5.3: Power functions
- Page ID
- 238572
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1.Use a graphing calculator, Desmos, graph \(y=x^2, y=x^3, y=x^4\), and \(y=x^5\) on the same coordinate plane. Draw these graphs below, or on separate graph paper.
Discuss with your group and write a "rule" in a sentence or two.
- If we can write the general form of a power function as \(f(x)=k x^p\), what can you say about all graphs that have a positive, even integer \(p\) ?
- What can you say about all graphs that have a positive, odd integer \(p\) ?
2. Rewrite \(y=x^{-2}, y=x^{-3}\), and \(y=x^{-p}\) as functions without negative integers. Describe in a sentence how to rewrite a negative-integer power function with only positive powers.
3. Use a graphing calculator, Desmos, graph \(y=x^{-2}, y=x^{-3}, y=x^{-4}\), and \(y=x^{-5}\) on the same coordinate plane. Draw these graphs below, or on separate graph paper.
Discuss with your group and write a "rule" in a sentence or two.
- What can you say about all graphs that have a negative, even integer \(p\) ?
- What can you say about all graphs that have a negative, odd integer \(p\) ?
4. Rewrite \(y=x^{\frac{1}{3}}\) and \(y=2 x^{\frac{1}{2}}\) without fractional exponents.
5. Use a graphing calculator, Desmos, graph \(y=x^{\frac{1}{2}}, y=x^{\frac{1}{3}}, y=x^{\frac{1}{4}}\), and \(y=x^{\frac{1}{5}}\) on the same coordinate plane. Draw these graphs below, or on separate graph paper.
Discuss with your group and write a "rule" in a sentence or two.
- What can you say about all graphs that have a fractional power with an even denominator?
- What can you say about all graphs that have a fractional power with an odd denominator?
6. Each part of this problem will explore a function's long-run behavior, that is, what happens to the value of \(f(x)\) as \(x\) becomes infinitely large.
- For the functions \(f(x)=x^2\) and \(g(x)=x^3\), make a table of values for \(x=10,20,30,40\), and 50 . What value does each function appear to be approaching as \(x\) increases?
- For the functions \(f(x)=x^{-2}\) and \(g(x)=x^{-3}\), make a table of values for \(x=10,20,30,40\), and 50. What value does each function appear to be approaching as \(x\) increases?
- Use a similar method to test the long-run behavior of \(y=x^{\frac{1}{2}}\) and \(y=x^{\frac{1}{4}}\).
- Discuss with your group and make a general rule about the long-run behavior of a power function \(y=x^p\). Make sure you clearly indicate any different cases depending on the value of \(p\).
7. For the functions \(f(x)=x^{-2}\) and \(g(x)=x^{-3}\), make a table of values for \(x=0.1,0.05,0.01,0.001\), 0.0001, and 0. What value does each function appear to be approaching as \(x\) approaches zero from the right?