FUNCTIONS Introduction to Functions Overview of Objectives, students should be able to: 1. Find the domain and range of a relation 2. Determine whether a relation is a function 3. Evaluate a function 4. Recognize functions in tables 5. Use functions as models and make future predictions. Objectives: Main Overarching Questions: How do you determine the domain and range of a relation? Compare/contrast relations and functions. How can you determine whether a relation is a function from a table of values? Activities and Questions to ask students: Find the domain and range of a relation 1. Ask students if they remember what the domain and range of a function represent. (x and y- values, input/output, independent/dependent variables). 2. Provide a list of ordered pairs, and have students identify the domain and range. Determine whether a relation is a function Recognize functions in tables 1. Ask students what the word function means in every day language. Solicit several definitions/descriptions. Building upon student responses, ask, Would it be correct to say that my paycheck is a function of the amount of time that I work, or that the amount of time that I work is a function of my paycheck? (Paycheck is a function of time) Relate this to x and y values (time x, paycheck y). If I get paid $8 per hour, what equation would represent my pay? 2. Show students several examples of x/y charts, some representing functions and some not representing functions. Ask students to work in pairs to determine which charts fit the definition of a function. Discuss the results as a class. Evaluate a function 1. Using the equation from the paycheck example above, have students evaluate the pay for several different inputs. Ask for general observations about the x- and y-values (ex. The pay depends on the number of hours worked). 2. In a function, the y value acts as a function of the x value. The y-value changes as a result of changes to the x-value, just like my paycheck increases or decreases depending upon how many hours I work. Just as each time produces a unique pay, each x-value produces a unique y-value in a function. In other words, for every x-value, there is exactly one y-value. Use functions as models and make future 1. Provide a quadratic equation that models the height of a trajectory over time. Ask students to evaluate the function for several different x-values (sufficient to see repeating y-values). Ask if
predictions. this equation would represent a function. 2. Have the students use the quadratic model to predict the height of the trajectory at specific points in time (1 sec, 2 sec, 3 sec, etc.). Discuss the significance of using equations to model functions and how they can be used to make predictions without continuing the pattern forever. 3. Provide a problem set for independent practice. Graphs of Functions Overview of Objectives, students should be able to: 1. Graph functions by plotting points or through a table 2. Use the vertical line test to identify functions 3. Obtain information about a function from its graph a. Find f(c), where c is a constant b. Find x in f(x) = c, where c is a constant. 4. Identify domain and range from a function s graph. 5. Recognize the connection between finding zeros, roots, x-intercepts, and solutions to a function. Objectives: Main Overarching Questions: How can you determine whether a relation represents a function from a graph? How can you identify the domain and range of a function from the graph? What does it mean to find the zeros of a function? Activities and Questions to ask students: Graph functions by plotting points or through a table 1. Using the paycheck example from the previous section, have students recreate a table of values and use those points to plot a line graph. (You may want to discuss the fact that this graph would be a ray, rather than a line, since time can t be negative.) 2. Ask students what the points on the graph represent. (Many students have a hard time making the connection between the solutions to an equation and points on the graph. Make sure to emphasize that x values are inputs to the function and y values are outputs). Have students input the function and graph using the graphing calculator. Show students how to generate a table of values using the calculator. Use the vertical line test to identify functions 1. Using the tables of values from the previous section, have students plot the points. Ask them to find similarities between the graphs of functions and the graphs of non-functions. 2. Show students how the vertical line test can be used to quickly determine whether or not a graph is a function. Provide several examples and non-examples in addition to the studentgenerated graphs (circles, parabolas, horizontal and vertical lines, etc.).
Obtain information about a function from its graph I. Find f(c), where c is a constant II. Find x in f(x) = c, where c is a constant. 3. Provide a brief problem set for small group or independent practice. Introduce the Stat Plot feature on the graphing calculator, and have students input the values from the x/y charts and generate a graph. Have students use the vertical line test to determine whether or not each set of values represents a function. 1. Using the line graph from the paycheck example, ask students if they can determine how much you would make if you worked x number of hours. (Example: How much would I make if I worked 7 hours?). Encourage students to use the graph only, rather than the equation. Be sure to use function notation ( f(7) = ). 2. Using the line graph, ask students to determine how long it would take to earn y dollars. (Example: How long would I need to work in order to earn $80?). Again, encourage students to draw conclusions from the graph, rather than the equation. Be sure to use function notation ( f ( ) = 80). 3. Provide a problem set for independent practice. Have students re-enter the paycheck function, and show them how to evaluate the function for a specific x-value using the calculate: value feature. Emphasize that when using this feature, the calculator marks that point on the graph. Identify domain and range from a function s graph. 1. Review the definitions of domain and range. Provide a table of values, and ask students to identify the domain and range. Have students plot the points from the table, and then ask how they would determine the domain and range if they only had the graph. 2. Provide a few examples of continuous graphs, and ask students to work in small groups or pairs to discuss ways to identify the domain and range of each graph. Have students share solutions and methods with the whole group. Recognize the connection between finding zeros, roots, x-intercepts, and solutions to a function. o Using the graphs from the previous exercise, ask students to identify the x-intercepts for each graph. What is the value of y at each of those points? o Recognizing that y = 0 at each x-intercept, how could we identify the x-intercepts of a function given its equation only? Provide several examples of function equations and have students work in small groups or pairs to determine the x-intercepts for each. o Explain to students that finding the x-intercepts is also called finding the zeros of a function, and that this terminology will be used throughout the rest of the course. o Provide a problem set for independent practice. Possible graphing calculator usage: Provide several additional function equations, and allow students to graph each equation using the calculator. Introduce the calculate: zero feature.
