Linear Functions - Definition, Equation, Graph, Examples

Linear functions are mathematical rules where the output changes at a steady, constant rate relative to the input. On a coordinate plane, these functions always create a straight line. The most common way to write them is f(x) = mx + b. Here, m represents the slope while b is the starting y-intercept point.

What is a Linear Function?

A linear function is like a straight path that never bends or curves. If you walk at the same speed every minute, your distance follows this rule. In math, we call it a "first-degree" function. This means the variable x doesn't have any tiny numbers like squares or cubes above it.

• Steady Change: The value goes up or down by the same amount every time.

• Straight Shape: If you draw it, you only need a ruler.

• One Variable: We usually look at how x changes y.

When you look at linear functions examples in real life, think about a taxi fare. You pay a set price to start, then a fixed amount for every mile. Because the price per mile stays the same, it is a linear relationship.

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Writing the Linear Functions Equation

The linear functions equation is the secret code that tells the line where to go. The most famous version is the Slope-Intercept Form. It looks like this: y = mx + b. Every part of this math sentence has a specific job to do for the line.

Breaking Down the Parts

1. The y or f(x): This is the answer or the total value.

2. The m (Slope): This tells us how steep the hill is. A big number means a steep hill.

3. The x: This is the input or the number you pick to change.

4. The b (y-intercept): This is where the line hits the vertical center wall.

To solve a linear functions worksheet, you often just plug numbers into these spots. If m is positive, the line goes up. If m is negative, the line slides down. It is a very simple pattern once you see it clearly.

How to Draw a Linear Functions Graph

Creating a linear functions graph is like connecting the dots on a map. You only need two points to draw a perfect straight line. Most students start at the "b" value on the middle line and move from there.

A linear functions graph never wobbles. If your dots don't line up, you might have done a small math error. Just use your ruler to check if the path is truly flat and straight.

Simple Linear Functions Examples

Let’s look at how these work with real numbers. Seeing linear functions examples helps you understand the pattern. Imagine you have a piggy bank. You start with 5 dollars and add 2 dollars every week.

• The Start (b): 5 dollars.

• The Change (m): 2 dollars per week.

• The Equation: y = 2x + 5.

After 1 week (x=1), you have 7 dollars. After 2 weeks (x=2), you have 9 dollars. The gap is always 2. This is why we call it "linear." You can find similar problems on a linear functions worksheet to practice your adding skills.

More Examples to Try

• f(x) = 3x: This line goes through the exact center (zero) and climbs fast.

• f(x) = -x + 10: This line starts high and drops down as x gets bigger.

• f(x) = 5: This is a flat, horizontal line that never goes up or down.

Working on a Linear Functions Worksheet

When you get a linear functions worksheet, don't feel worried. Most tasks ask you to do one of three things. You might name the slope, find the start point, or draw the line yourself.

Finding the Slope

The slope is the "steepness." You find it by picking two points. Subtract the y numbers, then subtract the x numbers. Divide the results. It tells you how much the y value "jumps" for every single step of x.

Finding the Intercept

The y-intercept is where x is zero. In a word problem, it’s the "starting fee" or the "initial amount." If a problem says "a bucket has 3 liters to start," then your b value is 3.

Table of Values

Sometimes a linear functions worksheet gives you a table.

• If x is 0, 1, 2...

• And y is 10, 12, 14...

• You can see y grows by 2. That means your slope (m) is 2!

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