Linear functions

y = f(x) = ax + b

Where:

a: the slope

b: the y-intercept

A function written as y = ax + b is said to be in slope-intercept form

What the coefficients mean

• The y-intercept b

Substitute x = 0 into the function:

y = f(0) = a ⨉ 0 + b = b

⇒ y = b is where the graph of y = f(x) crosses the y-axis ⇒ b is called the y-intercept

• The slope a:

Example: For the function y = f(x) = 2x + 3, we have the following table of values:

Each time x increases by 1 unit, y increases by 2 units. This is exactly the value of the slope a:

• x increases from 0 → 1, y increases from 3 → 5
• x increases from 1 → 2, y increases from 5 → 7
• x increases from 2 → 3, y increases from 7 → 9
• x increases from 3 → 4, y increases from 9 → 11

In general: the meaning of the slope a:

For the linear function y = f(x) = ax + b

⇒ f(x + 1) = a(x + 1) + b = ax + a + b = (ax + b) + a = f(x) + a

When x increases by 1 unit, y increases by a units ⇒ When x increases by n units, y increases by (a ⨉ n) units

⇒ This is the key sign that a function is linear

Formula for the slope a:

Where:

(x₁,y₁) and (x₂,y₂): two points that satisfy the equation of the function

= y₂ - y₁ = the change in y

= x₂ - x₁ = the change in x

• a > 0 ⇒ f(x) is an increasing function ⇒ the graph rises from left to right
• a < 0 ⇒ f(x) is a decreasing function ⇒ the graph falls from left to right

Problem: Tom is saving money to buy a new bicycle. He currently has $50 saved up and is saving $10 per week from his allowance. How much money will Tom have saved after 8 weeks?

Let x be the number of weeks that have passed and y be the amount of money Tom has saved

• Tom saves $10 every week

⇒ When x increases by 1 week, y increases by $10 ⇒ This is the sign of a linear function

⇒ Slope a = 10

• At the start (before any week has passed), Tom already has $50 saved

⇒ When x = 0, y = f(0) = 50 ⇒ y-intercept b = 50

Putting the two facts together ⇒ The linear function is y = 10x + 50

⇒ After 8 weeks, Tom will have saved f(8) = 10 ⨉ 8 + 50 = $130

The graph of a linear function is a straight line.

To draw the graph of a linear function, you first need to identify 2 points that satisfy the function equation, then connect those 2 points.

For example: y = f(x) = x + 2

We see that the y-intercept b = 2 ⇒ The graph passes through the point (0,2)

To determine the second point, we can substitute x = -1 into the equation ⇒ f(-1) = -1 + 2 = 1

⇒ The graph passes through the point (-1,1)

Let y = a₁x + b₁ and y = a₂x + b₂ be two linear functions whose graphs are the lines d₁ and d₂, respectively

Determining the relationship between these two lines on the coordinate plane means determining how many intersection points they have.

Setting the two y-values equal gives equation (1): a₁x + b₁ = a₂x + b₂

Formulas

d₁ // d₂ (parallel) ⇒ a₁ = a₂ and b₁ ≠ b₂

The two lines are parallel ⇒ No intersection point ⇒ Equation (1) has no solution

d₁ intersects d₂ ⇒ a₁ ≠ a₂

The two lines intersect ⇒ Exactly one intersection point ⇒ Equation (1) has exactly one solution

Special case: d₁ 丄 d₂ ⇒ a₁a₂ = -1

d₁ ☰ d₂ ⇒ a₁ = a₂ and b₁ = b₂

The two lines coincide ⇒ Infinitely many intersection points ⇒ Equation (1) has infinitely many solutions

Method 1: 3 steps to graph a linear inequality

Method 2: use Desmos

• Clue words for linear inequalities in word problems

Sarah has a budget of $500 to spend on clothes. She wants to buy some jeans that cost $50 each and some t-shirts that cost $20 each. She wants to buy at least 5 items in total.

Let the number of pairs of jeans Sarah buys be j and the number of t-shirts Sarah buys be t

• The total amount of money Sarah has is $500 ⇒ 50j + 20t 500
• Sarah wants to buy at least 5 items ⇒ j + t 5