Analyzing Functions

ANALYZING FUNCTIONS

The objective for this lesson on Analyzing Functions is, the student will construct and interpret a function to model a linear relationship between two quantities and determine the rate of change and initial value of the function from a description of a relationship or from two values, including reading these from a table or from a graph.

The skills students should have in order to help them in this lesson include, KEMS Grade Eight Lesson on Functions, KEMS Grade Eight Lesson on Comparing functions, and Understanding of slope and y-intercept and Graphing functions.

We will have three essential questions that will be guiding our lesson. Number 1, how can you identify the slope of a function using a graph? Number 2, how can you identify the slope using an equation in slope-intercept form? And number 3, how can you identify the y-intercept using an equation in slope-intercept form and what is the meaning of the y-intercept?

Begin by completing the warm-up working with different forms of functions to identify rates of change, to prepare for analyzing functions in this lesson.

SOLVE PROBLEM – INTRODUCTION

We will begin this lesson by completing an entire SOLVE problem as it relates to writing fractions. The SOLVE problem is, Joyce belongs to a DVD club. Each month she must pay a membership fee of ten dollars. After paying her fee, she can then purchase as many DVDs as she would like. Joyce’s cost per DVD is only four dollars. What is the function that represents the total monthly cost for the DVD club, represented by the variable y, based on the number of DVDs purchased, represented by the variable x?

We will begin by Studying the Problem. First we need to identify where the question is located within the problem and we will underline the question. What is the function that represents the total monthly cost for the DVD club, represented by the variable y, based on the number of DVDs purchased, represented by the variable x? Now that we have identified the question, we will put this question in our own words in the form of a statement. This problem is asking me to find the function that represents the total monthly cost for the DVD club, or y, based on the number of DVD’s purchased or x.

In Step O, we will Organize the Facts. First we need to identify the facts. Joyce belongs to a DVD club, fact. Each month she must pay a membership fee of ten dollars, fact. After paying her fee, she can then purchase as many DVDs as she would like, fact. Joyce’s cost per DVD is only four dollars, fact. What is the function that represents the total monthly cost for the DVD club, y, based on the number of DVDs purchased, x?

Now that we have identified the facts, we will eliminate the unnecessary facts. These are the facts that will not help us to find the function that represents the total monthly cost for the DVD club. Joyce belongs to a DVD club. Knowing that Joyce belongs to the DVD club will not help us to find the function that represents the total monthly cost for the DVD club, so we will eliminate this fact. Each month she must pay a membership fee of ten dollars. The membership fee is important for us to know to come up with the function that represents the total cost, so we will keep this fact. After paying her fee, she can then purchase as many DVDs as she would like. We need to know that she is able to purchase as many as she would like, in order to help us to write the function. So we will keep this fact as well. Joyce’s cost per DVD is only four dollars. Knowing the cost of each DVD that she purchases, will also help us to write the function. So we will keep this fact as well. What is the function that represents the total monthly cost for the DVD club, y, based on the number of DVDs purchased, x? We will need to know what y represents and what x represents. So we will keep these two facts that are located in the question as well.

Now that we have eliminated the unnecessary facts, let’s list the necessary facts. The membership fee is ten dollars each month. Each DVD is four dollars. We will use the variable y, to represent the total monthly cost for the DVD club. And we will use the variable x, to represent the number of DVDs purchased.

In Step L, we will Line Up a Plan. First we need to write in words what your plan of action will be. We need to create a function that can be used to identify Joyce’s total monthly cost for the DVD club. We will multiply the cost per DVD by the number of DVDs and add the monthly membership fee. What operation or operations will we use in our plan? We will use multiplication and addition.

Now let’s Verify Your Plan with Action. First estimate your answer. We can estimate that our answer will be a function in the form of an equation. Now let’s carry out your plan. We said in our plan that we wanted to create a function that can be used to identify Joyce’s total monthly cost for the DVD club. We will multiply the cost per DVD by the number of DVD’s and add the monthly membership fee. The rate of change, which will be the coefficient of x, is given as the cost per DVD, which was four dollars, and the monthly membership of ten dollars is only paid one per month and is added to the four x. So our function is y equals four x plus ten. Where y represents the total monthly cost for the DVD club, and x represents the number of DVDs purchased during that month.

In Step E, we will Examine Your Results.

Does your answer make sense? Here compare your answer to the question. Yes, because I wrote a function that represents Joyce’s monthly DVD club costs.

Is your answer reasonable? Here compare your answer to the estimate. Yes, because my answer is a function in the form of an equation.

And is your answer accurate? Here check your work. Yes. The answer is accurate.

We are now ready to write your answer in a complete sentence. The function for Joyce’s total monthly DVD club costs is y equals four x plus ten.

EXTEND THE SOLVE PROBLEM SLOPE AND Y-INTERCEPT WITH TABLES

We are now going to extend the SOLVE problem we just completed to working with Slope and y-intercept. Let’s review. What is the equation that represents Joyce’s total monthly cost for the DVD club from the first SOLVE problem? The equation is y equals four x plus ten. Where y represents the total monthly cost for the DVD club, and x represents the number of DVDs purchased.

Take a look at the table seen here. We are going to complete the table using Joyce’s function y equals four x plus ten. Notice that the input values or the x values are given in the table. We need to substitute these values into the equation to get the output values or the y values which represent the total monthly cost of the DVD club.

