ECON 120 | Graph Skills Preparation

Step 1

Begin with the axes

Before looking at a line or a point, ask two questions: What is being measured? and In what units? The axis labels provide the answers.

Meaning first: A number such as 4 has no clear meaning by itself. It might mean 4 dollars, 4 hours, or 4 meals. A graph becomes readable when each axis names both the variable and its unit.

Temperature and ice-cream sales

The arrows show the positive direction; the scale shows how much each step represents.

Horizontal axis

Temperature is measured in degrees Celsius. Moving right means a higher temperature.

Vertical axis

Sales are measured in ice creams per hour. Moving up means more sales.

Check the scale

What does the second numbered tick to the right of zero represent?

Step 2

A point combines two values

A point written as (x, y) tells you where to move along the horizontal axis first and then where to move along the vertical axis.

Read a point as a sentence: The point (20, 50) means: “When the temperature is 20°C, the shop sells 50 ice creams per hour.” The comma joins two pieces of information about the same observation.

Temperature: 20°C

Sales: 50 per hour

Point (20, 50)

Read the ordered pair

What does (10, 30) mean?

Step 3

Plot a table on the graph

Each row of the table becomes one point. Click the matching grid intersection for every temperature–sales pair.

Ice-cream sales table

Temperature (°C)	Sales per hour
0	10
5	20
10	30
15	40
20	50
25	60

The first column supplies x. The second column supplies y.

Plot the six points

Click near a grid intersection to add a point. Click a plotted point to remove it.

0 of 6 points plotted.

Step 4

Intercepts describe edge cases

An intercept is where a graph meets an axis. At that location, the variable measured on the other axis equals zero.

Meaning: An intercept answers an edge-case question: “What does the model predict when one variable is zero?”

Ice cream

The vertical intercept (0, 10) says that predicted sales are 10 per hour when temperature is 0°C.

Hot chocolate

The intercepts (0, 120) and (30, 0) describe sales at zero temperature and the temperature at zero sales.

Model boundary: A line can be extended mathematically, but its predictions may stop being realistic. Interpret an intercept within the context and useful range of the model.

Step 5

Direction describes a relationship

Direction tells whether the variables move together or in opposite directions.

Temperature: 15°C

Ice cream: 40/hour · Hot chocolate: 60/hour

Positive relationship

As temperature rises, predicted ice-cream sales rise.

Negative relationship

As temperature rises, predicted hot-chocolate sales fall.

Use the words carefully: “Positive” and “negative” describe movement, not whether something is good or bad. A relationship alone does not prove causation.

Step 6

Slope is a rate of change

Choose any two points on a straight line. Measure the vertical change, measure the horizontal change, and divide. The result is the same everywhere on that line.

Start with the movement: The symbol Δ (the Greek letter delta) means “change in.” Therefore, Δx means “change in x,” and Δy means “change in y.” slope = Δy ÷ Δx Always move from the first point to the second in the same direction when calculating both changes.

Positive constant slope: ice-cream sales

The line rises from left to right, so its slope is positive. The two graphs use different pairs of points on the same line.

Pair 1: A to B

Horizontal change: ΔT = 15 − 5 = 10°C.
Vertical change: ΔI = 40 − 20 = +20 per hour.
slope = +20 ÷ 10 = +2

Pair 2: C to D

Horizontal change: ΔT = 25 − 15 = 10°C.
Vertical change: ΔI = 60 − 40 = +20 per hour.
slope = +20 ÷ 10 = +2

What stayed constant? The triangles are in different locations, but both have a rise of 20 and a run of 10. Every 1°C increase is associated with 2 additional ice creams sold per hour.

Negative constant slope: hot-chocolate sales

The line falls from left to right, so its slope is negative. Again, two different pairs of points produce the same result.

Pair 1: A to B

Horizontal change: ΔT = 15 − 5 = 10°C.
Vertical change: ΔH = 60 − 100 = −40 per hour.
slope = −40 ÷ 10 = −4

Pair 2: C to D

Horizontal change: ΔT = 25 − 15 = 10°C.
Vertical change: ΔH = 20 − 60 = −40 per hour.
slope = −40 ÷ 10 = −4

Why the slope is constant: A straight line has the same steepness everywhere. Changing the two selected points can change the sizes of Δy and Δx, but their ratio remains unchanged. On a curve, the steepness—and therefore the slope—can change.

Check the idea

Two points farther apart on the same straight line produce a rise of 40 and a run of 20. What should happen to the slope?

Step 7

Constant slope: The change is the same at every step.

On a straight line, moving one equal step to the right always increases or decreases the vertical value by the same amount — the slope itself. The table and the graph should show the identical pattern.

