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Common Core: High School - Functions : High School: Functions

Study concepts, example questions & explanations for Common Core: High School - Functions

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All Common Core: High School - Functions Resources

6 Diagnostic Tests 82 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Model Periodicity, Amplitude, Frequency, And Midline Of Trigonometric Functions: Ccss.Math.Content.Hsf Tf.B.5

What is the period of the following function?

$y=\sin(4x)$

Possible Answers:

$\frac{\pi}{4}$

$\pi$

$\frac{\pi}{2}$

Correct answer:

$\frac{\pi}{2}$

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.

F(x)=af(bx-c)+d

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.

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Example Question #4 : Model Periodicity, Amplitude, Frequency, And Midline Of Trigonometric Functions: Ccss.Math.Content.Hsf Tf.B.5

What is the vertical shift of the function?

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

Possible Answers:

$\pi$

$\frac{\pi}{2}$

Correct answer:

$\pi$

Explanation:

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.

F(x)=af(bx-c)+d

where

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 $\pi$ .

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Example Question #2 : Model Periodicity, Amplitude, Frequency, And Midline Of Trigonometric Functions: Ccss.Math.Content.Hsf Tf.B.5

What is the vertical shift of the function?

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

Possible Answers:

$\pi$

$2\pi$

$\frac{\pi}{2}$

$-\pi$

Correct answer:

$-\pi$

Explanation:

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.

F(x)=af(bx-c)+d

where

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 $-\pi$ .

Report an Error

Example Question #6 : Model Periodicity, Amplitude, Frequency, And Midline Of Trigonometric Functions: Ccss.Math.Content.Hsf Tf.B.5

What is the horizontal shift of the function?

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

Possible Answers:

$\pi$

$\frac{\pi}{2}$

$-\pi$

Correct answer:

$\pi$

Explanation:

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.

F(x)=af(bx-c)+d

where

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 $\pi$ .

Report an Error

Example Question #8 : Model Periodicity, Amplitude, Frequency, And Midline Of Trigonometric Functions: Ccss.Math.Content.Hsf Tf.B.5

What is the horizontal shift of the function?

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

Possible Answers:

$-\pi$

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

$\pi$

$\frac{\pi}{2}$

Correct answer:

$-\pi$

Explanation:

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.

F(x)=af(bx-c)+d

where

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 $-\pi$ .

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Example Question #9 : Model Periodicity, Amplitude, Frequency, And Midline Of Trigonometric Functions: Ccss.Math.Content.Hsf Tf.B.5

What is the amplitude of the following function?

$y=2\cos(x)$

Possible Answers:

$2\pi$

$\pi$

Correct answer:

Explanation:

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.

F(x)=af(bx-c)+d

where

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.

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Example Question #2 : Model Periodicity, Amplitude, Frequency, And Midline Of Trigonometric Functions: Ccss.Math.Content.Hsf Tf.B.5

What is the period of the following function?

$y=\cos(2x)$

Possible Answers:

$2\pi$

$\pi$

$\frac{\pi}{2}$

Correct answer:

$\pi$

Explanation:

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.

F(x)=af(bx-c)+d

where

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.

Report an Error

Example Question #411 : High School: Functions

What is the vertical shift of the function?

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

Possible Answers:

$\pi$

$\frac{\pi}{2}$

$-\pi$

Correct answer:

$\pi$

Explanation:

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.

F(x)=af(bx-c)+d

where

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 $\pi$ .

Report an Error

Example Question #51 : Trigonometric Functions

What is the vertical shift of the function?

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

Possible Answers:

$\frac{\pi}{2}$

$-\pi$

$\pi$

Correct answer:

$-\pi$

Explanation:

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.

F(x)=af(bx-c)+d

where

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 $-\pi$ .

Report an Error

Example Question #1 : Restricting Domain Of Trigonometric Functions To Allow For Construction Of Inverse: Ccss.Math.Content.Hsf Tf.B.6

Find the exact value of the following statement.

$f(x)=\cos^{-1}(1)$

Possible Answers:

$2\pi$

$\pi$

$-\pi$

$\frac{\pi}{2}$

$\frac{\pi}{3}$

Correct answer:

$2\pi$

Explanation:

This question is testing ones ability to understand and identify inverses of trigonometric functions as they relate to the unit circle.

For the purpose of Common Core Standards, " Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed." concept (CCSS.MATH.CONTENT.HSF-TF.B.6). It is important to note that this standard is not directly tested on but is used for building a deeper understanding on trigonometric functions.

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: Identify what the question is asking for.

Since there is a trigonometric function raised to the negative one power, this question is talking about the inverse of the function. In other words, which angle on the unit circle results in a cosine equalling one?

$y=\cos^{-1}(a)\Rightarrow \cos(\theta)=a$

Therefore, theta needs to be solved for.

Step 2: Draw and label the unit circle.

Screen shot 2016 01 14 at 10.52.42 am

Step 3: Locate the angle that results in one for its cosine value.

Recall that

$(x,y)=(\cos(\theta),\sin(\theta))$

therefore look for the (x,y) that has x=1 . Looking at the unit circle from step 2, it is seen that at angle $2\pi$ the cosine equals one.

Thus,

$f(x)=\cos^{-1}(1)\Rightarrow f(x)=2\pi$

To verify the solution simply find the cosine of the angle theta.

$\cos(2\pi)=1\rightarrow\cos^{-1}(1)=2\pi$

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All Common Core: High School - Functions Resources

6 Diagnostic Tests 82 Practice Tests Question of the Day Flashcards Learn by Concept

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Common Core: High School - Functions : High School: Functions

Study concepts, example questions & explanations for Common Core: High School - Functions

All Common Core: High School - Functions Resources

Example Questions

Example Question #1 : Model Periodicity, Amplitude, Frequency, And Midline Of Trigonometric Functions: Ccss.Math.Content.Hsf Tf.B.5

Example Question #4 : Model Periodicity, Amplitude, Frequency, And Midline Of Trigonometric Functions: Ccss.Math.Content.Hsf Tf.B.5

Example Question #2 : Model Periodicity, Amplitude, Frequency, And Midline Of Trigonometric Functions: Ccss.Math.Content.Hsf Tf.B.5

Example Question #6 : Model Periodicity, Amplitude, Frequency, And Midline Of Trigonometric Functions: Ccss.Math.Content.Hsf Tf.B.5

Example Question #8 : Model Periodicity, Amplitude, Frequency, And Midline Of Trigonometric Functions: Ccss.Math.Content.Hsf Tf.B.5

Example Question #9 : Model Periodicity, Amplitude, Frequency, And Midline Of Trigonometric Functions: Ccss.Math.Content.Hsf Tf.B.5

Example Question #2 : Model Periodicity, Amplitude, Frequency, And Midline Of Trigonometric Functions: Ccss.Math.Content.Hsf Tf.B.5

Example Question #411 : High School: Functions

Example Question #51 : Trigonometric Functions

Example Question #1 : Restricting Domain Of Trigonometric Functions To Allow For Construction Of Inverse: Ccss.Math.Content.Hsf Tf.B.6

All Common Core: High School - Functions Resources

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