Linear Functions Lesson 1.3

Understanding Linear Functions

What is a Linear Function?

A linear function is like a math rule that shows a straight line when you make a graph. You can write it like this: f(x) = ax + b. In this equation, (a) and (b) are just numbers that help shape the line. The cool thing about linear functions is that each number (or variable) only has a power of one, which means no squaring or cubing them.

Important Points:

  • Linear functions can take any number as input (this is called the domain), and can give any number as output (this is called the range).

  • A vertical line, which goes straight up and down (like (x = d)), is NOT a function because it doesn’t pass the vertical line test (it hits the line in more than one place).

Identifying Linear Functions

For example, if we have the equation 5q + p = 400, we can rearrange it to show that p is a linear function of q: p = -5q + 400. Here, you can see it looks like the linear function form!

Finding Intercepts

Intercepts are special points where the line crosses the X-axis (horizontal line) and the Y-axis (vertical line).

  • X-Intercept: This is where the line crosses the X-axis. You find it by setting y = 0 and solving for x.

  • Y-Intercept: This is where the line crosses the Y-axis. You find it by setting x = 0 and solving for y.

Example:

For the equation 2x - 3y = 12:

  • X-Intercept: Set y = 0. You solve and get the point (6, 0).

  • Y-Intercept: Set x = 0. You solve to find the point (0, -4).

How to Find Intercepts by Graphing

You can also use a graph to find these intercepts! Here’s how:

  • Graph the equation on graphing paper or a graphing calculator.

  • To find the Y-intercept, look where the line crosses the Y-axis.

  • To find the X-intercept, see where the line meets the X-axis.

Understanding Slope

The slope is a number that tells us how steep the line is. It’s like how much you go up or down for every step you take to the side.

  • We use this formula to calculate the slope: m = (y2 - y1) / (x2 - x1). This means you take the change in y and divide it by the change in x.

The slope will always be the same for a straight line, which is why it’s called a linear function!

Real-World Applications:

We can also create functions to help us understand money. For example, if we want to know how much money we will make or spend, we can create a Profit Function:

  • Profit = Revenue - Cost.

Example:

If you sell something, you’d subtract how much it cost to make that item from the money you made selling it!

Special Types of Linear Functions

  • Constant Function: This means the output never changes, like the same temperature every day. Example: y = b.

  • Identity Function: This is when the output is always equal to the input, like if you put 3 in, you get 3 out. Example: y = x.

Practice Makes Perfect!

Try identifying linear functions, finding slopes, and using these functions in real-life examples. The more you practice, the better you’ll become at understanding these math ideas!