Mathematics
Grade 8
15 min
Interpret linear functions
Interpret linear functions
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1
Introduction & Learning Objectives
Learning Objectives
Identify the slope and y-intercept of a linear function from its graph, equation, or a table of values.
Describe the real-world meaning of the slope and y-intercept in contextual problems.
Graph a linear function given its equation or a set of points.
Write the equation of a linear function given its graph, two points, or a real-world scenario.
Compare and contrast properties (like slope and y-intercept) of two different linear functions represented in various forms.
Use a linear function to make predictions and solve problems within a given context.
Ever wonder how your phone's battery life decreases over time, or how the cost of a taxi ride changes with distance? 📉 These are all examples of linear relationships!
In this lesson, we'll dive in...
2
Key Concepts & Vocabulary
TermDefinitionExample
Linear FunctionA function whose graph is a straight line. It represents a constant rate of change between two quantities.The relationship between the number of hours worked and the money earned at a fixed hourly wage.
Coordinate PlaneA two-dimensional plane formed by the intersection of a horizontal number line (x-axis) and a vertical number line (y-axis). Points are located using ordered pairs (x, y).Plotting the point (3, 2) means moving 3 units right from the origin and 2 units up.
Slope (Rate of Change)A measure of the steepness and direction of a line. It describes how much the y-value changes for every unit change in the x-value. Often called 'rise over run'.If a line has a slope of 2, it means for every 1 unit moved to the right on the x-axis, the li...
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Core Formulas
Slope Formula
$m = \frac{y_2 - y_1}{x_2 - x_1}$
Use this formula to calculate the slope ($m$) of a line when you are given any two points $(x_1, y_1)$ and $(x_2, y_2)$ on the line. It represents the change in y divided by the change in x.
Slope-Intercept Form
$y = mx + b$
This is a standard way to write the equation of a linear function. '$m$' represents the slope of the line, and '$b$' represents the y-intercept (the point where the line crosses the y-axis at $(0, b)$). This form is very useful for graphing and interpreting linear relationships.
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Challenging
A phone company charges a monthly fee plus a per-gigabyte rate for data. Plan A is modeled by the equation C = 5g + 40. Plan B is shown in the table below. Which plan has a higher initial monthly fee, and which has a higher per-gigabyte rate?
| Gigabytes (g) | Cost (C) for Plan B |
|---------------|---------------------|
| 0 | $60 |
| 2 | $68 |
| 4 | $76 |
A.Plan A has a higher fee; Plan B has a higher rate.
B.Plan B has a higher fee; Plan A has a higher rate.
C.Plan A has a higher fee; Plan A has a higher rate.
D.Plan B has a higher fee; Plan B has a higher rate.
Challenging
A rental car company charges a flat fee plus a fee per mile. A 100-mile trip costs $80 and a 250-mile trip costs $125. What is the company's flat fee (the cost for 0 miles)?
A.$0.30
B.$30
C.$50
D.$45
Challenging
Two online streaming services have different pricing models. Service A charges $12 per month with no additional fees. Service B's cost is modeled by C = 2w + 5, where C is the total cost and w is the number of weeks. To compare them fairly, we model Service A's cost per week. Assuming 4 weeks in a month, which service has a higher weekly rate and which has a higher initial fee?
A.Service A has a higher rate; Service B has a higher initial fee.
B.Service B has a higher rate; Service A has a higher initial fee.
C.Service A has a higher rate; they have the same initial fee.
D.Service B has a higher rate; Service B has a higher initial fee.
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