12 Section 2.4 Sinusoidal Functions
a. If [latex]f(x)=\sin(x)[/latex] is the parent function of [latex]g(x)[/latex], describe the transformations that have been performed on [latex]f(x)[/latex] to get [latex]g(x)[/latex].
b. What is the amplitude of [latex]g(x)[/latex]?
c. What is the midline of [latex]g(x)[/latex]?
d. What is the period of [latex]g(x)[/latex]?
e. What is the range of [latex]g(x)[/latex]?
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Answers
a. Horizontal compression by a factor of 3, and shifted down 1.
b. [latex]a=1[/latex].
c. [latex]y=-1[/latex].
d. Since [latex]k=3[/latex], period [latex]P=\frac{2\pi}{3}[/latex].
e. Range is [latex][-2, 0][/latex]
2. Given [latex]h(x)=4\cos(2(x-3))+7[/latex].
a. Determine the period of [latex]h(x)[/latex].
b. Describe the horizontal shift of [latex]h(x)[/latex].
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Answers
a. Since [latex]k=2[/latex], period [latex]P=\frac{2\pi}{2}=\pi[/latex].
b. This function has been shifted 3 units to the right.
a. Determine the period of [latex]j(x)[/latex].
b. Describe the horizontal shift of [latex]j(x)[/latex].
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Answers
a. Since [latex]k=\frac{1}{3}[/latex], period [latex]P=\frac{2\pi}{\frac{1}{3}}=6\pi[/latex].
b. This function has been shifted 9 units to the left. While it may seem like this should be 3 units to the left, recall that when you have both a horizontal shift as well as a horizontal stretch/compression, you must put the function in the form [latex]k(x-b)[/latex] in order to determine the horizontal shift value of [latex]b[/latex].
In this case, [latex]\frac{1}{3}x+3[/latex] will become [latex]\frac{1}{3}(x+9)[/latex] when factored into the [latex]k(x-b)[/latex] form.