9.6: Linear Functions
Introduction
Banana Math
“The vegetation of the rainforest was very interesting,” Carmen commented in Mr. Thomas’ class as the rainforest discussion continued.
“There were so many different things growing,” Mark agreed.
The students began having a discussion about the things that had intrigued them about the plant life of the rainforest. One of the points that was brought up is that there are plants in the rainforest that can’t be found anywhere else in the world. Mr. Thomas spotted the opportunity and wrote the following problem.
You buy a banana tree that is 8 inches tall. It grows 4 inches per day. Its height (in inches) \begin{align*}h\end{align*} is a function of time (in days) \begin{align*}d\end{align*}.
You can express this function as an equation and a graph. That is the work that you will learn about in this lesson. Pay close attention and you will be ready to complete both of these parts by the end of the lesson.
What You Will Learn
In this lesson, you will learn how to complete the following skills.
- Write linear equations given two points on a line, \begin{align*}x\end{align*} and \begin{align*}y\end{align*} – intercepts, slope and \begin{align*}y\end{align*} – intercept, or a table of values.
- Use function notation to express a linear relationship between an independent and dependent variable.
- Graph functions given in function notation.
- Model and solve real - world problems using tables, equations and graphs of linear functions.
Teaching Time
I. Write Linear Equations Given Two Points on a Line, \begin{align*}\underline{x}\end{align*} and \begin{align*}\underline{y}\end{align*} – intercepts, slope and \begin{align*}\underline{y}\end{align*} – intercept, or a table of values
The \begin{align*}y=mx+b\end{align*} form of an equation was most useful in rapidly identifying both the slope, \begin{align*}m\end{align*}, and the \begin{align*}y\end{align*}-intercept, \begin{align*}b\end{align*}. In fact, if we know what the slope of an equation is and we know its \begin{align*}y\end{align*}-intercept, then we can just as easily write the equation. All you have to do is plug in the slope for \begin{align*}m\end{align*} in the \begin{align*}y=mx+b\end{align*} form and the \begin{align*}y\end{align*}-intercept for \begin{align*}b\end{align*}.
Example
\begin{align*}m = 4, y-intercept = 3\end{align*}
We know that we are going to use the form of the equation \begin{align*}y=mx+b\end{align*}, so we can substitute these values into the equation and write it.
\begin{align*}y=4x+3\end{align*}
This is the answer. The key is to always watch for negative signs and be sure to include them when you write your equation.
Sometimes, we can also be given the slope and a point that the line crosses through. We can also use this information to write an equation of a line. Let’s look at an example.
Example
\begin{align*}\text{Slope} = -2\end{align*}, the line passes through the point (0,-3)
With this example, we know the slope, so that can be easily substituted into the slope-intercept form. The point has a 0 for the \begin{align*}x\end{align*} value, so that means that we have been given the coordinate of the \begin{align*}y\end{align*} – intercept.
\begin{align*}y= -2x-3\end{align*}
This is the answer.
What if you only know two points and you don’t know the slope? It’s a similar operation that we can use in order to write the equation.
Do you recall that the slope formula is \begin{align*}m=\frac{{y_2}-{y_1}}{{x_2}-{x_1}}\end{align*}? In other words, given any two points \begin{align*}(x_1, y_1)\end{align*} and \begin{align*}(x_2, y_2)\end{align*}, we can use the slope formula to calculate the slope of the line that passes through those points. Even when you only know two points, finding the slope is just a matter of using the formula. But then what?
We will use the notion below.
if \begin{align*}m=\frac{{y_2}-{y_1}}{{x_2}-{x_1}}\end{align*} then \begin{align*}m=\frac{{y}-{y_1}}{{x}-{x_1}}\end{align*} because the slope is the same on any part of a line. In other words, we can use a formula similar to the slope formula for finding the equation. This time, however, we will leave \begin{align*}x\end{align*} and \begin{align*}y\end{align*} as variables because the relationship is true for any values of \begin{align*}x\end{align*} and \begin{align*}y\end{align*} in that equation.
