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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 #5 : Construct Linear And Exponential Functions, Arithmetic And Geometric Sequences: Ccss.Math.Content.Hsf Le.A.2

What is the function that describes the graph?

Possible Answers:

y=3x+3

y=x-1

y=x-3

y=3x-3

y=x+3

Correct answer:

y=3x-3

Explanation:

This question is testing one's ability to identify and construct an algebraic function given a graph.

For the purpose of Common Core Standards, "Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table)." falls within the Cluster A of "Construct and compare linear, quadratic, and exponential models and solve problems" concept (CCSS.MATH.CONTENT.HSF-LE.A.2).

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 -intercept.

Recall that the -intercept is the point at which the line crosses the -axis. In other words, it is at the point where x=0 .

$\textup{y-intercept}=-3$

Step 2: Identify the slope.

Recall that slope is known as rise over run or the change in the y values over the change in x values.

$\frac{\textup{rise}}{\textup{run}}=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$

The rise is three units up and the run is one unit to the right.

$\textup{slope}=\frac{\textup{rise}}{\textup{run}}=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}=\frac{0--3}{1-0}=\frac{3}{1}=3$

Step 3: Construct the equation using the slope-intercept form of a linear function.

The slope-intercept form of a linear function is,

$\\y=mx+b \\m=\textup{Slope} \\b=\textup{y-intercept}$ .

Substitute the slope and intercept into the general form to construct this particular function.

y=3x-3

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Example Question #6 : Construct Linear And Exponential Functions, Arithmetic And Geometric Sequences: Ccss.Math.Content.Hsf Le.A.2

What is the function that describes the graph?

Q10

Possible Answers:

$y=\frac{1}{2}x$

$y=\frac{1}{2}x-1$

y=2x+1

$y=\frac{1}{2}x+1$

Correct answer:

$y=\frac{1}{2}x+1$

Explanation:

This question is testing one's ability to identify and construct an algebraic function given a graph.

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 -intercept.

Recall that the -intercept is the point at which the line crosses the -axis. In other words, it is at the point where x=0 .

Q10

$\textup{y-intercept}=1$

Step 2: Identify the slope.

Recall that slope is known as rise over run or the change in the y values over the change in x values.

$\frac{\textup{rise}}{\textup{run}}=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$

The rise is one unit up and the run is two units to the right.

$\textup{slope}=\frac{\textup{rise}}{\textup{run}}=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}=\frac{2-1}{2-0}=\frac{1}{2}$

Step 3: Construct the equation using the slope-intercept form of a linear function.

The slope-intercept form of a linear function is,

$\\y=mx+b \\m=\textup{Slope} \\b=\textup{y-intercept}$ .

Substitute the slope and intercept into the general form to construct this particular function.

$y=\frac{1}{2}x+1$

Report an Error

Example Question #321 : High School: Functions

What is the function that describes the graph?

Q11

Possible Answers:

$y=\frac{1}{2}x+1$

$y=\frac{3}{4}x-1$

$y=\frac{4}{3}x$

$y=\frac{1}{2}x-2$

$y=\frac{3}{4}x$

Correct answer:

$y=\frac{3}{4}x$

Explanation:

This question is testing one's ability to identify and construct an algebraic function given a graph.

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 -intercept.

Recall that the -intercept is the point at which the line crosses the -axis. In other words, it is at the point where x=0 .

Q11

$\textup{y-intercept}=0$

Step 2: Identify the slope.

Recall that slope is known as rise over run or the change in the y values over the change in x values.

$\frac{\textup{rise}}{\textup{run}}=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$

The rise is three units up and the run is four units to the right.

$\textup{slope}=\frac{\textup{rise}}{\textup{run}}=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}=\frac{3-1}{4-0}=\frac{3}{4}$

Step 3: Construct the equation using the slope-intercept form of a linear function.

The slope-intercept form of a linear function is,

$\\y=mx+b \\m=\textup{Slope} \\b=\textup{y-intercept}$ .

Substitute the slope and intercept into the general form to construct this particular function.

