ALEX Classroom Resources

   View Standards     Standard(s): [MA2019] PRE-19 (9-12) 24 :

24. Compare and contrast families of functions and their representations algebraically, graphically, numerically, and verbally in terms of their key features.

Note: Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries (including even and odd); end behavior; asymptotes; and periodicity. Families of functions include but are not limited to linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, trigonometric, and their inverses.

[MA2019] PRE-19 (9-12) 25 :

25. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Extend from polynomial, exponential, logarithmic, and radical to rational and all trigonometric functions.

a. Find the difference quotient ^{f(x+Î"x)-f(x)}/_Î"x of a function and use it to evaluate the average rate of change at a point.

b. Explore how the average rate of change of a function over an interval (presented symbolically or as a table) can be used to approximate the instantaneous rate of change at a point as the interval decreases.

This video from MIT Open CourseWare on the CK-12 website will explore the meaning of exponential growth or decay by solving a differential equation that models such growth or decay.

   View Standards     Standard(s): [MA2019] PRE-19 (9-12) 20 :

20. Explain each step in solving an equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a clear-cut solution. Construct a viable argument to justify a solution method. Include equations that may involve linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, and trigonometric functions, and their inverses.

[MA2019] PRE-19 (9-12) 24 :

24. Compare and contrast families of functions and their representations algebraically, graphically, numerically, and verbally in terms of their key features.

Note: Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries (including even and odd); end behavior; asymptotes; and periodicity. Families of functions include but are not limited to linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, trigonometric, and their inverses.

This video will explain how to identify components of quadratic functions and solve the function for each variable. 

   View Standards     Standard(s): [MA2019] PRE-19 (9-12) 20 :

20. Explain each step in solving an equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a clear-cut solution. Construct a viable argument to justify a solution method. Include equations that may involve linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, and trigonometric functions, and their inverses.

[MA2019] PRE-19 (9-12) 24 :

24. Compare and contrast families of functions and their representations algebraically, graphically, numerically, and verbally in terms of their key features.

Note: Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries (including even and odd); end behavior; asymptotes; and periodicity. Families of functions include but are not limited to linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, trigonometric, and their inverses.

Travis found the following equation in his math book

d=rt−16t²

It is an equation to calculate velocity. In fact, it is a function. Being an avid sports player, Travis was very interested in figuring out how to use the equation, but he isn’t even sure what kind of a function it is. Can you identify this function? In this concept, you will learn to recognize a quadratic function as an equation in two variables with a specific form.

This informational material will help students identify key features of and solutions for quadratic functions.  Practice questions with a PDF answer key are provided.  

   View Standards     Standard(s): [MA2019] PRE-19 (9-12) 24 :

24. Compare and contrast families of functions and their representations algebraically, graphically, numerically, and verbally in terms of their key features.

Note: Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries (including even and odd); end behavior; asymptotes; and periodicity. Families of functions include but are not limited to linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, trigonometric, and their inverses.

[MA2019] PRE-19 (9-12) 25 :

25. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Extend from polynomial, exponential, logarithmic, and radical to rational and all trigonometric functions.

a. Find the difference quotient ^{f(x+Î"x)-f(x)}/_Î"x of a function and use it to evaluate the average rate of change at a point.

b. Explore how the average rate of change of a function over an interval (presented symbolically or as a table) can be used to approximate the instantaneous rate of change at a point as the interval decreases.

This interactive will model exponential growth using the following scenario:

An ancient legend tells of a wise man who advises a king during a time of famine. As a reward for his help, the man asks the miserly king for grains of rice every day.  He asks the king to put a single grain on the first square of a chessboard on the first day, two grains on the second square on the second day, double that amount on the third square on the third day, and so on. The amount of rice grows exponentially.

In this interactive, students will:

• • move the grains of rice to the squares on the chessboard to show how quickly a single grain of rice can become many grains.

After engaging with this interactive, students should be able to explain the key features of exponential functions. 

   View Standards     Standard(s): [MA2019] PRE-19 (9-12) 7 :

7. Determine numerically, algebraically, and graphically the limits of functions at specific values and at infinity.

a. Apply limits of functions at specific values and at infinity in problems involving convergence and divergence.

[MA2019] PRE-19 (9-12) 24 :

24. Compare and contrast families of functions and their representations algebraically, graphically, numerically, and verbally in terms of their key features.

Note: Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries (including even and odd); end behavior; asymptotes; and periodicity. Families of functions include but are not limited to linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, trigonometric, and their inverses.

This informational material will introduce students to the concept of limits, including end behavior, asymptotes, and limit notation. When learning about the end behavior of a rational function, students described the function as either having a horizontal asymptote at zero or another number or going to infinity. Limit notation is a way of describing this end behavior mathematically. There is a corresponding video available. Practice questions with a PDF answer key are provided. 

   View Standards     Standard(s): [MA2019] PRE-19 (9-12) 7 :

7. Determine numerically, algebraically, and graphically the limits of functions at specific values and at infinity.

a. Apply limits of functions at specific values and at infinity in problems involving convergence and divergence.

[MA2019] PRE-19 (9-12) 24 :

24. Compare and contrast families of functions and their representations algebraically, graphically, numerically, and verbally in terms of their key features.

