Common Core: High School - Functions - Model Periodicity, Amplitude, Frequency, and Midline of Trigonometric Functions: CCSS.Math.Content.HSF-TF.B.5

1

What is the amplitude of the following function?

$y=2\sin(x)$

$2\pi$

$\frac{\pi}{2}$

Explanation

This question is testing one's ability to identify the periodicity of a trigonometric function. This requires the understanding of trigonometric functions and their graphical and algebraic characteristics.

For the purpose of Common Core Standards, "Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline" falls within the Cluster B of "Model Periodic Phenomena with Trigonometric Functions" concept (CCSS.MATH.CONTENT.HSF-TF.B.5).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Write the general form of trigonometric shifts.

where

$\F(x)=\textup{New function after shift} \a=\textup{Amplitude} \b=\textup{period}\rightarrow \frac{2\pi}{|B|} \ \c=\textup{Horizontal shift} \ d=\textup{Vertical shift} \f(x)=\textup{Trigonometric function}$

Step 2: Algebraically identify the amplitude.

Given the function

$y=2\sin(x)$

identify the variables of the general form.

$\f(x)=\sin \a=2 \b=1\rightarrow \frac{2\pi}{|1|}=2\pi \c=0 \d=0$

Step 3: Graph the trigonometric function to verify.

The graph above verifies that the amplitude of this function is two because the range of the function on the graph goes from negative two to positive two meaning the distance from zero at its highest peak or lowest valley is two.

2

What is the horizontal shift of the function?

$f(x)=\sin(x-\pi)$

$\pi$

$-\pi$

$\frac{\pi}{2}$

Explanation

This question is testing one's ability to identify the periodicity of a trigonometric function. This requires the understanding of trigonometric functions and their graphical and algebraic characteristics.

For the purpose of Common Core Standards, "Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline" falls within the Cluster B of "Model Periodic Phenomena with Trigonometric Functions" concept (CCSS.MATH.CONTENT.HSF-TF.B.5).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Write the general form of trigonometric shifts.

where

$\F(x)=\textup{New function after shift} \a=\textup{Amplitude} \b=\textup{period}\rightarrow \frac{2\pi}{|B|} \ \c=\textup{Horizontal shift} \ d=\textup{Vertical shift} \f(x)=\textup{Trigonometric function}$

Step 2: Algebraically identify the horizontal shift.

Given the function

$y=\sin(x-\pi)$

identify the variables of the general form.

$\f(x)=\sin \a=1 \b=1\rightarrow \frac{2\pi}{|1|}=2\pi \c=\pi \d=0$

Step 3: Graph the trigonometric function to verify.

The graph above verifies that the horizontal shift of the function is $\pi$ .

3

What is the vertical shift of the function?

$f(x)=\cos(x)+\pi$

$\pi$

$-\pi$

$\frac{\pi}{2}$

Explanation

This question is testing one's ability to identify the periodicity of a trigonometric function. This requires the understanding of trigonometric functions and their graphical and algebraic characteristics.

For the purpose of Common Core Standards, "Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline" falls within the Cluster B of "Model Periodic Phenomena with Trigonometric Functions" concept (CCSS.MATH.CONTENT.HSF-TF.B.5).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Write the general form of trigonometric shifts.

where

$\F(x)=\textup{New function after shift} \a=\textup{Amplitude} \b=\textup{period}\rightarrow \frac{2\pi}{|B|} \ \c=\textup{Horizontal shift} \ d=\textup{Vertical shift} \f(x)=\textup{Trigonometric function}$

Step 2: Algebraically identify the vertical shift.

Given the function

$y=\cos(x)+\pi$

identify the variables of the general form.

$\f(x)=\cos \a=1 \b=1\rightarrow \frac{2\pi}{|1|}=2\pi \c=0 \d=\pi$

Step 3: Graph the trigonometric function to verify.

Q11

The graph above verifies that the vertical shift of the function is $\pi$ .

4

What is the vertical shift of the function?

$f(x)=\cos(x)-\pi$

$-\pi$

$\pi$

$\frac{\pi}{2}$

Explanation

This question is testing one's ability to identify the periodicity of a trigonometric function. This requires the understanding of trigonometric functions and their graphical and algebraic characteristics.

