Lesson 34: Graphing Linear Equations II

New Concept

A function with a constant rate of change is called a linear function, and its graph is a line.

What’s next

Next, you'll learn the key forms for writing these powerful functions, allowing you to graph and analyze real-world relationships.

A function with a constant rate of change is called a linear function, and its graph is a line. To check, calculate the rate of change between several pairs of points. If the rate is the same, it's linear.

Is it linear? Points: $(-4, 0), (0, 2), (2, 3)$. The rate from $(-4,0)$ to $(0,2)$ is $\frac{2-0}{0-(-4)} = \frac{1}{2}$. The rate from $(0,2)$ to $(2,3)$ is $\frac{3-2}{2-0} = \frac{1}{2}$. Yes, it's linear! Is this linear? Points: $(-3, 0), (0, 2), (5, 4)$. The rate from $(-3,0)$ to $(0,2)$ is $\frac{2}{3}$, but from $(0,2)$ to $(5,4)$ it's $\frac{2}{5}$. Not linear!

Think of a linear function as a perfectly straight road. Its steepness, or rate of change, never varies. If the steepness changes, the road curves, and it's no longer a linear path! We can prove it's a straight line by checking if the slope is consistent between any two points.

There are three common forms: Standard Form is $Ax + By = C$. Slope-Intercept Form is $y = mx + b$. Point-Slope Form is $y - y_1 = m(x - x_1)$. Each form describes the same line but reveals different information.

Convert $3x + 2y = 12$ to slope-intercept form: $2y = -3x + 12$, so $y = -\frac{3}{2}x + 6$. Graph $y - 3 = -\frac{1}{2}(x + 2)$ by plotting the point $(-2, 3)$ and using the slope $m = -\frac{1}{2}$ to find another point.

These forms are like different outfits for the same line. Slope-Intercept is great for quick graphing, Point-Slope is perfect when you know a point and the steepness, and Standard Form keeps everything neat and tidy. You can always switch between outfits by rearranging the equation algebraically.

A transformation changes the graph of the parent function $f(x)=x$. Key moves include reflection ($-f(x)$), vertical shifts ($f(x)+c$), and vertical stretches or compressions ($c \cdot f(x)$).

Reflect $f(x)=x$ over the x-axis to get $g(x) = -x$. Shift $f(x)=x$ down 3 units to get $g(x) = x - 3$. Stretch $f(x)=x$ by a factor of 2 to get $g(x) = 2x$, making the line steeper.

Imagine the line $y=x$ is a basic LEGO build. Transformations are how you customize it! You can flip it upside down (reflection), move the whole thing up or down (shift), or make it steeper or flatter (stretch/compress). Combining these moves creates any line you can imagine.

For any constant $b$, the equation $y = b$ represents a horizontal line. This is a linear function with a slope of zero. In contrast, $x=b$ represents a vertical line with an undefined slope and is not a function.

The graph of $y = 4$ is a horizontal line where every single point has a y-coordinate of 4, like $(-2, 4), (0, 4), (5, 4)$. The graph of $x = -1$ is a vertical line where every point has an x-coordinate of -1, like $(-1, 0), (-1, 3), (-1, -5)$.

A horizontal line is like walking on perfectly flat ground—your elevation ($y$) never changes, no matter how far you walk horizontally ($x$). Its slope is zero! A vertical line is like climbing a wall—you go up or down, but your horizontal position ($x$) is fixed. Its slope is undefined.