) = (x - 4)^2 - 11 \).
- The y-intercept of the function is \((0, 5)\).
- The function crosses the x-axis twice.
So the three options that are true are 1, 4, and 5.
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