The idea in one sentence: If the slope is +2, then every time temperature goes up by 1°C, sales go up by exactly 2. In real terms, each extra degree reliably brings in 2 more ice-cream sales — that steady, predictable connection between the two variables is what the slope measures.

Ice cream: slope = +2

Each row increases by 10.

Temperature rises by 5°C each row, so sales rise by 5 × 2 = 10 each row.

Temperature (°C)	Sales per hour	Change from row above
0	10	—
5	20	+10
10	30	+10
15	40	+10
20	50	+10
25	60	+10

Every row adds the same +10, because the slope (+2 per °C) times the row's step size (5°C) is always +10.

The same “plus 10” on the graph

Every step is 5°C wide (labeled once, since it repeats) and 10 sales tall — matching the table exactly.

Hot chocolate: slope = −4

Each row decreases by 20.

Temperature rises by 5°C each row, so sales fall by 5 × 4 = 20 each row.

Temperature (°C)	Cups per hour	Change from row above
0	120	—
5	100	−20
10	80	−20
15	60	−20
20	40	−20
25	20	−20

Every row subtracts the same 20, because the slope (−4 per °C) times the row's step size (5°C) is always −20.

The same “minus 20” on the graph

Every step is 5°C wide (labeled once, since it repeats) and 20 cups shorter — matching the table exactly.

Why this matters: Once you know a line's slope, you can build its whole table by repeated addition (for positive slopes) or subtraction (for negative slopes) — you never need two new points to find the next row. This is also why the equation works: each extra unit of x increases or decreases y by exactly one more “slope” unit.

Check the pattern

A line has a slope of +6. If temperature rises by 3°C from one row to the next, how much does the row's y-value rise by?

Step 8

An equation summarizes the graph

Change an intercept or slope and watch the corresponding line move. The equation and graph update together.

Ice-cream equation

I = 10 + 2T

Intercept: 10

Slope: +2

At 0°C, predicted sales are 10 per hour. Each additional 1°C adds 2 sales per hour.

Hot-chocolate equation

H = 120 − 4T

Intercept: 120

Slope: −4

At 0°C, predicted sales are 120 per hour. Each additional 1°C subtracts 4 sales per hour.

Step 9

Explain the whole graph

Imagine that someone gives you a graph and asks, “What can you tell from it?” Start by finding out what the graph is about. Then use its points, intercepts, slope, and equation to tell the full story.

A useful reading order: What do the axes measure? What does one point mean? Where does the line begin? Does it rise or fall? How fast does it change? What does the equation say? Finally, what does all of this mean in the real world?

Story 1: Ice-cream sales

What are we talking about? The horizontal axis shows outdoor temperature in degrees Celsius. The vertical axis shows how many ice creams the shop sells per hour. So this graph is about the relationship between temperature and ice-cream sales.

What can we learn from the graph? The line begins at (0, 10). This means the model predicts 10 ice creams sold per hour when the temperature is 0°C. The point (20, 50) gives another part of the story: at 20°C, the model predicts 50 sales per hour.

As we move to the right, the line rises. Temperature and ice-cream sales have a positive relationship. The slope is +2, so each 1°C increase is linked to 2 more ice creams sold per hour.

The equation I = 10 + 2T says the same thing in a shorter way: start with 10 sales and add 2 for every additional degree.

The whole story: In this model, warmer weather is linked to higher ice-cream sales. The graph does not prove that temperature is the only cause of sales, but it shows a clear positive relationship.

Story 2: Hot-chocolate sales

What are we talking about? The horizontal axis again shows outdoor temperature in degrees Celsius. This time, the vertical axis shows cups of hot chocolate sold per hour. The graph is about the relationship between temperature and hot-chocolate sales.

What can we learn from the graph? The line begins at (0, 120). This means the model predicts 120 cups sold per hour when the temperature is 0°C. The point (20, 40) means that at 20°C, the model predicts 40 cups sold per hour. The other intercept, (30, 0), means predicted sales reach zero at 30°C.

As we move to the right, the line falls. Temperature and hot-chocolate sales have a negative relationship. The slope is −4, so each 1°C increase is linked to 4 fewer cups sold per hour.

The equation H = 120 − 4T tells us to start with 120 sales and subtract 4 for every additional degree.

The whole story: In this model, warmer weather is linked to lower hot-chocolate sales. Temperature is not the only thing that affects sales, but the graph shows a clear negative relationship.

Choose the strongest graph story

Which statement best explains the ice-cream graph?

Step 10

Calculate areas from linear graphs

A region above or below a straight line may form a right triangle. Read the base and height as differences between coordinates, then apply the triangle formula.

Area of a right triangle: ½ × base × height. Base and height are lengths, so subtract their endpoint values first.