Example
Write the equation of the line that passes through the points \begin{align*}(3,7)\end{align*} and \begin{align*}(5,11)\end{align*}. First we will find the slope using the slope formula for \begin{align*}{x_1}=3,{y_1}=7,{x_2}=5,{y_2}=11\end{align*}.
\begin{align*}m &=\frac{{y_2}-{y_1}}{{x_2}-{x_1}}\\ m &=\frac{11-7}{5-3}\\ m &=\frac{4}{2}\\ m &=2\end{align*}
Now plug in our known values of \begin{align*}m, x_1\end{align*}, and \begin{align*}y_1\end{align*}.
\begin{align*}m &=\frac{y-{y_1}}{x-{x_1}}\\ \frac{2}{1} &=\frac{y-3}{x-7}\end{align*}
Do you see that we have a proportion? This can be solved by cross multiplying.
\begin{align*}\frac{2}{1} &=\frac{y-3}{x-7}\\ 1(y-3) &=2(x-7)\\ y-3 &=2x-14\\ y &=2x-11\end{align*}
This is the answer.
What about when you have been given a table of values? Well, there is a way to figure out the equation quite simply when you have a table of values. Take a look at the example.
Example
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
0 | 5 |
1 | 7 |
2 | 9 |
3 | 11 |
4 | 13 |
First, notice that the \begin{align*}y\end{align*} – intercept is the value that has an \begin{align*}x\end{align*} value of 0. With an \begin{align*}x\end{align*} value of 0, we know that the \begin{align*}y\end{align*} – intercept is 5.
Now we need to figure out the slope. Look at the y values in the table. Can you see a pattern? If you look carefully, you will see that the values jump by +2 each time. This is the slope. Think about how the line would move when graphed. The pattern of the \begin{align*}y\end{align*} values represents the slope of the line.
\begin{align*}y=2x+5\end{align*}
This is the answer.
II. Use Function Notation to Express a Linear Relationship Between an Independent and Dependent Variable
In science, an independent variable is a parameter that is manipulated or chosen by a scientist while the dependent variable is a parameter that is measured. Scientists oftentimes look for a correlation between an independent variable and a dependent variable—they want to know if the dependent variable depends on the independent variable. For example, a scientist might measure the speed at which a car is moving and the force upon impact when the car hits a wall. The scientist can manipulate the speed of the car—she can make the car move slower or faster. She would then measure the force of impact related to the given speed. Then, a conclusion can be drawn about their relatedness and cars, in this case, might be designed based on that relation. The independent variable will be shown in the left column of a t-table and on the \begin{align*}x\end{align*}-axis of a graph. The dependent variable will be shown in the right column of a t-table and on the \begin{align*}y\end{align*}-axis of a graph.
If you think about this it makes perfect sense. We have talked about how the function of \begin{align*}y\end{align*} depends on the rest of the equation. This brings us back to the function notation that we worked on in earlier sections. Remember?
\begin{align*}f(x)=4x+1\end{align*}
Here we know that the function of \begin{align*}x\end{align*} is dependent on 4 times that value, \begin{align*}x\end{align*} and one.
Once you understand the connection between independent variables and dependent variables, we can move on to graphing these linear equations.
III. Graph Functions Given in Function Notation
As with any other linear equation, functions can be graphed using a t-table or using the slope-intercept form. It is important to place the independent variable on the \begin{align*}x\end{align*}-axis and the dependent variable on the \begin{align*}y\end{align*}-axis or the results can be misinterpreted. Let’s look at an example.
Example
You buy a orange tree that is 12 inches tall. It grows 3 inches per day. Its height (in inches) \begin{align*}h\end{align*} is a function of time (in days) \begin{align*}d\end{align*}.
\begin{align*}h=3d+12\end{align*}
Let’s use slope-intercept form to show its graph. We know that the slope is 3 and the \begin{align*}y\end{align*} – intercept is 12. That gives us enough information to graph this line.
IV. Model and Solve Real – World Problems Using Tables, Equations and Graphs of Linear Functions
There are so many ways that we can use functions in real life. Think about all of the things that you do in a day. If you drive to work, then the time that it takes you to drive is a function of the speed of your car. If you are paid hourly, then the amount of money that you make is a function of the number of hours that you work. Each and every day we use functions. Let’s take a look at another example.
Example
A group of students measure the length of their classmates’ arms and legs and found the following data:
arm(in) | legs(in) |
---|---|
25 | 30 |
27 | 33.2 |
26 | 31.6 |
Assume there is a linear relationship. Write and graph the linear function that describes this data.
