The best examples of graphing linear functions: 3 practical examples you’ll actually use

Before we get into formulas, let’s begin with situations you already know. Many everyday relationships are linear, or close enough that a line is a good model. Some classic examples of graphing linear functions include:

• Your total pay based on hours worked at a fixed hourly rate
• The cost of a ride-share with a base fee plus a per-mile charge
• Temperature changing at a steady rate over time
• A phone plan with a monthly fee plus a cost per gigabyte of data
• The distance a car travels at a constant speed
• The amount of water in a tank as it fills or drains at a steady rate

All of these can be modeled with a linear function and graphed as a straight line. In this article, we’ll focus on examples of graphing linear functions: 3 practical examples in depth, then connect them to several more real-world cases so you see the same pattern again and again.

Example 1: Hourly pay – the most relatable linear function

Imagine you get a part-time job that pays $18 per hour with no bonus or starting fee. If we let:

• \(x\) = number of hours you work
• \(y\) = total pay in dollars

then your pay is given by the linear function:

[
y = 18x
]

This is one of the cleanest examples of graphing linear functions because there’s no starting amount; you earn nothing if you work zero hours.

Step 1: Identify slope and intercept

The general slope–intercept form of a linear function is:

[
y = mx + b
]

where:

• \(m\) is the slope (rate of change)
• \(b\) is the y-intercept (starting value when \(x = 0\))

In our function \(y = 18x\):

• Slope \(m = 18\). This means your pay increases by $18 for every extra hour.
• Intercept \(b = 0\). If you don’t work, you don’t get paid.

Step 2: Make a small table of values

Pick a few easy values of \(x\):

• If \(x = 0\), then \(y = 18(0) = 0\) → point \((0, 0)\)
• If \(x = 2\), then \(y = 18(2) = 36\) → point \((2, 36)\)
• If \(x = 5\), then \(y = 18(5) = 90\) → point \((5, 90)\)

You don’t need many points for a line—two points define it—but three helps you check for mistakes.

Step 3: Plot and interpret the graph

If you sketch these points on a coordinate plane and draw a straight line through them, you get a line rising steadily from the origin. Every point on that line represents a possible combination of hours worked and total pay.

How to read the graph in context:

• The point \((4, 72)\) means: 4 hours → $72
• The point \((10, 180)\) means: 10 hours → $180
• The steepness of the line shows how fast your pay grows. A steeper line would mean a higher hourly wage.

This is a textbook example of graphing a linear function where the graph directly answers practical questions. For instance: “How many hours do I need to work to earn at least \(200?” On the graph, you’d find \)200 on the y-axis, move horizontally to hit the line, then drop down to the x-axis to read the approximate hours.

Example 2: Phone data plan – base fee plus usage

Now let’s look at a slightly more interesting situation: a monthly phone plan.

Suppose a carrier in 2024 offers a plan that costs \(25 per month plus \)8 per gigabyte (GB) of data you use beyond a basic allowance. Let:

• \(x\) = number of extra GB of data used
• \(y\) = total monthly cost in dollars

The linear function is:

[
y = 8x + 25
]

This is another one of our best examples of graphing linear functions: 3 practical examples, because you can clearly see both a starting cost and a usage-based cost.

Step 1: Understand slope and intercept in real terms

For \(y = 8x + 25\):

• Slope \(m = 8\). Every extra gigabyte increases your bill by $8.
• Intercept \(b = 25\). Even if you don’t use any extra data, you still pay $25.

This kind of model shows up constantly in pricing, from streaming services to ride-shares.

Step 2: Build a quick table

Choose some values of \(x\):

• If \(x = 0\): \(y = 8(0) + 25 = 25\) → point \((0, 25)\)
• If \(x = 2\): \(y = 8(2) + 25 = 41\) → point \((2, 41)\)
• If \(x = 5\): \(y = 8(5) + 25 = 65\) → point \((5, 65)\)

Step 3: Graph and use the line to make decisions

On the graph, your line will cross the y-axis at 25 (the base fee) and rise with slope 8.

Now the graph becomes a decision tool:

• Want to keep your bill under $60?
  Look for \(y = 60\) on the vertical axis, move across to the line, and then down to see how many GB of extra data that allows.

• Curious how your cost changes if you double your extra data?
  Compare points like \((2, 41)\) and \((4, 57)\). You can see the cost jump clearly.

This is one of the best examples of graphing linear functions because it mirrors real 2024–2025 pricing structures, where you often pay a fixed fee plus a variable charge. You’ll find similar linear models in sample problems from resources like the Khan Academy algebra library and in many high school curricula.

Example 3: Temperature change – tracking a steady trend

For the third of our examples of graphing linear functions: 3 practical examples, let’s look at temperature change, which is especially common in science classes.

Imagine a lab experiment where a beaker of water is heated on a hot plate. From measurements, you notice the temperature increases at a nearly constant rate of 3°F per minute. At time \(t = 0\) minutes, the water temperature is 68°F.

Let:

• \(x\) = time in minutes
• \(y\) = temperature in °F

The linear function is:

[
y = 3x + 68
]

Step 1: Interpret slope and intercept scientifically

In this function:

• Slope \(m = 3\). The temperature rises 3°F every minute.
• Intercept \(b = 68\). The starting temperature is 68°F at time zero.

This kind of linear modeling is common in introductory science and statistics courses, such as those described by the National Center for Education Statistics.

