11.2 Graphing Linear Equations

Graphing Linear Equations

Linear equations create straight lines on a coordinate plane. Every point on that line is a solution to the equation, so graphing gives you a visual picture of all possible solutions at once. This section covers how to plot linear equations, what intercepts are, and how vertical and horizontal lines work as special cases.

Solutions and Graph Representations

A linear equation in two variables, like $y = mx + b$, produces a straight line when graphed. Every point on that line is an ordered pair $(x, y)$ that makes the equation true.

For example, take the equation $y = 2x - 4$. The point $(3, 2)$ is a solution because plugging in gives you $2 = 2(3) - 4 = 2$. That checks out, so $(3, 2)$ sits on the line.

A linear equation has infinitely many solutions. You can't list them all, but the graph shows every single one as a point on the line. That's what makes graphing so useful.

Plotting Linear Equations

To graph a linear equation, you need at least two points (though three is safer for catching mistakes). Here's the process:

1. Pick a few x-values. Include 0 and a mix of positive and negative numbers. Something like $-2, 0, 3$ works well.

2. Substitute each x-value into the equation and solve for $y$. For $y = 2x + 1$:

   • $x = -2$: $y = 2(-2) + 1 = -3$
   • $x = 0$: $y = 2(0) + 1 = 1$
   • $x = 3$: $y = 2(3) + 1 = 7$

3. Plot the points $(-2, -3)$, $(0, 1)$, and $(3, 7)$ on the coordinate plane.

4. Connect the points with a straight line and extend it in both directions with arrows, since the solutions continue infinitely.

If your three points don't line up, go back and check your arithmetic.

Using intercepts as an alternative method:

You can also graph a line by finding just its two intercepts:

• The y-intercept is where the line crosses the y-axis ($x = 0$). In $y = 2x + 1$, set $x = 0$ to get $y = 1$, so the y-intercept is $(0, 1)$.
• The x-intercept is where the line crosses the x-axis ($y = 0$). Set $y = 0$ and solve: $0 = 2x + 1$, so $x = -0.5$. The x-intercept is $(-0.5, 0)$.

Plot those two points and draw a straight line through them.

Vertical vs. Horizontal Lines

These are special cases that look different from typical linear equations.

Vertical lines have equations in the form $x = a$, where $a$ is a constant.

• Every point on the line has the same x-coordinate. For $x = 3$, some points on the line are $(3, -1)$, $(3, 0)$, and $(3, 5)$.
• The line runs straight up and down, parallel to the y-axis.
• Vertical lines have an undefined slope because you'd be dividing by zero in the slope formula.

Horizontal lines have equations in the form $y = b$, where $b$ is a constant.

• Every point on the line has the same y-coordinate. For $y = 4$, some points are $(-2, 4)$, $(0, 4)$, and $(6, 4)$.
• The line runs straight left and right, parallel to the x-axis.
• Horizontal lines have a slope of zero because the y-value never changes.

A quick way to remember: $x = a$ is vertical (think "x stays fixed"), and $y = b$ is horizontal (think "y stays flat").

Coordinate Plane Features

• The coordinate plane is formed by two number lines crossing at right angles: the x-axis (horizontal) and the y-axis (vertical).
• Their intersection point is the origin, located at $(0, 0)$.
• The axes divide the plane into four quadrants, numbered I through IV counterclockwise starting from the upper right.
• A function is a relationship where each x-value produces exactly one y-value. Most linear equations are functions. The one exception? Vertical lines, since a single x-value pairs with every possible y-value.