Use the table to determine ordered pairs. Then you can find the slope. Find the slope using the slope formula for \begin{align*}x_1=25, y_1 = 30, x_2 = 27, y_2 = 33.2\end{align*}.
\begin{align*}m &=\frac{{y_2}-{y_1}}{{x_2}-{x_1}}\\ m &=\frac{33.2-30}{27-25}\\ m &=\frac{3.2}{2}\\ m &=1.6 \\ m &=\frac{8}{5}\end{align*}
Now substitute in our known values of \begin{align*}m, x_1,\end{align*} and \begin{align*}y_1\end{align*}.
\begin{align*}m &=\frac{{y}-{y_1}}{{x}-{x_1}}\\ \frac{8}{5} &=\frac{y-30}{x-25}\\ 5(y-30) &=8(x-25)\\ 5y-150 &=8x-200 \\ 5 &=8x-50\\ y &=\frac{8}{5}x-10\end{align*}
Now we can graph the data.
Now let’s go back to the problem from the introduction and use what we have learned.
Real-Life Example Completed
Banana Math
Here is the original problem once again. Reread it and then complete both parts of the solution. Write an equation and draw a graph for this problem.
“The vegetation of the rainforest was very interesting,” Carmen commented in Mr. Thomas’ class as the rainforest discussion continued.
“There were so many different things growing,” Mark agreed.
The students began having a discussion about the things that had intrigued them about the plant life of the rainforest. One of the points that was brought up is that there are plants in the rainforest that can’t be found anywhere else in the world. Mr. Thomas spotted the opportunity and wrote the following problem.
You buy a banana tree that is 8 inches tall. It grows 4 inches per day. Its height (in inches) \begin{align*}h\end{align*} is a function of time (in days) \begin{align*}d\end{align*}.
Remember there are two parts to this solution.
Solution to Real – Life Example
First, we need to write an equation. We can use the \begin{align*}h\end{align*} to represent the height of the banana tree. We can use the \begin{align*}d\end{align*} to represent the number of days. The 8 is the height that the tree started with. Here is our equation.
\begin{align*}h=4d+8\end{align*}
Let’s use slope-intercept form to show its graph.
\begin{align*}m=4,b=8\end{align*}
Notice that we only need the first quadrant of the coordinate plane because negative values would not make sense.
Vocabulary
Here are the vocabulary words in this lesson.
- Independent Variable
- a value that is not dependent on another value. It is the \begin{align*}x\end{align*} value in a table.
- Dependent Variable
- a value that is dependent on the equation. It is the \begin{align*}y\end{align*} value in a table.
- Function Notation
- an equation where the value of \begin{align*}x\end{align*} is dependent on the equation involving \begin{align*}x\end{align*}.
Time to Practice
Directions: Write the equation of a line with the following slopes and \begin{align*}y\end{align*} – intercepts.
- \begin{align*}\text{slope} = 2, \ y \ \text{int} = 4\end{align*}
- \begin{align*}\text{slope} = -3, \ y \ \text{int} = 2\end{align*}
- \begin{align*}\text{slope} = -4, \ y \ \text{int} = 4\end{align*}
- \begin{align*}\text{slope} = 3, \ y \ \text{int} = -5\end{align*}
- \begin{align*}\text{slope} = \frac{1}{2}, \ y \ \text{int} = -2\end{align*}
- \begin{align*}\text{slope} = -\frac{1}{3}, \ y \ \text{int} = 2\end{align*}
- \begin{align*}\text{slope} = 1, \ y \ \text{int} = 8\end{align*}
- \begin{align*}\text{slope} = -2, \ y \ \text{int} = 4\end{align*}
- \begin{align*}\text{slope} = -1, \ y \ \text{int} = -1\end{align*}
- \begin{align*}\text{slope} = 5, \ y \ \text{int} = -2\end{align*}
Directions: Write the following horizontal or vertical line equations.
- A horizontal line with a \begin{align*}b\end{align*} value of 7.
- A horizontal line with a \begin{align*}b\end{align*} value of -4.
- A vertical line with an \begin{align*}x\end{align*} value of 2
- A vertical line with an \begin{align*}x\end{align*} value of -5
Directions: Write the equation of a line that passes through the following points.
- (3, -3) and (-3, 1)
- (2, 3) and (0, -3)
- (2, 3) and (3, 4)
- (2, 3) and (3, 4)
Directions: Use what you have learned to solve each problem.
- A migrating monarch butterfly travels 1100 miles. If it flies 30 miles per day, the distance \begin{align*}d\end{align*} it still has to travel is a function of days \begin{align*}t\end{align*} it has traveled. Write a function rule for the situation.
- A writer gets paid a writer’s fee of $3000 plus $1.50 for each copy of the book that is sold. Create a function rule for this situation.
- How many books does the writer need to sell to earn $10,000 total?