Have you ever thought about how to represent a function with a graph? Take a look at this dilemma about banana plants.
Let's say that 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) @$\begin{align*}d\end{align*}@$ .
We can write that linear equation as @$\begin{align*}h = 4d+8\end{align*}@$ .
Can you graph this function?
You will learn how to use function notation to graph functions like this one.
Guidance
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. The function of @$\begin{align*}y\end{align*}@$ depends on the rest of the equation. Then we can use function notation to represent this situation.
@$\begin{align*}f(x)=4x+1\end{align*}@$
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.
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 how to do this.
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.
Determine each slope and y-intercept by using the given information.
Example A
@$\begin{align*}f(x)=3x+2\end{align*}@$
Solution: Slope = 3, y-intercept = 2
Example B
@$\begin{align*}f(x)=-2x-9\end{align*}@$
Solution: Slope = -2, y-intercept = -9
Example C
@$\begin{align*}f(x)=-x+3\end{align*}@$
Solution: Slope = -1, y-intercept = 3
Now let's go back to the dilemma from the beginning of the Concept.
Here is the equation that we are going to graph. This equation describes the situation with the banana plant.
@$\begin{align*}h=4d+8\end{align*}@$
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.
Guided Practice
Here is one for you to try on your own.
Do you have enough information to graph this situation?
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.
Video Review
Explore More
Directions: Graph each function.
- @$\begin{align*}f(x)=3x+1\end{align*}@$
- @$\begin{align*}f(x)=2x+2\end{align*}@$
- @$\begin{align*}f(x)=5x-1\end{align*}@$
- @$\begin{align*}f(x)=x-3\end{align*}@$
- @$\begin{align*}f(x)=-2x+1\end{align*}@$
- @$\begin{align*}f(x)=-2x-5\end{align*}@$
- @$\begin{align*}f(x)=-4x+9\end{align*}@$
- @$\begin{align*}f(x)=4x+8\end{align*}@$
- @$\begin{align*}f(x)=x-10\end{align*}@$
- @$\begin{align*}f(x)=2x+6\end{align*}@$
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.
- What is the slope of 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.
- What is the slope of this situation?
- How many books does the writer need to sell to earn $10,000 total?