Prints clean in black and white.
Win Elements · Algebra 1 · Notes

Which Axis Is Which, and What a Point Means

Skill 0.12 · Core Standard: pre 8.F.3 About 10 minutes

A graph of a real situation is a sentence. Each axis supplies a noun, and every point on the line is one complete statement.

Skill 0.07 treated a point as a location: two numbers, two moves, one spot on a grid. Here the same point carries meaning. It reports a fact about a phone bill, a distance, or a price.

Throughout these notes one situation runs underneath: a phone plan costing $20 each month plus $5 for every gigabyte of data used.

1. Each axis names one quantity

The horizontal axis carries the input, the quantity you choose. The vertical axis carries the output, the quantity that results. Labeling both with units before reading anything settles most confusion in advance.

$0 $20 $40 $60 0 2 4 6 8 dollars charged → the output, on the vertical axis gigabytes used → the input, on the horizontal axis $20 before any data is used
Figure 1. The line meets the vertical axis at $20, which is the bill before a single gigabyte is used. That number lives on the cost axis because it is a cost.

2. Reading one point in both directions

To read a point, travel to each axis separately. Going straight down reaches the input. Going straight across reaches the output. Put the two together and you have a sentence about the situation.

$0 $20 $40 $60 0 2 4 6 8 (4, 40) across to the output down to the input One point says: 4 gigabytes costs 40 dollars.
Figure 2. The two dashed paths never mix, because each one ends at a different axis carrying a different unit.
Key idea

Read the input on the horizontal axis and the output on the vertical axis. A point is a pair of facts, one from each axis, describing the same moment.

3. The mistake: reading the starting value off the wrong axis

The starting value is where the line meets the vertical axis: the output when the input is zero. Students often read a number off the horizontal axis instead, reporting an amount of data when the question asked for a cost.

Watch out: the starting value is an output

Ask what unit the answer should carry. That question alone picks the axis for you.

READ OFF THE HORIZONTAL an input, not a cost reads 1.2 READ OFF THE VERTICAL the cost before anything is used $20 The starting value is a cost, so it must be read on the cost axis.

Sense check: the starting value answers "what is the cost before anything happens", so the answer must be in dollars. A number of gigabytes cannot be an answer to it.

4. Where it starts and how fast it climbs

Two separate facts live in this graph. The height at the very left says what the bill is before any data. The steepness says what each extra gigabyte adds. Changing one does not change the other.

$0 $20 $40 $60 0 2 4 6 8 1 more gigabyte $5 more starts at $20 Where the line starts and how fast it climbs are two different facts.
Figure 3. The step is drawn between two plotted points, so the $5 comes from the graph itself rather than from the formula.

5. Turning points into sentences

Every point on this line can be read aloud the same way. Practicing the sentence is what keeps the two axes straight.

POINT INPUT, OUTPUT WHAT IT SAYS (0, 20) 0 GB, $20 no data used, bill is still $20 (2, 30) 2 GB, $30 two gigabytes costs $30 (4, 40) 4 GB, $40 four gigabytes costs $40 (6, 50) 6 GB, $50 six gigabytes costs $50
Figure 4. Every two gigabytes adds $10, which agrees with $5 per gigabyte. The table and the steepness describe the same line.

6. Worked examples

Example A

On the phone plan graph, what does the point (6, 50) mean?

The first number sits on the horizontal axis, which carries gigabytes. The second sits on the vertical axis, which carries dollars.

Using 6 gigabytes costs $50. Sense check: $20 plus six lots of $5 is $50.

Example B

A classmate says the plan starts at 1.2 because that is where the line appears to begin. What went wrong?

They read a number off the horizontal axis, so their answer is measured in gigabytes. The starting value is a cost, and costs live on the vertical axis.

The plan starts at $20, not 1.2. Sense check: the question asks what you pay, so the answer must be in dollars.

Example C

Reading the graph, the cost at 2 gigabytes is $30 and at 3 gigabytes is $35. What does that tell you?

The input rose by 1 and the output rose by 5. That step is the same anywhere along a straight line.

Each gigabyte adds $5. Sense check: from 2 GB to 4 GB the cost goes $30 to $40, which is two steps of $5.

Words to know
Input
The quantity you choose, plotted on the horizontal axis.
Output
The quantity that results, plotted on the vertical axis.
Starting value
The output when the input is zero, read on the vertical axis.
Steepness
How much the output changes for each step of input.
Units
What each axis measures in, such as dollars or gigabytes.

7. A look ahead

Coming next: Skill 1.01, terms, factors and coefficients

Skill 1.01 opens Unit 1 and leaves the review material behind. It gives precise names to the parts of an expression. Two traps wait there: calling everything after a plus sign a coefficient, and missing the invisible 1 in front of a lone variable.

Check your understanding
  1. State which axis carries the input and which carries the output.
  2. Explain what the point (2, 30) says about the phone plan.
  3. Explain why the starting value cannot be read off the horizontal axis.
  4. Describe how to find what one extra gigabyte costs, using only the graph.
  5. Explain why labeling both axes with units prevents most reading errors.
What to carry away