The Algebra of Functions Overview of Objectives, students should be able to: 1. Find the domain of the function in equation form a. Find the domain of a linear, quadratic, polynomial, rational, and radical function. b. Find the domain of a square root function c. Find the domain of a rational function 2. Use the algebra of functions to combine functions a. Find the sum function and its domain b. Find the difference function and its domain c. Find the product function and its domain d. Find the quotient function and its domain Objectives: Find the domain of the function in equation form o Find the domain of a linear, quadratic, polynomial, rational, and radical function. o Find the domain of a square root function o Find the domain of a rational function Main Overarching Questions: i. What could cause the domain of a function to be restricted? i. When you add, subtract, multiply, or divide functions, how is the domain affected? Activities and Questions to ask students: Ask students to recall how to determine the domain and range of a relation (look at the x/y values on the table, look at the x/y values from the points on the graph). Because functions are special types of relations, every function also has a domain and a range. Provide a linear function, and then ask students to discuss in pairs how they would determine the domain of that function. Ask individuals to share out responses after a few minutes. (Create a t-chart, plot points, etc.) Lead the class in creating a short table of values for the linear function. Ask students, Would this table represent all of the possible x and y values for this function? How many different numbers could I substitute for x? How many possible output values are there for y? Guide students toward understanding that you could plug in an unlimited number of x-values, so the domain would be unrestricted. Provide the students with the equation for a horizontal line (ex. f(x) = 5). What is the domain for this function? What is the range? Provide a simple quadratic function for students (ex. f(x) = x^2 + 5). Ask students to work in pairs again to determine a method for finding the domain. Have students share their methods with the entire group. Guide students through a similar process as the linear function, creating a table of values and/or
Use the algebra of functions to combine functions o Find the sum function and its domain o Find the difference function and its domain o Find the product function and its domain o Find the quotient function and its domain graphing. Is there any x-value that would be invalid as an input for this function? Would this be the case for any quadratic function? Provide a similar example and use a similar process for a polynomial function. Provide an example of a basic rational function (ex. f(x) = 4/(x-2). Ask students to work in pairs to determine the domain of the function. Ideally, students will recognize a problem when they try to substitute x = 2. Ask students why the error occurred. Can x = 2 in this function? Why not? Discuss the implications for the domains of rational functions. Guide students to the conclusion that any input value that causes the denominator to equal zero should be excluded from the domain of a rational function. Provide several additional examples from whole group or small group practice. Provide an example of a basic radical function (ex. f( x) = x 1). Have students work in pairs to determine the domain of the function. Ideally, students will recognize a problem when they try to input values that are less than 1. Ask students why the error occurred. Discuss the implications for the domains of radical functions. Guide students to the conclusion that any input value that causes the radicand to be negative should be excluded from the domain of a radical function. Provide several additional examples for whole group or small group practice. Briefly review the situations that can create restrictions on the domain of a function (zero in the denominator, negative radicand). Provide a problem set for independent practice. After students have identified the domains for each of the sample functions above, allow them to graph them in the calculator. Assist them in making connections between the domains and the graphs. 1. Review the terms sum, difference, product and quotient. Solicit student definitions for each. 2. Sometimes it is useful to combine two or more functions into a single function. Given that f(x) = 2x + 1 and g(x) = -3x -4, how could we simplify f(x) + g(x)? Have students work in pairs to come up with an answer. Ask individuals to share methods with the entire group. 3. What were the domains of the original functions f(x) and g(x)? What is the domain of f(x) + g(x)? 4. Using the same functions, ask pairs to determine the difference and share methods. Emphasize that you are subtracting the entire value of g(x), showing students how to distribute the subtraction sign to each term. 5. Discuss the domain of the difference. Is it any different than the domain of the sum? 6. Using the same functions, ask pairs to determine the product and share their methods with
