To analyze the function \( h(x) = x^2 + 20x - 17 \), we will check each statement provided.
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The vertex of h is (–10, –117).
To find the vertex, we can use the vertex formula \( x = -\frac{b}{2a} \). Here, \( a = 1 \) and \( b = 20 \): \[ x = -\frac{20}{2 \cdot 1} = -10. \] Now we substitute \( x = -10 \) back into the function to find the y-coordinate: \[ h(-10) = (-10)^2 + 20(-10) - 17 = 100 - 200 - 17 = -117. \] Therefore, the vertex is indeed \( (-10, -117) \), so this statement is True.
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The vertex form of the function is h(x) = (x + 20)² – 17.
The correct vertex form of a quadratic function is \( h(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. We already found the vertex to be \( (-10, -117) \). To rewrite \( h(x) \) in vertex form, complete the square: \[ h(x) = x^2 + 20x - 17. \] Factor out the quadratic and linear parts: \[ h(x) = (x^2 + 20x + 100) - 100 - 17 = (x + 10)^2 - 117. \] Thus, the vertex form is \( h(x) = (x + 10)^2 - 117 \), not \( (x + 20)^2 - 17 \). So this statement is False.
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The maximum value of the function is –17.
Since the parabola opens upwards (the coefficient of \( x^2 \) is positive), there is a minimum value at the vertex, which we found to be \( -117 \). Thus, the function does not have a maximum value, and this statement is False.
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To graph the function h, shift the graph of f(x) = x² left 10 units and down 117 units.
Since the vertex is \( (-10, -117) \) and the function opens upwards, the transformation involves shifting the base graph \( f(x) = x^2 \) left 10 units to reach \( (0, 0) \) to \( (-10, 0) \) and then down 117 units to reach \( (-10, -117) \). This statement is True.
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The axis of symmetry of function h is x = 20.
The axis of symmetry is given by \( x = -10 \) because the vertex is at \( (-10, -117) \). Thus, this statement is False.
In summary, the true statements about the function \( h(x) = x^2 + 20x - 17 \) are:
- The vertex of h is (–10, –117).
- To graph the function h, shift the graph of f(x) = x² left 10 units and down 117 units.