Step 2: Create a few data points

• If \(x = 0\): \(y = 3(0) + 68 = 68\) → \((0, 68)\)
• If \(x = 4\): \(y = 3(4) + 68 = 80\) → \((4, 80)\)
• If \(x = 10\): \(y = 3(10) + 68 = 98\) → \((10, 98)\)

Step 3: Use the graph to predict and compare

Once you graph the line, you can:

• Predict when the water will reach a certain temperature.
  For example, to reach 95°F, you can visually find 95 on the y-axis, move across to the line, then down to find the time.

• Compare different heating rates.
  If another hot plate heats at 5°F per minute starting at 60°F, its function would be \(y = 5x + 60\). On the same graph, you’d see which line is steeper and when one beaker becomes hotter than the other.

This makes the third of our best examples of graphing linear functions: 3 practical examples especially useful: it connects algebra to experiments and real data trends.

More real examples: distance, savings, and ride-share costs

The three core cases above give you a strong feel for how linear functions behave. But to really own the idea, it helps to see more real examples of graphing linear functions that follow the same pattern.

Here are several more scenarios, written in plain language so you can quickly turn them into linear functions and graphs:

Constant speed driving

A car travels at a steady 60 miles per hour on a highway.

• \(x\) = time in hours
• \(y\) = distance in miles
• Function: \(y = 60x\)

Graphing this linear function gives a line through the origin with slope 60. Doubling the time doubles the distance, and every point on the line represents a possible trip.

Saving money with a fixed monthly deposit

You open a savings account with an initial deposit of \(150, then add \)50 every month.

• \(x\) = number of months after opening the account
• \(y\) = total amount saved in dollars
• Function: \(y = 50x + 150\)

Graphing this linear function shows a line starting at 150 on the y-axis, climbing by 50 each month. This mirrors many personal finance examples used in educational resources from universities like Harvard’s math-related outreach materials.

Ride-share pricing

A ride-share app charges a \(3.50 base fee plus \)1.80 per mile.

• \(x\) = number of miles
• \(y\) = total cost in dollars
• Function: \(y = 1.8x + 3.5\)

Graphing this linear function helps you compare costs between services or estimate your fare before you travel.

These additional cases reinforce that examples of graphing linear functions almost always come down to the same idea: a starting value plus a steady rate of change.

How to graph any linear function step by step

By now you’ve seen several examples of graphing linear functions: 3 practical examples in detail and a handful of extra real-world cases. Let’s zoom out and summarize the process you can use on any linear function.

1. Put the function in slope–intercept form

Try to rewrite your function as \(y = mx + b\). For instance:

• \(2y - 4x = 10\)
  Solve for \(y\):
  \(2y = 4x + 10\)
  \(y = 2x + 5\)

Now you can see the slope and intercept clearly.

2. Identify slope and y-intercept

• Slope \(m\): how fast \(y\) changes when \(x\) increases by 1.
• Intercept \(b\): the value of \(y\) when \(x = 0\).

Connect them to the story: rate of pay, base fee, starting temperature, initial distance, and so on.

3. Plot the intercept

Start by plotting \((0, b)\) on the graph. That’s your anchor point.

4. Use the slope to find another point

If \(m = \frac{rise}{run}\), move “run” units horizontally and “rise” units vertically from your intercept. For example, if \(m = 3\), that’s the same as \(\frac{3}{1}\): right 1, up 3.

5. Draw the line

Once you have at least two accurate points, draw a straight line through them, extending in both directions. Every point on that line is a solution to the linear function.

This same method works for all the examples of graphing linear functions we’ve discussed: hourly pay, phone plans, temperature changes, savings accounts, driving distance, and ride-share costs.

If you want more practice, many U.S. school districts and colleges recommend free online tools and practice sets, including those highlighted by the U.S. Department of Education. These often include interactive graphs where you can adjust slope and intercept and immediately see how the line changes.

FAQ: Common questions about examples of graphing linear functions

What are some everyday examples of graphing linear functions?

Everyday examples of graphing linear functions include:

• Total pay vs. hours worked at a constant hourly wage
• Total cost vs. miles driven in a taxi or ride-share with a base fee
• Temperature vs. time when something heats or cools at a steady rate
• Savings vs. time when you deposit the same amount each month

All of these can be written in the form \(y = mx + b\) and graphed as straight lines.

How can I tell if a situation is an example of a linear function?

Ask yourself two questions:

1. Is there a fixed starting amount? (like a base fee or initial savings)
2. Does the quantity change by the same amount for each equal step in \(x\)? (like $10 more per hour, or 3°F per minute)

If both are true, you probably have a linear function and a good example of a linear relationship to graph.

Why do we use linear functions so often in math and science?

Linear functions are simple, but they model a lot of real behavior reasonably well, at least over limited ranges. Scientists and educators often start with linear models before moving to more complex ones. For instance, early physics courses use constant speed and constant acceleration models that are linear or nearly linear. This approach is reflected in many introductory materials from universities and science organizations.

Where can I find more practice problems and examples of graphing linear functions?

You can find more practice and examples of graphing linear functions in:

• Online algebra courses and practice sets from reputable education platforms
• Open educational resources linked by the U.S. Department of Education
• High school and college algebra textbooks, many of which are summarized or supported by university math department pages

Look for topics like “slope–intercept form,” “linear models,” and “rate of change.”

If you walk away with one big idea, let it be this: all of these examples of graphing linear functions: 3 practical examples and beyond are really the same story repeated with different characters. You start with a beginning value, you add (or subtract) at a steady rate, and the graph is a straight line that tells that story visually. Once you see that pattern, linear graphs stop being abstract and start feeling like a language you already speak.