For the purpose of Common Core Standards, "Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline" falls within the Cluster B of "Model Periodic Phenomena with Trigonometric Functions" concept (CCSS.MATH.CONTENT.HSF-TF.B.5).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Write the general form of trigonometric shifts.

where

$\F(x)=\textup{New function after shift} \a=\textup{Amplitude} \b=\textup{period}\rightarrow \frac{2\pi}{|B|} \ \c=\textup{Horizontal shift} \ d=\textup{Vertical shift} \f(x)=\textup{Trigonometric function}$

Step 2: Algebraically identify the vertical shift.

Given the function

$y=\cos(x)-\pi$

identify the variables of the general form.

$\f(x)=\cos \a=1 \b=1\rightarrow \frac{2\pi}{|1|}=2\pi \c=0 \d=-\pi$

Step 3: Graph the trigonometric function to verify.

Q12

The graph above verifies that the vertical shift of the function is $-\pi$ .

5

What is the amplitude of the following function?

$y=4\sin(x)$

$2\pi$

Explanation

This question is testing one's ability to identify the periodicity of a trigonometric function. This requires the understanding of trigonometric functions and their graphical and algebraic characteristics.

For the purpose of Common Core Standards, "Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline" falls within the Cluster B of "Model Periodic Phenomena with Trigonometric Functions" concept (CCSS.MATH.CONTENT.HSF-TF.B.5).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Write the general form of trigonometric shifts.

where

$\F(x)=\textup{New function after shift} \a=\textup{Amplitude} \b=\textup{period}\rightarrow \frac{2\pi}{|B|} \ \c=\textup{Horizontal shift} \ d=\textup{Vertical shift} \f(x)=\textup{Trigonometric function}$

Step 2: Algebraically identify the amplitude.

Given the function

$y=4\sin(x)$

identify the variables of the general form.

$\f(x)=\sin \a=4 \b=1\rightarrow \frac{2\pi}{|1|}=2\pi \c=0 \d=0$

Step 3: Graph the trigonometric function to verify.

The graph above verifies that the amplitude of this function is four because the range of the function on the graph goes from negative four to positive four meaning the distance from zero at its highest peak or lowest valley is four.

6

What is the period of the following function?

$y=\cos(2x)$

$\pi$

$2\pi$

$\frac{\pi}{2}$

Explanation

This question is testing one's ability to identify the periodicity of a trigonometric function. This requires the understanding of trigonometric functions and their graphical and algebraic characteristics.

For the purpose of Common Core Standards, "Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline" falls within the Cluster B of "Model Periodic Phenomena with Trigonometric Functions" concept (CCSS.MATH.CONTENT.HSF-TF.B.5).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Write the general form of trigonometric shifts.

where

$\F(x)=\textup{New function after shift} \a=\textup{Amplitude} \b=\textup{period}\rightarrow \frac{2\pi}{|B|} \ \c=\textup{Horizontal shift} \ d=\textup{Vertical shift} \f(x)=\textup{Trigonometric function}$

Step 2: Algebraically identify the period.

Given the function

$y=\cos(2x)$

identify the variables of the general form.

$\f(x)=\cos \a=1 \b=2\rightarrow \frac{2\pi}{|2|}=\pi \c=0 \d=0$

Step 3: Graph the trigonometric function to verify.

Q10

The graph above verifies that the period of this function is $\pi$ because the flow of the graph only begins to repeat its cycle after $\pi$ units on the - axis.

7

What is the period of the following function?

$y=\sin(4x)$

$\frac{\pi}{2}$

$\pi$

$\frac{\pi}{4}$

Explanation

This question is testing one's ability to identify the periodicity of a trigonometric function. This requires the understanding of trigonometric functions and their graphical and algebraic characteristics.

For the purpose of Common Core Standards, "Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline" falls within the Cluster B of "Model Periodic Phenomena with Trigonometric Functions" concept (CCSS.MATH.CONTENT.HSF-TF.B.5).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Write the general form of trigonometric shifts.

where

$\F(x)=\textup{New function after shift} \a=\textup{Amplitude} \b=\textup{period}\rightarrow \frac{2\pi}{|B|} \ \c=\textup{Horizontal shift} \ d=\textup{Vertical shift} \f(x)=\textup{Trigonometric function}$

Step 2: Algebraically identify the period.

Given the function

$y=\sin(4x)$

identify the variables of the general form.

