ARE 201 · Fall 2026
Math Review
Functions, lines, graphs, equilibrium, and growth rates
What We Will Review
1
Functions
What a function is and how variables are related.
2
Lines
How to interpret slope and intercept.
3
Graphs
How to read, draw, and manipulate a graph.
4
Models
How to solve an economic model with algebra or a graph.
5
Growth
How to compute and interpret percentage change.
Goal: translate comfortably among words, equations, tables, and graphs.
What Is a Function?
A function is a rule that describes how an output depends on one or more inputs . In economics, functions often help us represent cause-and-effect relationships.
Grade = 0.10(Classwork) + 0.15(Quizzes) + 0.50(Unit Exams) + 0.25(Final)
The Linear Function We Will Use Most
A variable is a quantity that can take on a range of values. In y = mx + b, x and y are variables.
y = mx + b
x
Input Shown on the horizontal axis.
y
Output Shown on the vertical axis.
m
Slope How much y changes when x increases by one unit.
b
y-intercept The value of y when x = 0.
Read it in words: start at b , then change by m for each one-unit increase in x .
From an Equation to a Table
y = 9 + 3x
2
Plug that value into the equation.
3
Record the corresponding value of y .
If x = 2, then y = 9 + 3(2) = 15 .
x y
0 9 1 12 2 15 3 18 4 21 5 24 6 27
Interactive: Plot the Points
y = 9 + 3x
Your task
Work from left to right. Move over the graph, then click the correct height for the highlighted x-value .
Completed: 0 / 7
Next: x = 0
Reset
Move your pointer over the graph. The live coordinate will follow it.
0 1 2
3 4 5 6
0 5 10
15 20 25 30
x
y
(0, 9)
From the Table to a Graph
Each row becomes one point
For example, the row x = 2 and y = 15 becomes the point (2, 15) .
Why use graphs? They express equations visually and display statistics or data. Here, connecting the points reveals the shape of the function.
What can we already see?
The line begins at y = 9 and rises by 3 whenever x increases by 1.
0 1 2
3 4 5 6
0 5 10
15 20 25 30
x
y
y = 9 + 3x
run = 2
rise = 6
Interpreting the Slope
The slope describes both the shape of a line and the relationship between two variables .
Positive slope
As x rises, y rises.
Zero slope
As x changes, y does not.
Negative slope
As x rises, y falls.
Economics translation: the sign tells us the direction of the relationship; the size tells us how strongly y responds to x.
Calculating the Slope: Rise over Run
slope =
Δy Δx y2 − y1 x2 − x1
Use two points from our line
Start: (x1 , y1 ) = (1, 12)
End: (x2 , y2 ) = (5, 24)
Δy = 24 − 12 = 12
Δx = 5 − 1 = 4
slope = 12 ÷ 4 = 3
Interpretation: every one-unit increase in x is associated with a three-unit increase in y.
What Changes the Shape—and What Shifts the Line?
Change the slope, keep b fixed
m = 0.5
m = 1
m = 2
same b
A larger |m| makes the line steeper ; a smaller |m| makes it flatter .
Change the intercept, keep m fixed
b = 2
b = 4
b = 0
Changing b shifts the line up or down in parallel ; its steepness does not change.
Interactive: Slope & Intercept Lab
−4 −2 0 2 4
6 4 2 −2 −4 −6
x y
y = x
Match this target
y = 2x − 3
Check
New target
Move m to rotate the line; move b to shift it.
Reset line
Solving Economic Models
We can solve the same economic problem in two equivalent ways:
1
Use algebra Set equations equal and solve the system.
2
Use a graph Find the point where the two lines intersect.
Example: the pizza market
Equilibrium condition: the quantity buyers demand equals the quantity sellers supply, so Qd = Qs .
Algebra Method: Solve for Equilibrium Price
1
16 − 2P = 2 + 5P
Set demand equal to supply.
2
14 − 2P = 5P
Subtract 2 from both sides.
3
14 = 7P
Add 2P to both sides.
4
P* = 14 ÷ 7 = 2
Divide both sides by 7.
Result: the equilibrium price is P* = 2 .
Plug the Price Back In to Find Quantity
Use P* = 2 in either equation. Using both equations gives us a useful check.
Demand
Qd = 16 − 2P
= 16 − 2(2)
Qd = 12
Supply
Qs = 2 + 5P
= 2 + 5(2)
Qs = 12
✓ Qd = Qs = 12, so the market is in equilibrium.
Prepare the Equations for a Graph
Economics graphs place quantity on the horizontal axis and price on the vertical axis . Solve each equation for the vertical variable, P.
Before drawing: demand has a negative slope; supply has a positive slope.
Graph Method: Find the Intersection
The intersection gives the same answer as algebra:Q* = 12, P* = 2.
0 4 8
12 16
0 2 4
6 8
Quantity
Price
Demand
Supply
Equilibrium
Q* = 12
P* = 2
Interactive: Find Market Equilibrium
0 5 10 15 20 0 1 2 3 4 Quantity
Price
Demand Supply
P = 1.0
Goal
Move the price until Qd = Qs and the two dots overlap.
P = 1.0
Choose a price
Shortage of 7 units.
Reset
Show equilibrium
Growth Rates Are Percentage Changes
A growth rate tells us how large a change is relative to where we started .
Percentage change =
Change in quantity Original quantity
=
New − Original Original
π
The price level or inflation
Always identify the original value first. It belongs in the denominator.
Example: A Pay Raise
1. Find the absolute change
$12 − $10 = $2
2. Divide by the original value
$2 $10
Interpretation: the hourly wage increased by 20 percent .
Interactive: Percentage Change Speed Round
Round 1 of 5
Score: 0
Restart
Calculate the percentage change
Your hourly wage changes:
20%
25%
2%
−20%
Remember: divide the change by the original value.
Next question
Five Moves to Remember
1
A function maps inputs to an output.
2
For y = mx + b, m is the slope and b is the y-intercept .
3
Calculate slope with rise over run : Δy ÷ Δx.
4
Solve a model by setting equations equal or finding their graphical intersection .
5
Percentage change is change ÷ original × 100%.
The big skill: move between words, equations, tables, and graphs—and explain what the answer means.