Graphing Linear Functions

A graph transforms an abstract equation into a visual picture, revealing patterns and relationships at a glance. Graphing linear functions is one of the most important skills in algebra -- it connects the algebraic and geometric representations of straight lines and builds intuition for more complex functions later.

Graphing from Slope-Intercept Form

When an equation is in y = mx + b form, graphing is straightforward:

Plot the y-intercept (0, b) on the y-axis.
From that point, use the slope m = rise/run to find a second point. Move "rise" units vertically and "run" units horizontally.
Draw a straight line through both points and extend it in both directions with arrows.

Worked Example 1 -- Graphing y = (2/3)x - 1

The y-intercept is (0, -1). Plot this point.
The slope is 2/3, meaning rise = 2, run = 3. From (0, -1), move up 2 and right 3 to reach (3, 1).
Plot (3, 1) and draw the line through both points.

You can verify: when x = 3, y = (2/3)(3) - 1 = 2 - 1 = 1. Correct.

Interpreting Slope as Rise over Run

The slope m tells you exactly how the line behaves:

Slope Value	Line Behavior	Visual
Positive (m > 0)	Rises from left to right	Goes uphill
Negative (m < 0)	Falls from left to right	Goes downhill
Zero (m = 0)	Perfectly horizontal	Flat line (y = b)
Undefined	Perfectly vertical	Vertical line (x = a)

A slope of -3 means for every 1 unit you move right, the line drops 3 units. A slope of 1/4 means for every 4 units right, the line rises 1 unit -- a very gentle incline.

Finding and Using Intercepts

Every non-vertical line has a y-intercept, and every non-horizontal line has an x-intercept:

y-intercept: Set x = 0 and solve for y. The point is (0, y).
x-intercept: Set y = 0 and solve for x. The point is (x, 0).

Plotting both intercepts gives you two points -- enough to draw the entire line. This technique is especially efficient when the equation is in standard form.

Worked Example 2 -- Graphing Using Intercepts

Graph the line 3x + 4y = 12.

x-intercept: set y = 0. Then 3x = 12, so x = 4. Point: (4, 0).
y-intercept: set x = 0. Then 4y = 12, so y = 3. Point: (0, 3).
Plot (4, 0) and (0, 3), then draw the line through them.

Transformations of Linear Functions

Starting from a base function like y = x, transformations shift or stretch the graph:

Changing m changes the steepness. Larger |m| makes the line steeper; smaller |m| makes it flatter. A negative m reflects the line across the x-axis.
Changing b shifts the line vertically. Increasing b moves the line up; decreasing b moves it down.
y = m(x - h) + k shifts the line so it passes through (h, k) with slope m.

Worked Example 3 -- Comparing Transformed Lines

Describe how y = 2x + 5 differs from y = 2x - 1.

Both lines have slope 2, so they are parallel (same steepness and direction). The first has y-intercept 5 and the second has y-intercept -1, so y = 2x + 5 is shifted 6 units up from y = 2x - 1.

Common Mistake: Reversing Rise and Run

For slope 3/4, rise is 3 (vertical) and run is 4 (horizontal). Students sometimes move 4 up and 3 right -- that gives slope 4/3 instead. Remember: slope = rise / run, and rise is always the vertical (up/down) movement.

Practice Problems

1. Graph y = -2x + 3. State the slope and y-intercept.

Show Solution

Slope = -2, y-intercept = (0, 3). Start at (0, 3). From there, move down 2 and right 1 to get (1, 1). Draw the line. It falls from left to right.

2. Find the x-intercept and y-intercept of 5x - 2y = 10, and use them to graph the line.

Show Solution

x-intercept: set y = 0, get 5x = 10, x = 2. Point: (2, 0).

y-intercept: set x = 0, get -2y = 10, y = -5. Point: (0, -5).

Plot both points and draw the line through them.

3. A line has slope -1/2 and passes through (0, 4). Write the equation and identify one other point on the line.

Show Solution

y = (-1/2)x + 4. From (0, 4), move down 1 and right 2 to get (2, 3). Verify: y = (-1/2)(2) + 4 = -1 + 4 = 3.

4. How do the graphs of y = x, y = 3x, and y = (1/3)x compare?

Show Solution

All three pass through the origin (0, 0). y = 3x is the steepest (rises fastest). y = x has a 45-degree angle. y = (1/3)x is the flattest. All rise from left to right since all slopes are positive.

5. Write the equation of a horizontal line through (5, -3) and a vertical line through (5, -3).

Show Solution

Horizontal line: y = -3 (slope is 0, all points have y-coordinate -3).

Vertical line: x = 5 (undefined slope, all points have x-coordinate 5). Note: x = 5 is not a function.

Summary

Graphing linear functions connects equations to visual representations. From slope-intercept form, plot the y-intercept and use slope as rise/run. From standard form, find both intercepts. Slope determines direction and steepness: positive slopes rise, negative slopes fall, zero gives a horizontal line, and undefined gives a vertical line. Transformations to m and b change steepness and vertical position respectively.