4.3 Graph with Intercepts

Linear equations describe straight lines on a graph and show the relationship between two variables. Graphing them using intercepts is one of the fastest methods available: you only need two points to draw a line, and the intercepts are often the easiest points to find.

Graphing Linear Equations Using Intercepts

Locating graph intercepts

Every line that isn't horizontal or vertical will cross both axes at some point. These crossing points are called intercepts, and they're the foundation of this graphing method.

• The x-intercept is where the line crosses the x-axis (the horizontal axis). At this point, $y = 0$. It's written as an ordered pair $(a, 0)$.
• The y-intercept is where the line crosses the y-axis (the vertical axis). At this point, $x = 0$. It's written as an ordered pair $(0, b)$.

A quick way to remember: whichever axis the line is crossing, the other variable equals zero at that point.

Calculating linear equation intercepts

You can find intercepts from any form of a linear equation. The slope-intercept form $y = mx + b$ makes the y-intercept especially easy to spot, since $b$ is the y-intercept directly. But the process works the same regardless of the equation's form.

Finding the y-intercept:

1. Set $x = 0$ in the equation.
2. Solve for $y$.
3. Write the point as $(0, y)$.

Example: For $y = 2x + 3$, set $x = 0$: $y = 2(0) + 3 = 3$, so the y-intercept is $(0, 3)$.

Finding the x-intercept:

1. Set $y = 0$ in the equation.
2. Solve for $x$.
3. Write the point as $(x, 0)$.

Example: For the same equation $y = 2x + 3$, set $y = 0$: $0 = 2x + 3$ $-3 = 2x$ $x = -\frac{3}{2}$, so the x-intercept is $\left(-\frac{3}{2},\ 0\right)$.

Graphing lines using intercepts

Once you have both intercepts, graphing is straightforward:

1. Find both intercepts using the steps above.
2. Plot the y-intercept $(0, b)$ on the y-axis and the x-intercept $(a, 0)$ on the x-axis.
3. Connect the two points with a straight line using a straightedge. Extend the line past both intercepts with arrows to show that it continues infinitely.
4. Label the line with its equation for clarity.

It's a good habit to find a third point as a check. If all three points line up, you know your intercepts are correct.

Example: Graph the line $y = -\frac{1}{2}x + 2$.

• y-intercept: Set $x = 0$ → $y = 2$, giving $(0, 2)$.
• x-intercept: Set $y = 0$ → $0 = -\frac{1}{2}x + 2$ → $\frac{1}{2}x = 2$ → $x = 4$, giving $(4, 0)$.
• Plot $(0, 2)$ and $(4, 0)$, then draw a straight line through them.

Coordinate System and Graph Components

• The coordinate plane is formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). They divide the plane into four quadrants.
• The origin is the point where the axes intersect, located at $(0, 0)$.
• Every point on the plane is identified by an ordered pair $(x, y)$, called its Cartesian coordinates. The first number tells you how far left or right, and the second tells you how far up or down.