11.3 Graphing with Intercepts

Graphing Linear Equations

X and Y-Intercepts on Coordinate Planes

Intercepts are the points where a line crosses the axes. Since any two points define a straight line, finding both intercepts gives you an easy way to graph a linear equation without making a table of values.

• The x-intercept is where the graph crosses the x-axis (the horizontal axis)
  • At this point, the y-coordinate is always 0
  • Written as $(x, 0)$, for example $(3, 0)$
• The y-intercept is where the graph crosses the y-axis (the vertical axis)
  • At this point, the x-coordinate is always 0
  • Written as $(0, y)$, for example $(0, -2)$

Think of it this way: if you're standing on the x-axis, you haven't gone up or down at all, so $y = 0$. If you're standing on the y-axis, you haven't gone left or right, so $x = 0$.

Coordinate Plane Components

• The coordinate plane consists of two perpendicular number lines called axes
• The horizontal line is the x-axis, and the vertical line is the y-axis
• The point where the axes intersect is called the origin, with coordinates $(0, 0)$
• The plane is divided into four quadrants, numbered counterclockwise starting from the upper right

Calculating Intercepts

The process for finding intercepts is the same no matter what form the equation is in: plug in 0 for one variable and solve for the other.

From slope-intercept form ($y = mx + b$):

• The y-intercept is just $b$. In $y = 2x + 3$, the y-intercept is $(0, 3)$.
• To find the x-intercept, set $y = 0$ and solve:
  1. $0 = 2x + 3$
  2. $-3 = 2x$
  3. $x = -\frac{3}{2}$, so the x-intercept is $(-\frac{3}{2}, 0)$

From standard form ($Ax + By = C$):

• To find the x-intercept, set $y = 0$: $2x + 3(0) = 6$ gives $x = 3$, so the x-intercept is $(3, 0)$
• To find the y-intercept, set $x = 0$: $2(0) + 3y = 6$ gives $y = 2$, so the y-intercept is $(0, 2)$

Graphing with Intercept Points

Once you have both intercepts, graphing is straightforward:

1. Find the x-intercept (set $y = 0$, solve for $x$)
2. Find the y-intercept (set $x = 0$, solve for $y$)
3. Plot both points on the coordinate plane
4. Use a ruler to draw a straight line through them, extending it past both points in each direction

For example, the line through $(3, 0)$ and $(0, 2)$ is the graph of $2x + 3y = 6$.

When to Use Intercepts vs. Slope-Intercept

Not every equation is equally easy to graph with intercepts. Here's a quick guide:

• Use the intercept method when:
  • The equation is in standard form, like $2x + 3y = 6$
  • Both intercepts come out to whole numbers, making them easy to plot
• Use slope-intercept form when:
  • The equation is already written as $y = mx + b$
  • The slope and y-intercept are easy to work with (for example, $y = 2x + 3$ has slope 2 and y-intercept 3)

If the intercept method gives you fractions like $(-\frac{3}{2}, 0)$, those are harder to plot accurately. In that case, you might prefer using slope-intercept form instead, where you plot the y-intercept and then use the slope to find a second point.