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.
Compare the slopes of two linear functions represented in different forms to determine which has a greater rate of change.
Compare the y-intercepts of two linear functions represented in different forms.
Determine which of two linear functions has a greater value at a specific input (x-value).
Translate a linear function from one representation (graph, table, equation) to another.
Solve real-world problems by comparing linear functions presented in various forms.
Ever wonder which 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 linear functions when they're shown as grap...
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Key Concepts & Vocabulary
TermDefinitionExample
Linear FunctionA function whose graph is a straight line, representing a constant rate of change. Its output (y) changes by a constant amount for each unit change in its input (x).The equation y = 2x + 3 describes a linear function.
Slope (Rate of Change)A measure of the steepness and direction of a line, calculated as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. It represents how much the y-value changes for every 1-unit change in the x-value.In the equation y = 2x + 3, the slope is 2, meaning y increases by 2 for every 1-unit increase in x.
Y-interceptThe point where the graph of a function crosses the y-axis. It is the value of y when x = 0, representing the starting value or initial condition.In the eq...
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Core Formulas
Slope Formula
$m = \frac{y_2 - y_1}{x_2 - x_1}$
This formula is used to calculate the slope (m) of a line when you are given any two distinct points $(x_1, y_1)$ and $(x_2, y_2)$ on the line. It helps find the rate of change from tables or graphs.
Slope-Intercept Form
$y = mx + b$
This is the most common and useful form for linear equations. 'm' directly represents the slope (rate of change), and 'b' directly represents the y-intercept (the value of y when x is 0, or the starting point). It's ideal for quickly identifying and comparing key features.
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Challenging
Function A is represented by the graph of a line passing through (0, 10) and (5, 0). Function B is represented by the table below. For what positive x-value is the output of Function A exactly 12 greater than the output of Function B?
| x | y |
|---|---|
| 0 | 2 |
| 1 | 1 |
| 2 | 0 |
A.x = 2
B.x = 4
C.x = 5
D.x = 10
Challenging
Tank A starts with 100 gallons of water and drains at a rate of 5 gallons per minute, represented by V = 100 - 5t. Tank B's volume is shown on a graph as a line segment from (0, 120) to (30, 0). Which tank will be empty first and by how many minutes?
A.Tank A, by 4 minutes
B.Tank B, by 4 minutes
C.Tank A, by 10 minutes
D.Tank B, by 10 minutes
Challenging
Function F passes through the points (-4, 15) and (2, -3). Function G is represented by the table below. Which statement correctly compares the steepness of the two functions?
| x | y |
|---|---|
| -5| -10 |
| 0 | 0 |
| 5 | 10 |
A.Function F is steeper than Function G.
B.Function G is steeper than Function F.
C.Both functions have the same steepness.
D.Function G is not a linear function.
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