Comparing Rates, Slopes & Functions

📖 Three Ways to Show the Same Thing

A linear function can be shown as an equation, a graph, or a table — and the GED will put two or three of these in front of you at the same time. Your job is to pull out the information you need from whichever form it's in.

Every comparison question on the GED is really asking one of three things:

Question Type 1

"Which charges more per pound / hour / mile?"

This is asking for the rate — the slope. How much does the cost go up with each unit?

Question Type 2

"Which has the lowest one-time fee / starting charge?"

This is asking for the starting value — the y-intercept. What's the cost before anything happens?

Question Type 3

"Which costs less for 6 pounds / 10 hours?"

This is asking for the total at a specific input. Substitute the number into each function and compare.

The Key Rule

Read the question before doing any math

The most common mistake: finding the rate when the question asked for the total, or vice versa.

📐 Finding the Rate from Each Form

The rate (cost per unit, growth per week, etc.) hides in different places depending on the form. Here's where to look:

📋 From an Equation

In y = mx + b, the rate is m — the number multiplied by x.

y = 4x + 10 → rate = $4 per unit

📈 From a Graph

Pick the two labeled points. Calculate:
rise ÷ run
= (change in y) ÷ (change in x)

(0,$10) to (5,$30): rise=20, run=5 → $4/unit

📊 From a Table

Find how much y changes per step of x. Subtract two consecutive y values and divide by the step size.

$10→$14→$18 per unit: $14−$10 = $4/unit

Finding the starting value is even easier: look for the value when x = 0. In the equation it's b. In the graph it's where the line crosses the y-axis. In the table it's the first row.

🔢 Worked Example — Shipping Companies

Three shipping companies. Each is shown a different way. We'll answer all three question types.

The Three Representations

📋 FastShip — Equation

y = 4x + 10

x = pounds shipped
y = total cost ($)

📈 QuickBox — Graph

📊 LoadRun — Table

Lbs (x)	Cost (y)
0	$15
1	$18
2	$21
3	$24
4	$27

These three boxes all show different shipping companies. Each company uses a different format — an equation, a graph, and a table. Below you'll see the three most common question types, worked out step by step.

Question Type 1 — Finding the Rate

Which company charges the least per pound?

How to find the rate from each form:

📋 FastShip (equation): The number in front of x is the slope — that's the rate. y = 4x + 10 → $4 per pound

📈 QuickBox (graph): Pick two points where the line crosses grid intersections. Rise ÷ Run = change in cost ÷ change in pounds → $5 per pound

📊 LoadRun (table): The cost goes up by the same amount each row. $18 − $15 = $3 per pound

Answer

✓ LoadRun charges the least — $3 per pound

The rate is how much the cost goes up with each pound. From the equation it's the slope. From the graph it's rise ÷ run. From the table it's the change between any two rows.

Question Type 2 — Finding the Starting Value

Which company has the lowest one-time handling fee?

How to find the starting value from each form:

📋 FastShip (equation): The number added at the end is b — the starting value. y = 4x + 10 → $10 fee

📈 QuickBox (graph): Find where the line crosses the y-axis (the vertical axis, when x = 0). That point is → $5 fee

📊 LoadRun (table): Look at the row where x = 0. That's the cost before any pounds are added → $15 fee

Answer

✓ QuickBox has the lowest one-time fee — $5

The starting value is always the cost when x = 0 — before any units are counted. In an equation it's the last number (b). On a graph it's where the line begins on the left. In a table it's the first row.

Question Type 3 — Comparing Totals at a Specific Amount

A customer ships 6 pounds. Which company costs the least?

Substitute x = 6 into each function and compare:

📋 FastShip: y = 4(6) + 10 = 24 + 10 = $34

📈 QuickBox: y = 5(6) + 5 = 30 + 5 = $35

📊 LoadRun: y = 3(6) + 15 = 18 + 15 = $33

Answer

✓ LoadRun costs the least for 6 pounds — $33

Even though LoadRun has the highest starting fee ($15), its low per-pound rate ($3) makes it cheapest for larger shipments. A high starting fee doesn't always mean the most expensive overall — it depends on how many pounds are shipped.

⚠️ Common Mistakes

Mistake 1

Answering the wrong question

"Which has the lowest fee?" vs "Which is cheapest for 8 units?" — read carefully before calculating.

Mistake 2

Confusing rate with starting fee

The lowest starting fee doesn't mean the cheapest rate. The lowest rate doesn't mean cheapest overall. Always check what's being asked.

Mistake 3

Wrong slope calculation from graph

Rise ÷ run — not run ÷ rise. The y-values go on top. Check your labeled points carefully.

Mistake 4

Not using x=0 for starting value

The starting fee/value is always when x = 0 — before any units. Don't use x = 1 or any other row.

Comparing Rates,Slopes & Functions

📖 Three Ways to Show the Same Thing

📐 Finding the Rate from Each Form

🔢 Worked Example — Shipping Companies

⚠️ Common Mistakes

✏️ Practice Questions

Comparing Rates,
Slopes & Functions