the whole group. Emphasize the fact that each term in the first function must be multiplied by each term in the second function by distributing (you may need to review FOIL and distributive property). 7. Discuss the domain of the product. Has it changed from the original functions and if so, how? 8. Repeat the process for the quotient of the functions. Discuss changes to the domain, emphasizing that any value that makes the denominator zero should be excluded from the domain. 9. Model additional examples using quadratic, polynomial, radical, and rational functions. For each, ask students to determine the domain of the original functions and the domain of the resulting function. Re-emphasize that denominators of zero and negative radicands will result in restrictions to the domain. 10. Provide a problem set for independent practice. Quadratic Functions and their Graphs Overview of Objectives, students should be able to: Find the intercepts a. x-intercepts by solving a quadratic equation b. y-intercept Find the vertex of a quadratic function a. Use the formula to find the vertex b. Determine if the vertex is a maximum or minimum point Determine if a quadratic function s graph opens up or down Recognize the connection between finding zeros, roots, x-intercepts, and solutions to a quadratic function. Main Overarching Questions: How can you find the x-intercepts of a quadratic function on the graph and from the equation? How can you find the y-intercepts of a quadratic function from the graph and from the equation? How can you find the vertex of a quadratic function from the graph and from the equation? How can you determine whether the vertex of a quadratic function represents a minimum or a maximum value? How can you tell whether the graph of a quadratic function should open up or down given its equation? Explain how to find the zeros of a quadratic function. Objectives: Activities and Questions to ask students:
Find the intercepts o x-intercepts by solving a quadratic equation o y-intercept o by looking at graph Find the vertex of a quadratic function o Use the formula to find the vertex o Determine if the vertex is a maximum or minimum point 1. Draw a parabola on the board (or provide a graphed parabola) and ask students to identify the y-intercept. When the graph crosses the y-axis, what is the value of x? 2. Provide a quadratic function, and ask students to find the y-intercept (plug in zero for x). Provide a few additional examples for independent or small group practice. After students have found solutions, ask them if there is a shortcut for determining the y-intercept of a quadratic function (just use the c value). 3. Using the original parabola, ask students to identify the x-intercepts. When the graph crosses the x-axis, what is the value of y? 4. Provide a quadratic function, and ask students to find the x-intercepts (plug in zero for y, or f(x) ). Ask students to recall methods for solving quadratic equations from previous lessons (factoring, completing the square, quadratic formula). Which method would be most efficient for this problem? 5. Emphasize the idea that finding the x-intercepts of a quadratic function is the same as solving a quadratic equation. This is also called finding the zeros, finding the roots, or finding the solutions of a quadratic function. 6. Guide students through several additional examples, asking them to determine which method to use for each. Provide a problem set for independent practice. After students have identified intercepts using the equations, allow them to graph the functions and check their answers using the calculate: value and calculate: zero features. 1. Quadratic functions can be used to model many different real-world problems. One of the most common is the path of a trajectory. For example, the height of a punted football could 2 be modeled with the quadratic function f( x) = 0.01x + 1.18x+ 2, where the horizontal distance in feet from the kicker s foot is x, and f(x) represents the height of the ball in feet. 2. If I were to graph this function, where would the y-intercept be? What about the x- intercepts? Allow students to work in pairs to find x- and y-intercepts. What do the intercepts represent? 3. Plot the x- and y-intercepts on a graph, and then ask students to predict where the vertex would be located. (You may need to review that the vertex is the highest or lowest point on the graph). Guide students to the conclusion that the vertex would be exactly halfway between the two x-intercepts (at x-value 59). 4. Would the vertex be located above or below the x-intercepts? Assist students in understanding that because the ball is being kicked upward, the vertex would be above the x-intercepts.
Determine if a quadratic function s graph opens up or down 5. How can we tell how high the ball actually went? Solicit student responses, guiding students to the conclusion that the height can be found by plugging in the corresponding x-value (59 feet). (Highest point, 36.81 feet). 6. Provide students with the formula y = -b/2a. Ask them to use the a and b values from the football example to solve the formula. What do they notice? (The result is the same as the x- value of the vertex). Clarify that this is the formula for finding the x-value of the vertex of any quadratic function, and that it is usually faster to use this formula than to determine the midpoint of the x-intercepts. 7. In the football example, would you say that the vertex was the maximum or minimum of the function? From the graph, it is relatively easy to recognize that it is a maximum because it is the highest point on the graph. Is there a way to tell whether the vertex is a maximum or minimum without actually graphing? Solicit student responses, guiding them to the conclusion that (-a) results in a downward facing graph with a maximum value at its vertex, and that (a) results in an upward facing graph with a maximum value at its vertex. 8. Provide a problem set for small group or independent practice, asking students to identify the vertex, the direction of the graph, and whether the vertex represents a maximum or a minimum. Allow students to graph the football equation in the calculator. Introduce the calculate: minimum/maximum feature and allow students to verify their vertices using the calculator. Covered in previous section. Provide several equations for quadratic functions (some with positive a values, and some with negative). Allow students to graph the functions in the calculator, and ask them to draw a conclusion about the relationship between the equation and the direction of the graph.