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Flashcards: Common Core: High School - Functions

Graph the function and its asymptotes.

$f(x)=\frac{1}{2x-1}$ $\displaystyle f(x)=\frac{1}{2x-1}$

Question2

$\text{Asymptotes: } x=0, y=0$ $\displaystyle \text{Asymptotes: } x=0, y=0$

Question11

$\text{Asymptotes: } x=0, y=0$ $\displaystyle \text{Asymptotes: } x=0, y=0$

Question5

$\text{Asymptotes: } x=0, y=0$ $\displaystyle \text{Asymptotes: } x=0, y=0$

Question9

$\text{Asymptotes: } x=0, y=0$ $\displaystyle \text{Asymptotes: } x=0, y=0$

Question12

$\text{Asymptotes: } x=\frac{1}{2}, y=0$ $\displaystyle \text{Asymptotes: } x=\frac{1}{2}, y=0$

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Flashcards: Common Core: High School - Functions

Graph the function and its asymptotes.

$f(x)=\frac{1}{2x-1}$ $\displaystyle f(x)=\frac{1}{2x-1}$

Question2

$\text{Asymptotes: } x=0, y=0$ $\displaystyle \text{Asymptotes: } x=0, y=0$

Question11

$\text{Asymptotes: } x=0, y=0$ $\displaystyle \text{Asymptotes: } x=0, y=0$

Question5

$\text{Asymptotes: } x=0, y=0$ $\displaystyle \text{Asymptotes: } x=0, y=0$

Question9

$\text{Asymptotes: } x=0, y=0$ $\displaystyle \text{Asymptotes: } x=0, y=0$

Question12

$\text{Asymptotes: } x=\frac{1}{2}, y=0$ $\displaystyle \text{Asymptotes: } x=\frac{1}{2}, y=0$

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Flashcards: Common Core: High School - Functions

Graph the function and its asymptotes.

$f(x)=\frac{1}{2x-1}$ $\displaystyle f(x)=\frac{1}{2x-1}$

Your answer:

Question2

$\text{Asymptotes: } x=0, y=0$ $\displaystyle \text{Asymptotes: } x=0, y=0$

Your answer:

Question11

$\text{Asymptotes: } x=0, y=0$ $\displaystyle \text{Asymptotes: } x=0, y=0$

Your answer:

Question5

$\text{Asymptotes: } x=0, y=0$ $\displaystyle \text{Asymptotes: } x=0, y=0$

Your answer:

Question9

$\text{Asymptotes: } x=0, y=0$ $\displaystyle \text{Asymptotes: } x=0, y=0$

Correct answer:

Question12

$\text{Asymptotes: } x=\frac{1}{2}, y=0$ $\displaystyle \text{Asymptotes: } x=\frac{1}{2}, y=0$

Explanation:

This question is testing one's ability to graph a rational function and identify its asymptotes.

For the purpose of Common Core Standards, "graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior." falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7).

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: Use algebraic techniques to identify asymptotes of the function.

Recall that asymptotes occur at locations where the domain and range of the function do not exist. For vertical asymptotes this occurs if there is a fraction where the denominator contains the variable:

$y=\frac{1}{a-x} \rightarrow a-x\neq 0\rightarrow a\neq x$ $\displaystyle y=\frac{1}{a-x} \rightarrow a-x\neq 0\rightarrow a\neq x$

Given the function

$f(x)=\frac{1}{2x-1}$ $\displaystyle f(x)=\frac{1}{2x-1}$

$\\2x-1\neq0 \\x\neq\frac{1}{2}$ $\displaystyle \\2x-1\neq0 \\x\neq\frac{1}{2}$

thus a vertical asymptote is at $x=\frac{1}{2}$ $\displaystyle x=\frac{1}{2}$.

Step 2: Identify the horizontal asymptote by examining the end behavior of the function.

Given the function

$f(x)=\frac{1}{2x-1}$ $\displaystyle f(x)=\frac{1}{2x-1}$

to find the end behavior, substitute in large values for x. When large values of x are put into the function the denominator becomes larger.

Recall that when n is some large value, the fraction approaches zero.

$\frac{1}{n}\rightarrow 0$ $\displaystyle \frac{1}{n}\rightarrow 0$

Therefore, the horizontal asymptote is at y=0 $\displaystyle y=0$.

Step 3: Graph the function using technology and plot the asymptotes.

Question12

Flashcards: Common Core: High School - Functions

What is the period of the following function?

$y=\sin(4x)$ $\displaystyle y=\sin(4x)$

$\displaystyle 2$

$\frac{\pi}{4}$ $\displaystyle \frac{\pi}{4}$

$\pi$ $\displaystyle \pi$

$\displaystyle 1$

$\frac{\pi}{2}$ $\displaystyle \frac{\pi}{2}$

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Flashcards: Common Core: High School - Functions

What is the period of the following function?

$y=\sin(4x)$ $\displaystyle y=\sin(4x)$

Your answer:

$\displaystyle 2$

Your answer:

$\frac{\pi}{4}$ $\displaystyle \frac{\pi}{4}$

Your answer:

$\pi$ $\displaystyle \pi$

Your answer:

$\displaystyle 1$

Correct answer:

$\frac{\pi}{2}$ $\displaystyle \frac{\pi}{2}$

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.

$\displaystyle F(x)=af(bx-c)+d$

where

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

Step 2: Algebraically identify the period.

Given the function

$y=\sin(4x)$ $\displaystyle y=\sin(4x)$

identify the variables of the general form.

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

Step 3: Graph the trigonometric function to verify.

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

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Flashcards: Common Core: High School - Functions

All Common Core: High School - Functions Resources

Flashcards: Common Core: High School - Functions

Flashcards: Common Core: High School - Functions

Flashcards: Common Core: High School - Functions

Flashcards: Common Core: High School - Functions

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