$\f(x)=\sin \a=1 \b=4\rightarrow \frac{2\pi}{|4|}=\frac{\pi}{2} \c=0 \d=0$

Step 3: Graph the trigonometric function to verify.

The graph above verifies that the period of this function is $\frac{\pi}{2}$ because the flow of the graph only begins to repeat its cycle after $\frac{\pi}{2}$ units on the - axis.

8

What is the amplitude of the following function?

$y=2\cos(x)$

$\pi$

$2\pi$

Explanation

This question is testing one's ability to identify the periodicity of a trigonometric function. This requires the understanding of trigonometric functions and their graphical and algebraic characteristics.

For the purpose of Common Core Standards, "Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline" falls within the Cluster B of "Model Periodic Phenomena with Trigonometric Functions" concept (CCSS.MATH.CONTENT.HSF-TF.B.5).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Write the general form of trigonometric shifts.

where

$\F(x)=\textup{New function after shift} \a=\textup{Amplitude} \b=\textup{period}\rightarrow \frac{2\pi}{|B|} \ \c=\textup{Horizontal shift} \ d=\textup{Vertical shift} \f(x)=\textup{Trigonometric function}$

Step 2: Algebraically identify the amplitude.

Given the function

$y=2\cos(x)$

identify the variables of the general form.

$\f(x)=\cos \a=2 \b=1\rightarrow \frac{2\pi}{|1|}=2\pi \c=0 \d=0$

Step 3: Graph the trigonometric function to verify.

The graph above verifies that the amplitude of this function is two because the range of the function on the graph goes from negative two to positive two meaning the distance from zero at its highest peak or lowest valley is two.

9

What is the vertical shift of the function?

$f(x)=\sin(x)+\pi$

$\pi$

$\frac{\pi}{2}$

Explanation

This question is testing one's ability to identify the periodicity of a trigonometric function. This requires the understanding of trigonometric functions and their graphical and algebraic characteristics.

For the purpose of Common Core Standards, "Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline" falls within the Cluster B of "Model Periodic Phenomena with Trigonometric Functions" concept (CCSS.MATH.CONTENT.HSF-TF.B.5).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Write the general form of trigonometric shifts.

where

$\F(x)=\textup{New function after shift} \a=\textup{Amplitude} \b=\textup{period}\rightarrow \frac{2\pi}{|B|} \ \c=\textup{Horizontal shift} \ d=\textup{Vertical shift} \f(x)=\textup{Trigonometric function}$

Step 2: Algebraically identify the vertical shift.

Given the function

$y=\sin(x)+\pi$

identify the variables of the general form.

$\f(x)=\sin \a=1 \b=1\rightarrow \frac{2\pi}{|1|}=2\pi \c=0 \d=\pi$

Step 3: Graph the trigonometric function to verify.

The graph above verifies that the vertical shift of the function is $\pi$ .

10

What is the horizontal shift of the function?

$f(x)=\sin(x+\pi)$

$-\pi$

$\pi$

$\frac{\pi}{2}$

$-\frac{\pi}{2}$

Explanation

This question is testing one's ability to identify the periodicity of a trigonometric function. This requires the understanding of trigonometric functions and their graphical and algebraic characteristics.

For the purpose of Common Core Standards, "Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline" falls within the Cluster B of "Model Periodic Phenomena with Trigonometric Functions" concept (CCSS.MATH.CONTENT.HSF-TF.B.5).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Write the general form of trigonometric shifts.

where

$\F(x)=\textup{New function after shift} \a=\textup{Amplitude} \b=\textup{period}\rightarrow \frac{2\pi}{|B|} \ \c=\textup{Horizontal shift} \ d=\textup{Vertical shift} \f(x)=\textup{Trigonometric function}$

Step 2: Algebraically identify the horizontal shift.

Given the function

$y=\sin(x+\pi)$

identify the variables of the general form.

$\f(x)=\sin \a=1 \b=1\rightarrow \frac{2\pi}{|1|}=2\pi \c=-\pi \d=0$

Step 3: Graph the trigonometric function to verify.

The graph above verifies that the horizontal shift of the function is $-\pi$ .

Model Periodicity, Amplitude, Frequency, and Midline of Trigonometric Functions: CCSS.Math.Content.HSF-TF.B.5

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