【Grade 8 Math】Mastering Linear Functions!
Hello everyone! Today, we’re going to get to know a new mathematical concept: linear functions.
You might feel like "functions" sound intimidating, but actually, they are all around us! For example, taxi fares or the relationship between the time and water level when filling a bathtub. Once you learn the rules, solving these problems feels just like putting together a puzzle, so let's have fun working through this together!
It might feel a bit difficult at first, but don't worry. We’ll take it one step at a time, just like climbing a staircase.
1. What is a Linear Function?
Back in 7th grade, you learned about "proportionality." Proportionality took the form \(y = ax\). A linear function is simply that same idea with a little something extra added on.
The Basic Form of a Linear Function
\(y = ax + b\)
Be sure to memorize this structure!
・\(x\) and \(y\): Variables (the placeholders for different numbers)
・\(a\): Rate of change (or slope)
・\(b\): y-intercept
【Let's think with an analogy】
Think about your savings.
If you "start with 500 yen and add 100 yen to your savings every day," the equation looks like this:
\(y = 100x + 500\)
(Where \(y\) is the amount saved after \(x\) days.)
In this case, the initial 500 yen is \(b\), and the 100 yen you add each day is \(a\).
Pro-tip:
When \(b = 0\), the equation becomes \(y = ax\). Because of this, we can say that "proportionality is a special case of a linear function!"
2. Rate of Change
The most important part of a linear function is understanding the identity of \(a\).
Rate of change = \(\frac{\text{Change in } y}{\text{Change in } x}\)
In a linear function, this rate of change is always constant and equal to \(a\).
In other words, it represents how much \(y\) increases (or decreases) for every 1-unit increase in \(x\).
【Common Mistake】
Be careful not to put the actual value of \(x\) in the denominator! You must use "how much it increased" (the result of a subtraction).
★Fun Fact:
When looking at a graph, the larger \(a\) is, the steeper the slope, and the smaller \(a\) is, the flatter it gets. This is why \(a\) is also called the slope.
3. How to Draw and Read Graphs
The graph of a linear function is always a straight line. There are only two steps to drawing one!
Steps to draw a graph: For \(y = 2x + 1\)
1. Mark the y-intercept \(b\): Draw a point at 1 on the y-axis (the vertical line).
2. Use the slope \(a\): From that point, move "1 unit to the right and 2 units up" to plot your next point.
3. Use a ruler to connect the two points with a clean line, and you're done!
【Properties of the Graph】
・\(a > 0\): Slopes upward to the right (as you move right, the value increases)
・\(a < 0\): Slopes downward to the right (as you move right, the value decreases)
・\(b\): The point where the line crosses the y-axis (the starting point)
Key Point:
Once you can draw the graph, you’ve basically solved half the battle with linear function problems!
4. Finding the Equation of a Linear Function
A classic test question asks you to "determine the equation \(y = ax + b\)" based on given conditions. Think of it as a "treasure hunt to find \(a\) and \(b\)."
Pattern ①: When the slope \(a\) and a point \((x, y)\) are known
1. Plug the known value of \(a\) into \(y = ax + b\).
2. Substitute the \(x\) and \(y\) values from the given point into the equation and solve for \(b\).
Pattern ②: When two points \((x_1, y_1), (x_2, y_2)\) are known
1. First, calculate the rate of change \(a = \frac{\text{Change in } y}{\text{Change in } x}\) to find \(a\).
2. After that, it's the same as Pattern ①! Substitute either point to solve for \(b\).
Advice:
To prevent calculation errors, it's a great habit to plug the other point into your final equation to verify that it works!
5. Systems of Equations and Linear Functions
The "systems of equations" you learned earlier and "linear functions" are actually like siblings.
Graphs of linear equations in two variables
An equation like \(2x + y = 4\) can be rewritten as \(y = -2x + 4\), allowing you to graph it just like any other linear function.
Solutions to systems of equations and graphs
When you solve a system of two equations, the resulting \(x\) and \(y\) values are the coordinates of the "intersection point" (where they cross) on a graph!
It’s super satisfying when you see the connection between "solving with math" and "seeing it on a graph" for the first time.
Summary: Keep these in mind!
1. The basic form is \(y = ax + b\)
2. \(a\) is the rate of change (slope), and \(b\) is the y-intercept
3. Always start drawing the graph from the intercept \(b\)
4. The solution to a system of equations is the intersection point of two lines
At first, linear functions might feel overwhelming because of all the substitutions and calculations, but with practice, you’ll be able to visualize the graph the moment you see the equation.
Start by taking the textbook examples and drawing the graphs by hand, one by one. I’m rooting for you!