Mathematics
Grade 8
15 min
Compare linear functions: graphs, tables, and equations
Compare linear functions: graphs, tables, and equations
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1
Introduction & Learning Objectives
Learning Objectives
Identify the slope and y-intercept of a linear function from its graph, table, or equation.
Convert a linear function from one representation (graph, table) to another (equation).
Compare the slopes of two linear functions presented in different forms.
Compare the y-intercepts of two linear functions presented in different forms.
Determine which of two linear functions has a greater rate of change or initial value.
Explain the meaning of slope and y-intercept in real-world contexts when comparing linear functions.
Have you ever wondered which cell phone plan is better, or which car travels faster? 🚗💨 We can use math to compare them!
In this lesson, you'll learn how to compare different linear functions, whether they're shown as a picture (gra...
2
Key Concepts & Vocabulary
TermDefinitionExample
Linear FunctionA function whose graph is a straight line, representing a constant rate of change.The relationship between the number of hours worked and the money earned at a constant hourly wage.
Slope (Rate of Change)A measure of the steepness of a line, indicating how much the y-value changes for every unit change in the x-value. It's often described as '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 line goes up 2 units on the y-axis.
Y-intercept (Initial Value)The point where the graph of a linear function crosses the y-axis. It represents the value of y when x is 0.In the equation $y = 3x + 5$, the y-intercept is 5. This means when x=0, y=5.
GraphA visual representation of a function on...
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Core Formulas
Slope-Intercept Form
$y = mx + b$
This is the standard form for a linear equation, where 'm' represents the slope (rate of change) and 'b' represents the y-intercept (initial value). Use this to easily identify and compare these key features.
Slope Formula from Two Points
$m = \frac{y_2 - y_1}{x_2 - x_1}$
Use this formula to calculate the slope 'm' of a linear function when you are given any two points $(x_1, y_1)$ and $(x_2, y_2)$ from its graph or table.
Identifying Slope from a Graph
$m = \frac{\text{Rise}}{\text{Run}}$
To find the slope from a graph, pick two clear points on the line. Count the vertical change (rise) between them and the horizontal change (run). The ratio of rise to run is the slope. Remember: upward rise is positive...
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Challenging
Function A is represented by the table below. Function B is represented by the graph of a line passing through (0, 8) and (4, 0). At what x-value will the two functions have the same y-value?
| x | y |
|---|---|
| 0 | 2 |
| 1 | 3 |
| 2 | 4 |
A.x = 1
B.x = 2
C.x = 3
D.x = 4
Challenging
Function 1 is defined by the equation y = -3x + 7. Function 2 represents the remaining distance on a 10-mile hike, where a person walks at a constant speed of 2 miles per hour. Which statement is true?
A.Function 1 has a greater initial value and a faster rate of change than Function 2.
B.Function 2 has a greater initial value but a slower rate of change (in terms of absolute value) than Function 1.
C.Function 1 and Function 2 have the same initial value.
D.Function 2 has a greater initial value and its rate of change is less steep than Function 1.
Challenging
Function A is a line passing through (0, -2) and (3, 7). Function B is represented by the table below, but one y-value is missing.
| x | y |
|---|---|
| -2| -10|
| 0 | -4 |
| 2 | k |
What value of 'k' would make the slope of Function B equal to the slope of Function A?
A.2
B.3
C.-1
D.0
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