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Chapter 9: Problem 9

For each function \(f(x)=a(x-h)^{2}+k\) graphed, is each of the constants \(h\) and \(k\) positive, negative, or zero?

Short Answer

Expert verified
Answer: The constants h and k in the quadratic function f(x) = a(x-h)^2 + k can be positive, negative, or zero. The constant h determines the horizontal shift of the parabola, whereas the constant k determines the vertical shift.

Step by step solution

01

Constants \(h\) and the horizontal shift

Constant \(h\) is responsible for the horizontal shift of the parabola. If \(h\) is positive, the parabola is shifted to the right. If \(h\) is negative, the parabola is shifted to the left. If \(h\) is zero, the parabola is not shifted horizontally. So, \(h\) can be positive, negative, or zero.
02

Constants \(k\) and the vertical shift

Constant \(k\) is responsible for the vertical shift of the parabola. If \(k\) is positive, the parabola is shifted upward. If \(k\) is negative, the parabola is shifted downward. If \(k\) is zero, the parabola is not shifted vertically. So, \(k\) can be positive, negative, or zero.
03

Conclusion

The constants \(h\) and \(k\) in the quadratic function \(f(x)=a(x-h)^{2}+k\) can be positive, negative, or zero. The constant \(h\) determines the horizontal shift, while \(k\) determines the vertical shift of the parabola graphed. The parabola can be shifted in any direction based on the values of \(h\) and \(k\).

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