Note: Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries (including even and odd); end behavior; asymptotes; and periodicity. Families of functions include but are not limited to linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, trigonometric, and their inverses.

In a previous read, students have learned what a limit is and how it is used to describe certain properties about functions. This interactive will challenge students to evaluate the limits of the function f(x) graphically.

In this interactive, the function f(x) will be represented by the green function.

Students will: 

• move the red dot in order to simulate x-values getting closer to the value under the blue dot.
• move the blue dot in order to change the x-value that they are approaching.

   View Standards     Standard(s): [MA2019] PRE-19 (9-12) 7 :

7. Determine numerically, algebraically, and graphically the limits of functions at specific values and at infinity.

a. Apply limits of functions at specific values and at infinity in problems involving convergence and divergence.

[MA2019] PRE-19 (9-12) 24 :

24. Compare and contrast families of functions and their representations algebraically, graphically, numerically, and verbally in terms of their key features.

Note: Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries (including even and odd); end behavior; asymptotes; and periodicity. Families of functions include but are not limited to linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, trigonometric, and their inverses.

This self-checking online assessment has 10 questions that will help students practice evaluating limits using graphs of functions. There are hints available on the screen, and there is an online scratchpad that students can use to work the problems.

   View Standards     Standard(s): [MA2019] PRE-19 (9-12) 24 :

24. Compare and contrast families of functions and their representations algebraically, graphically, numerically, and verbally in terms of their key features.

Note: Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries (including even and odd); end behavior; asymptotes; and periodicity. Families of functions include but are not limited to linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, trigonometric, and their inverses.

[MA2019] PRE-19 (9-12) 26 :

26. Graph functions expressed symbolically and show key features of the graph, by hand and using technology. Use the equation of functions to identify key features in order to generate a graph.

a. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

b. Graph trigonometric functions and their inverses, showing period, midline, amplitude, and phase shift.

   View Standards     Standard(s): [MA2019] PRE-19 (9-12) 24 :

24. Compare and contrast families of functions and their representations algebraically, graphically, numerically, and verbally in terms of their key features.

Note: Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries (including even and odd); end behavior; asymptotes; and periodicity. Families of functions include but are not limited to linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, trigonometric, and their inverses.

[MA2019] PRE-19 (9-12) 26 :

26. Graph functions expressed symbolically and show key features of the graph, by hand and using technology. Use the equation of functions to identify key features in order to generate a graph.

a. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

b. Graph trigonometric functions and their inverses, showing period, midline, amplitude, and phase shift.

   View Standards     Standard(s): [MA2019] PRE-19 (9-12) 7 :

7. Determine numerically, algebraically, and graphically the limits of functions at specific values and at infinity.

a. Apply limits of functions at specific values and at infinity in problems involving convergence and divergence.

[MA2019] PRE-19 (9-12) 24 :

24. Compare and contrast families of functions and their representations algebraically, graphically, numerically, and verbally in terms of their key features.

Note: Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries (including even and odd); end behavior; asymptotes; and periodicity. Families of functions include but are not limited to linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, trigonometric, and their inverses.

Finding limits for the vast majority of points for a given function is as simple as substituting the number that x approaches into the function. Since this turns evaluating limits into an algebra-level substitution, most questions involving limits focus on the cases where substituting does not work. How can you decide if substitution is an appropriate analytical tool for finding a limit?

This informational material will explain how to find a limit of a function using algebraic substitution and when this method is appropriate. There is a corresponding video available. Practice questions with a PDF answer key are provided.

   View Standards     Standard(s): [MA2019] PRE-19 (9-12) 7 :

7. Determine numerically, algebraically, and graphically the limits of functions at specific values and at infinity.

a. Apply limits of functions at specific values and at infinity in problems involving convergence and divergence.

[MA2019] PRE-19 (9-12) 24 :

24. Compare and contrast families of functions and their representations algebraically, graphically, numerically, and verbally in terms of their key features.

Note: Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries (including even and odd); end behavior; asymptotes; and periodicity. Families of functions include but are not limited to linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, trigonometric, and their inverses.

This online interactive will challenge students' knowledge of limits of functions using the following scenario:

You and your friend want to understand what the definition of a limit means. Here is an informal definition of a limit.

Notation and Informal Definition of a Limit of a Function

means that as x approaches (or gets very close to) a,  the function f(x) gets very close to the value L.  

• In the interactive, students will move the red dots in order to manipulate the line and simulate the definition of a limit.

   View Standards     Standard(s): [MA2019] PRE-19 (9-12) 7 :

7. Determine numerically, algebraically, and graphically the limits of functions at specific values and at infinity.

a. Apply limits of functions at specific values and at infinity in problems involving convergence and divergence.

[MA2019] PRE-19 (9-12) 24 :

24. Compare and contrast families of functions and their representations algebraically, graphically, numerically, and verbally in terms of their key features.

Note: Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries (including even and odd); end behavior; asymptotes; and periodicity. Families of functions include but are not limited to linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, trigonometric, and their inverses.

This video from Khan Academy on the CK-12 website will introduce students to the concept of limits by using two different functions as examples to demonstrate how to find the limits of a function.