Linear Functions and Graphs: Complete Guide

1. Basic Concepts of Linear Functions

A linear function is an equation that forms a straight line when graphed, written in the form:

y = mx + b

Where:

m = slope (rate of change)
b = y-intercept (where the line crosses the y-axis)
x = independent variable
y = dependent variable

1.1 Slope (m)

Slope Formula: m = (y₂ - y₁)/(x₂ - x₁)

Types of Slopes:

Positive Slope: Line goes up from left to right
Negative Slope: Line goes down from left to right
Zero Slope: Horizontal line (y = constant)
Undefined Slope: Vertical line (x = constant)

2. Intercepts

Y-intercept (b): The point where the line crosses the y-axis (x = 0)

X-intercept: The point where the line crosses the x-axis (y = 0)

To find y-intercept: Substitute x = 0 into the equation
To find x-intercept: Substitute y = 0 and solve for x

3. Forms of Linear Equations

1. Slope-Intercept Form: y = mx + b

2. Point-Slope Form: y - y₁ = m(x - x₁)

3. Standard Form: Ax + By = C

4. Direct Variation

A direct variation is a special type of linear function that passes through the origin (0,0)

Equation: y = kx

Where k is the constant of proportionality

Properties of Direct Variation:

Always passes through (0,0)
No y-intercept (except at origin)
k represents the slope

5. Arithmetic Sequences

An arithmetic sequence is a sequence where the difference between consecutive terms is constant.

General Term Formula: aₙ = a₁ + (n-1)d

Where:

aₙ = nth term
a₁ = first term
n = position number
d = common difference

6. Solving Linear Equations

Steps to solve linear equations:

Combine like terms on each side
Get all variables on one side
Get all constants on the other side
Solve for the variable

Example:

2x + 5 = 12

2x = 7

x = 3.5

7. Applications

Linear functions are used to model:

Distance vs. Time (constant speed)
Cost vs. Quantity (fixed rate)
Temperature conversions
Simple interest
Population growth (simple)

Key Tips for Problem Solving:

Always identify the slope and y-intercept when given an equation
Use the slope to understand the rate of change
The y-intercept often represents the starting value or fixed cost
Check if a relationship is proportional by seeing if it passes through (0,0)

Linear Functions: Practice Problems and Solutions

Section 1: Identifying Slopes and Y-intercepts

Problem 1: Y-intercept Identification

Find the y-intercept in the equation: y = 5x - 4

Answer: -4

In slope-intercept form (y = mx + b):

The equation y = 5x - 4 is already in slope-intercept form
m = 5 (slope)
b = -4 (y-intercept)

Problem 2: Slope Calculation

Find the slope of the line that passes through the points (2, 4) and (6, 12)

Answer: 2

Using the slope formula: m = (y₂ - y₁)/(x₂ - x₁)

m = (12 - 4)/(6 - 2)
m = 8/4
m = 2

Section 2: Converting to Slope-Intercept Form

Problem 3: Standard to Slope-Intercept Form

Write in slope-intercept form: 5x - 3y = -9

Answer: y = (5/3)x + 3

Steps to convert:

Start with: 5x - 3y = -9
Subtract 5x from both sides: -3y = -5x - 9
Divide everything by -3: y = (5/3)x + 3

Section 3: Arithmetic Sequences

Problem 4: Finding Common Difference

Identify the common difference in the sequence: 97, 86, 75, 64, ...

Answer: -11

To find the common difference:

86 - 97 = -11
75 - 86 = -11
64 - 75 = -11

The sequence decreases by 11 each time, so the common difference is -11.

Section 4: Linear Relationships

Problem 5: Direct Variation

What does the Direct Variation Equation look like?

Answer: y = kx

Properties of direct variation:

k is the constant of proportionality
The graph always passes through the origin (0,0)
No y-intercept except at the origin
k represents the slope of the line

Section 5: Solving Linear Equations

Problem 6: Solving Equations

Solve for x: 6(2x + 1) = 18

Answer: x = 1

Steps to solve:

6(2x + 1) = 18
12x + 6 = 18
12x = 12
x = 1

Section 6: Real-World Applications

Problem 7: Word Problem

A fitness club opens with 80 members. Each month the membership increases by 15 members. Which equation represents the relationship between the number of months the club has been opened, x, and the total fitness club membership, y?

Answer: y = 15x + 80

Breaking down the problem:

Initial value (y-intercept) = 80 members
Rate of change (slope) = 15 members per month
x represents the number of months
y represents the total membership
Therefore: y = mx + b = 15x + 80

Tips for Problem Solving

Always identify if the question is asking for slope, y-intercept, or both
When solving equations, maintain balance by performing the same operation on both sides
In word problems, identify the initial value (y-intercept) and rate of change (slope)
For arithmetic sequences, check if the difference between consecutive terms is constant
In direct variation problems, remember that the graph must pass through the origin

Quiz 1

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