$y=\frac{3}{4}x$

Report an Error

Example Question #322 : High School: Functions

What is the function that describes the graph?

Q12

Possible Answers:

y=-x-2

y=-x+2

y=x-2

y=x+2

y=-x

Correct answer:

y=-x+2

Explanation:

This question is testing one's ability to identify and construct an algebraic function given a graph.

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 -intercept.

Recall that the -intercept is the point at which the line crosses the -axis. In other words, it is at the point where x=0 .

Q12

$\textup{y-intercept}=2$

Step 2: Identify the slope.

Recall that slope is known as rise over run or the change in the y values over the change in x values.

$\frac{\textup{rise}}{\textup{run}}=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$

The rise is one unit down and the run is one unit to the right.

$\textup{slope}=\frac{\textup{rise}}{\textup{run}}=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}=\frac{1-2}{1-0}=\frac{-1}{1}=-1$

Step 3: Construct the equation using the slope-intercept form of a linear function.

The slope-intercept form of a linear function is,

$\\y=mx+b \\m=\textup{Slope} \\b=\textup{y-intercept}$ .

Substitute the slope and intercept into the general form to construct this particular function.

y=-x+2

Report an Error

Example Question #1 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

Which value for proves that the function f(x)=e^x will increases faster than the function g(x)=3x+1 ?

Possible Answers:

$x\geq3$

$x\leq2$

$x\geq2$

$x\geq1$

$x\leq1$

Correct answer:

$x\geq2$

Explanation:

This question is testing one's ability to identify the end behavior of two functions as they relate to one another by constructing graphs.

For the purpose of Common Core Standards, "Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function" falls within the Cluster A of "Construct and compare linear, quadratic, and exponential models and solve problems" concept (CCSS.MATH.CONTENT.HSF-LE.A.3).

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 technology to graph f(x) .

Screen shot 2016 01 14 at 7.53.13 am

Step 2: Use technology to graph g(x) .

Screen shot 2016 01 14 at 7.54.37 am

Step 3: Compare the graphs of f(x) and g(x) .

Screen shot 2016 01 14 at 7.54.10 am

Graphically, it appears that x=2 is the point where g(x) increases more rapidly than f(x) . Substitute two into both functions to algebraic verify the assumption.

f(2)=e^2=(2.718)^2=7.389

g(2)=3(2)+1=7

Since

$7.389>7\Rightarrow f(2)>g(2)\Rightarrow x>2$

Step 4: Answer the question.

For values $x\geq2$ , $f(x)\geq g(x)$ .

Report an Error

Example Question #321 : High School: Functions

Which value for proves that the function f(x)=e^x will increases faster than the function g(x)=x ?

Possible Answers:

$\text{ For all values of }x\text{, } f(x)=e^x \text{ is larger than } y=x.$

$x\geq0$

$x\geq8$

$x\geq1$

$x\geq2$

Correct answer:

$\text{ For all values of }x\text{, } f(x)=e^x \text{ is larger than } y=x.$

Explanation:

This question is testing one's ability to identify the end behavior of two functions as they relate to one another by constructing graphs.

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 technology to graph f(x) .

Step 2: Use technology to graph g(x) .

Q2 2

Step 3: Compare the graphs of f(x) and g(x) .

Q2 3

Graphically, it appears that f(x) is larger than g(x) for all values of .

Report an Error

Example Question #3 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

Which value for proves that the function f(x)=e^x will increases faster than the function g(x)=x+1 ?

Possible Answers:

$x\geq1$

$x\geq2$

$\text{ For all values of }x\text{, } f(x)=e^x \text{ is larger than } y=x+1.$

$x\geq0$

$x\geq-1$

Correct answer:

$\text{ For all values of }x\text{, } f(x)=e^x \text{ is larger than } y=x+1.$

Explanation:

This question is testing one's ability to identify the end behavior of two functions as they relate to one another by constructing graphs.

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 technology to graph f(x) .

Step 2: Use technology to graph g(x) .

