MLF - Graphing Linear Functions (Lesson)

Graphing Linear Functions

Linear functions can be graphed to provide a visual model. Graphing a linear function can help to see the domain and range and to learn more information about other values in the function. There are a few different ways to graph a linear function.

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Graphing Lines by Making a Table

Like we did in the cell phone problem, we can graph lines by using a table. In order to graph a line, you always need at least 2 points. 

Example: You are renting a bike for the day. A bike rental includes a deposit of $25 plus an hourly rate of $5. Let's first create a linear function that represents this relationship. 

$y=5x+25$

Now we can plug in values for x, the independent variable, to find what y is the output. When we are working with graphs, we often use x and y rather than x and f(x). But the important thing to remember is that f(x) and y both represent the OUTPUT! 

Input (x)	Substitute for x and solve for y	Write your ordered pair (input, output)
1	y = 5(1) + 25 y = 30	(1, 30)
2	y = 5(2) + 25 y = 35	(2, 35)
4	y = 5(4) + 25 y = 45	(4, 45)

Now let's consider the domain and range. Let's say that you could only rent a bike for a maximum of 6 hours. What would the graph look like then?

The minimum x-value is 0 and the maximum is 6. The resulting graph will look like this:

A graph can help you find the domain and range! The y-value of the maximum point is 55, so the maximum total cost would be $55.

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Graphing Lines by Finding x- and y-intercepts

Lines can be written in different forms. Two of these forms are slope-intercept form and standard form. Let's take a look at standard form first. The standard form of an equation is: $Ax + By = C$, where A, B, and C are all numbers. If a equation is in standard form, you can easily graph the line by finding the x- and y- intercepts.  

 

Example: Find the x- and y- intercepts of $2x + y = 7$.

 

 

Example: Your teacher gave you money to purchase supplies at the school store for your class. Rulers are $2 each and binders are $3 each. Your teacher has given you $24. Write an equation in standard form that describes this situation. Then, graph the equation by finding the intercepts. 

The function for this relationship is $2x+3y=24$.

x-intercept	y-intercept
$2x+3(0)=24$ $2x=24$ $x=12$	$2(0)+3y=24$ $3y=24$ $y=8$

Plot the two intercepts and connect them to graph the line.

Notice the domain and range for this scenario. 

Domain: [0, 12]

Range: [0, 8]

You can purchase at most 12 rulers (and 0 binders) or you can purchase at most 8 binders (and zero rulers).

How many binders can you purchase if you buy 6 rulers?

Answer: You can buy 4 rulers. Find the point (4, 6) on the graph.

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Graphing by Finding Slope and Y-Intercept

In Slope-Intercept form, $f(x) = mx + b$, m is the slope and (0, b) is the y-intercept.

Example:

$f(x)=-(\frac{3}{2})x-4$

slope (m): $-(\frac{3}{2})$

y-intercept: (0, -4)

Use that information to graph the line:

 

Watch this video for examples of equations in standard form:

 

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Slope and Intercepts Practice

1. Find the x and y intercepts: $2x - 5y = 10$

2. Find the x and y intercepts: $-2x + y = 6$

3. Find the x and y intercepts: $y = -2x + 4$

4. Find the slope and y-intercept: $y= 2x - 7$

5. Find the slope and y-intercept: $2x + y= 5$

6. Find the slope and y-intercept: $y = 1/2x - 7$

Your friend rides her scooter to school each day.  Sometimes she gets tired and walk some of the way. She can go 8 miles per hour on her scooter and can walk at 2 miles per hour. The distance to the school is 4 miles. 

7. Determine a linear function to represent the relationship.

8. Graph the linear function.

9. What is an appropriate domain and range for the scenario?

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