5.2 Graphing Linear Equations

• I can create a table of values and graph a linear relation.

• I can describe and analyze the pattern shown in a linear graph.

• I can use a graph to estimate values between (interpolate) and beyond (extrapolate) the points.

This table of values shows the number of movie tickets sold over a few days.

Linear Relation

• A linear relation is a relationship between two variables where the graph of the equation forms a straight line. The equation of a linear relation can typically be written in the form:

  • • • 𝑦 = 𝑚𝑥 + 𝑏

Where:

• 𝑚 is the rate of change

• 𝑏 is the y-intercept  or the value of 𝑦 when 𝑥 = 0

Table of Values

A table of values is a way to organize pairs of values (usually 𝑥 and 𝑦) that satisfy a given equation. 

Each value of 𝑥 is paired with a corresponding value of 𝑦, showing the relationship between the two variables.

Slope (m):

• The slope of a line is a measure of its steepness. It tells you how much 𝑦 changes for every change in 𝑥.

• The slope is often described as "rise over run" (how much the line rises vertically for each unit it moves horizontally).

Ordered Pair (or Coordinate)

An ordered pair is a pair of numbers written in the form (𝑥,𝑦) that represent a point on the coordinate plane. 

The first number, 𝑥, corresponds to the position on the horizontal axis (x-axis), and the second number, 𝑦, corresponds to the position on the vertical axis (y-axis).

Interpolate - estimate a value between two known values

Extrapolate - predict a value by extending a pattern beyond known values

When solving equations, it is important that we recognize what each part of the equation represents.

Graphing Equations

Senario: You earn $15 per hour at a job.

Step 1: Write the equation

When you are using equations, you must know what each variable represents. This starts with looking at the given information.

Write an equation in the format of y = mx + b

1. Identify each part of the equation.

• • • Let x = ________________

    • Let y = ________________

    • Slope (m)

      • • How much does y increase/decrease each time x changes.

        • m = _________

    • y-intercept/constant (b)

• What is y when x = 0

• When x=0, y= ________

• Does this make sense?

___________________________________

2. The equation for this scenario would be

__________________________

Is there anything for the y-intercept? ________________.

Step 2: Complete the table of values

Use this information to set up your table of values.

1. Fill in what the x-axis and y-axis represent

2. Enter what values you want to use for x-axis. 

3. Enter what you want x to be equal to.

Step 4: Use the graph to solve problems

1. On your graph, indicate how much the earnings would be in 2.5 hours? Look between (2, 30) and (3, 45).  Place a dot at the proper coordinate.

2. What is the coordinate?

3. How much will you earn after 2.5 hours?

4. What coordinate would this be? Extend the line past known points and read the graph to find the coordinates. How much do you earn in 6 hours?

5. How can we show on our graph that the pattern continues indefinitely?

To show that the pattern continues indefinitely, extend the line at the same slope and put an arrow on the end of it indicating that it extends past the given graph.

Bike Rental Bonanza

Scenario:

You’re running a bike rental shop. You charge a $10 flat fee plus $5/hour for a bike rental.

1. What is the equation for this scenario?

let x = _____________

let y = _____________

m = ________

y-intercept = __________

Thinking back to our special equation for the slope-intercept, create an equation in the same format

2. Create a table of values for 0 to 5 hours

3. Graph the relation.

4. Use the graph to:

• Interpolate: Estimate the cost at 2.5 hours

• Extrapolate: Estimate the cost at 8 hours.

5. Reverse it: If someone paid $45, how many hours did they ride for? 

1. Explain the difference between interpolation and extrapolation.

2. A taxi company charges $4 plus $2 per km. Write the equation for total cost. 

   1. • Let x= ______________

      • Let y = ______________

      • m = ______________

      • b = ______________

The equation would be ____________________________

3. Use the equation to estimate the cost to a customer who is traveling 10 km using the taxi service. Show your work.

4. Is this interpolation or extrapolation? Explain how you know.

Submit the last page of your work from the handout above. 

For those who wish to complete extra material, feel free to use complete this document.  Please see Ms. Dunwoody for the answers. You will need a sheet of looseleaf to show your work.