Q2 2

Step 3: Compare the graphs of f(x) and g(x) .

Q4 3

Graphically, it appears that f(x) is larger than g(x) for all values of .

Report an Error

Example Question #1 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

Which value for proves that the function f(x)=e^x will increases faster than the function g(x)=3x ?

Possible Answers:

$x\geq1$

$x\geq2.5$

$x\geq0$

$x\geq 2$

$x\geq1.51$

Correct answer:

$x\geq1.51$

Explanation:

This question is testing one's ability to identify the end behavior of two functions as they relate to one another by constructing graphs.

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 technology to graph f(x) .

Step 2: Use technology to graph g(x) .

Q2 2

Step 3: Compare the graphs of f(x) and g(x) .

Q6 3

Graphically, it appears that x=1.51 is the point where g(x) increases more rapidly than f(x) . Substitute this value into both functions to algebraic verify the assumption.

$f(1.51)=e^{1.51}=(2.718)^{1.51}=4.527$

g(1.5)=3(1.5)=4.5

Since

$4.527>4.5\Rightarrow f(1.51)>g(1.51)\Rightarrow x\geq1.51$

Step 4: Answer the question.

For values $x\geq1.51$ , $f(x)\geq g(x)$ .

Report an Error

Example Question #5 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

Which value for proves that the function f(x)=e^x will increases faster than the function g(x)=2x+1 ?

Possible Answers:

$x\geq1.36$

$x\geq 2$

$x\geq1.6$

$x\geq1$

$x\geq0$

Correct answer:

$x\geq1.36$

Explanation:

This question is testing one's ability to identify the end behavior of two functions as they relate to one another by constructing graphs.

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 technology to graph f(x) .

Step 2: Use technology to graph g(x) .

Q2 2

Step 3: Compare the graphs of f(x) and g(x) .

Q3 3

Graphically, it appears that x=1.36 is the point where g(x) increases more rapidly than f(x) . Substitute this value into both functions to algebraic verify the assumption.

$f(1.36)=e^{1.36}=(2.718)^{1.36}=3.896$

g(1.36)=2(1.36)+1=3.72

Since

$3.896>3.72\Rightarrow f(1.36)>g(1.36)\Rightarrow x\geq1.36$

Step 4: Answer the question.

For values $x\geq1.36$ , $f(x)\geq g(x)$ .

Report an Error

Example Question #1 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

Which value for proves that the function f(x)=e^x will increases faster than the function g(x)=4x-1 ?

Possible Answers:

$x\geq1.76$

$x\geq1$

$x\geq2$

$x\geq0$

$x\geq1.89$

Correct answer:

$x\geq1.89$

Explanation:

This question is testing one's ability to identify the end behavior of two functions as they relate to one another by constructing graphs.

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 technology to graph f(x) .

Step 2: Use technology to graph g(x) .

Q2 2

Step 3: Compare the graphs of f(x) and g(x) .

Q5 3

Graphically, it appears that x=1.89 is the point where g(x) increases more rapidly than f(x) . Substitute this value into both functions to algebraic verify the assumption.

$f(1.89)=e^{1.89}=(2.718)^{1.89}=6.619$

g(1.89)=4(1.89)-1=6.56

Since

$6.619>6.56\Rightarrow f(1.89)>g(1.89)\Rightarrow x\geq1.89$

Step 4: Answer the question.

For values $x\geq1.89$ , $f(x)\geq g(x)$ .

Report an Error

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All Common Core: High School - Functions Resources

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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 #5 : Construct Linear And Exponential Functions, Arithmetic And Geometric Sequences: Ccss.Math.Content.Hsf Le.A.2

Example Question #6 : Construct Linear And Exponential Functions, Arithmetic And Geometric Sequences: Ccss.Math.Content.Hsf Le.A.2

Example Question #321 : High School: Functions

Example Question #322 : High School: Functions

Example Question #1 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

Example Question #321 : High School: Functions

Example Question #3 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

Example Question #1 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

Example Question #5 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

Example Question #1 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

All Common Core: High School - Functions Resources

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