Let’s start with our first x value provided which is zero. This means that zero DVDs were purchased. When we substitute zero for x in our equation we will multiply four times zero and add ten. Four times zero plus ten equals ten dollars. So when zero DVDs are purchased the monthly DVD club cost is ten dollars. Let’s now substitute the value of x for one. Four times one plus ten equals fourteen dollars. Our next value for x is two. Four times two plus ten equals eighteen dollars. Our next value for x is three. Four times three plus ten equals twenty two dollars. And our final value for x in our table is five. Four times five plus ten equals thirty dollars. We have now completed the table for Joyce’s function showing what the cost is monthly for the DVD club, based on the number of DVDs purchased.

How do you find the rate of change for this scenario? You need to find the difference in y-values of any two points and divide by the difference in x-values of the same two points. Then check to be sure that the rate applies to all of the points. What is the slope of the function in the function table? Let’s choose two points and find out. We can choose the points zero, ten from the table, and one, fourteen. We need to find the difference in the y-values of these two points and divide by the difference in the x-values. Fourteen minus ten over one minus zero. Fourteen minus ten equals four and one minus zero equals one. Now we need to divide these two values. Four divided by one equals four. The slope is four. Note that you can choose any two values from the table, but the slope should be four no matter what points were chosen. What does the slope mean in the context of the problem? Defend your answer. This slope is a rate of change. Each DVD costs four dollars, therefore, the number of DVDS purchased should be multiplied by this rate to find the amount owed for DVDs bought.

When a graph of a function intersects the y-axis, what is this point called? We call this point the y-intercept. What must the x-value of an ordered pair be at a point where the graph crosses the y-axis? The x-value must be zero where a graph crosses the y-axis. Explain your answer. The x-value of each point on the y-axis is zero. So that is why the x-value would be zero when a point crosses the y-axis.

So looking at the table, what set of values represents the y-intercept? We can see that when x equals zero y equals ten. So the point is zero, ten. Explain your answer. This is the starting cost before any DVDs are purchased. The y-intercept generally represents the starting point, or in this case, the price before the four dollars per DVD rate is applied. Ten dollars represents the cost that is paid only once each month to maintain a membership in the DVD club.

How could we find the y-intercept if it is not given in the table? Well we just discussed that the x-value has to be zero at the y-intercept. We can use the function to substitute zero for x and solve for the y-value. Let’s do that now. Y equals four times zero plus ten. We substituted zero in for the x-value in our equation. Four times zero plus ten equals ten. So y equals ten when x equals zero.

Let’s take a look at another function together. This function is y equals negative two x minus four. Complete the table by filling in the output values. We will substitute each x-value into the equation to find y or the output value. Negative two times negative one minus four equals negative two. When x is negative one, y is negative two. Let’s substitute the value for x in as zero. Negative two times zero minus four equals negative four. When x equals zero y equals negative four.

Now let’s substitute in the value of x as one. Negative two times one minus four equals negative six. When x is positive one, y equals negative six.

Now let’s see what y equals when the x-value is two. Negative two times two minus four equals negative eight. When x is positive two, y equals negative eight.

What is the slope of this table? The slope is negative two. Explain how you determined the slope. We need to take two points from our table and find the difference in the y-values and the difference in the x-values. We will divide the difference of the y-values by the difference of the x-values to find the slope. Let’s choose two points now. The points are negative one, negative two, and zero, negative four. The difference in the y-values for these two points is negative four minus negative two. And the difference in the x-values for these two points can be represented by zero minus negative one. Negative four minus negative two equals negative two. And zero minus negative one equals one. Negative two divided by one equals negative two. This is how we can find the slope in the table.

What is the y-intercept? The y-intercept is zero, negative four. How did you find it? The x-value for a y-intercept is always zero, and the point zero, negative four is an ordered pair in the function table.

Let’s take a look at one more function together. This is the function y equals one-half x plus three. Let’s complete the table by filling in the output values for the given input values or x-values in the table. Our first x-value is negative two. We need to substitute negative two for x in our equation in order to find the output value or y-value. One-half times negative two plus three equals two. When x is negative two, y equals two.

Now let’s find the y-value when x equals positive two. One-half times two plus three equals four. When x is two, y equals four.

How about when x equals four? One-half times four plus three equals five. When x equals four, y equals five.

And what about when x equals six? One-half times six plus three equals six. When x is equal to six, y is equal to six.

What is the slope for this table? The slope is one-half. Explain how you determined the slope. Remember that we need to take two points from the table and we need to find the difference of the y-values and the difference of the x-values. And then divide the difference of the y-values by the difference of the x-values in order to find the slope. We can use the points, two, four and six, six. Remember that we can choose any two points from the table. They do not have to be two points right next to each other in the table. Six minus four divided by six minus two. Six minus four equals two and six minus two equals four. So we will divide two by four. When we divide two by four we get one-half. The slope from this table is one-half.

What is the y-intercept for the table? The y-intercept is zero, three. How did you find it? You need to substitute zero in for the x-value and solve for y. Since zero is not an x-value that was given to us in the table. Let’s take the equation y equals one-half x plus three and substitute in zero for x. One-half times zero plus three equals three. So when x equals zero, y equals three.

SLOPE AND Y INTERCEPT WITH GRAPHS

We are now going to take a look at Slope and y-intercepts with graphs. Take a moment to copy the function and the table values from the DVD club SOLVE problem. Remember the function for this problem was y equals four x plus ten.