Area below a line

Base: 4 − 0 = 4
Height: 12 − 5 = 7

½ × 4 × 7 = 14

Area above a line

Base: 5 − 0 = 5
Height: 11 − 3 = 8

½ × 5 × 8 = 20

Check the method

A shaded right triangle extends from x = 2 to x = 8 and from y = 4 to y = 10. What is its area?

Step 11

Curves can be concave or convex

First ask: does the curve rise or fall? Then ask: does it become steeper or flatter?

Slope on a curve: The green line touches the curve at one point. It shows how steep the curve is there. A curve can have a different slope at each point, so its slope is not constant.

Concave: diminishing increases

Real-world story

At first, warmer weather may bring a large jump in ice-cream sales. Later, the shop may be close to capacity. Sales still rise, but by smaller amounts.

These graphs show the same curve at two temperature levels.

Lower temperature

slope ≈ +3.5 Sales rise quickly.

Higher temperature

slope ≈ +0.4 Sales still rise, but slowly.

Why is it concave? Both slopes are positive, so sales rise. But +0.4 is smaller than +3.5, so the curve becomes flatter. This is called diminishing increases: sales keep increasing, but each equal temperature increase adds less than the one before.

Convex: diminishing decreases

Real-world story

At first, warmer weather may cause a large drop in hot-chocolate sales. Later, sales are already low. They still fall, but by smaller amounts.

These graphs show the same curve at two temperature levels.

Lower temperature

slope ≈ −3.5 Sales fall quickly.

Higher temperature

slope ≈ −0.4 Sales still fall, but slowly.

Why is it convex? Both slopes are negative, so sales fall. But −0.4 is closer to zero than −3.5, so the curve becomes flatter. This is called diminishing decreases: sales keep decreasing, but each equal temperature increase subtracts less than the one before.

Remember: A straight line has the same slope everywhere. A curve does not. Diminishing increases means rising by smaller amounts. Diminishing decreases means falling by smaller amounts.

Check the real-world meaning

Hot-chocolate sales fall by 30 units after one temperature increase but by only 5 units after the same-sized increase at a higher temperature. What does this suggest?

Step 12

Mixed graph-skills practice

Try several kinds of problems. Use the feedback to check both your calculation and your interpretation.

1. Multiple choice: read a point

On a graph with temperature on the horizontal axis and ice-cream sales per hour on the vertical axis, what does the point (15, 40) mean?

2. Interactive challenge: match a line

Use the sliders to make your solid line match the dotted target line Y = 10 + 2X.

Intercept: 0

Slope: 0

Your line: Y = 0 + 0X

3. Calculate a slope

A straight line passes through A(4, 18) and B(10, 42). Find Δx, Δy, and the slope.

Δx

Δy

Slope

4. Calculate a triangle’s area

Use the coordinates on the graph to find the triangle’s base, height, and area.

Base

Height

Area

5. Multiple choice: classify a curve

Ice-cream sales rise by 12 units after one temperature increase, then by 7 units, and then by 3 units after equal temperature increases. Which description fits best?

6. Write a graph story

Let T be temperature in °C and H be hot chocolates sold per hour. Explain the graph summarized by H = 90 − 3T. Mention the variables, intercept, direction, slope, and real-world meaning.

Write two or three sentences, then compare your response with the model answer.

Step 13

Extra practice

Use tables, lines, triangles, and curves to practice reading the information shown by a graph.

7. Plot a graph from a table

Plot all four rows. Each row becomes one point. The graph will connect your points in order.

X	Y
0	10
5	20
10	30
15	40

Click a point again to remove it.

0 of 4 points plotted.

8. Complete a table from an equation

Use H = 100 − 5T to find hot-chocolate sales at each temperature.

Temperature T (°C)	Hot chocolates H per hour
0	Enter H
10	Enter H
20	Enter H

9. Calculate a positive slope

Use points A and B on the graph. Find ΔX, ΔY, and the slope.

ΔX

ΔY

Slope

10. Calculate a negative slope

Use points A and B on the graph. Find ΔX, ΔY, and the slope.

ΔX

ΔY

Slope

11. Interpret the intercepts

The graph shows temperature T and hot-chocolate sales H. What do its two intercepts mean?

12. Choose the slope’s units

The horizontal axis measures temperature in °C. The vertical axis measures cups sold per hour. What are the slope’s units?

13. Find the slope error

A student uses (40 − 20) ÷ (5 − 15) and gets −2 for the line from A(5, 20) to B(15, 40). What went wrong?

14. Determine the equation from a graph

Read the vertical intercept and calculate the slope from the two labeled points.

15. Read a concave curve

Which statement best describes this graph?

16. Read a convex curve

Which statement best describes this graph?

Preparation complete

You can now translate axes, scales, points, tables, intercepts, relationships, slopes, equations, areas, and curves into meaningful statements.