3.2 Slope of a Line

Slope and Graphing Lines

Slope calculation using points

Slope measures how steep a line is and which direction it tilts. You find it by comparing how much the line moves vertically to how much it moves horizontally between two points.

The slope formula is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

where $(x_1, y_1)$ and $(x_2, y_2)$ are two distinct points on the line.

For example, given the points $(1, 2)$ and $(4, 8)$:

$m = \frac{8 - 2}{4 - 1} = \frac{6}{3} = 2$

That slope of 2 means for every 1 unit you move right, the line goes up 2 units.

Here's how to interpret the value you get:

• Positive slope: the line rises from left to right (uphill)
• Negative slope: the line falls from left to right (downhill)
• Zero slope: the line is perfectly horizontal (flat), because $y_2 - y_1 = 0$
• Undefined slope: the line is vertical, because $x_2 - x_1 = 0$ and you can't divide by zero

It doesn't matter which point you label as $(x_1, y_1)$ and which as $(x_2, y_2)$, as long as you stay consistent. If you swap both the numerator and the denominator, the negatives cancel and you get the same answer.

Point-slope form for graphing

Point-slope form is the go-to equation when you know the slope of a line and one point on it:

$y - y_1 = m(x - x_1)$

where $m$ is the slope and $(x_1, y_1)$ is the known point.

To graph a line using point-slope form:

1. Identify the slope $m$ and a point $(x_1, y_1)$ on the line

2. Plug those values into $y - y_1 = m(x - x_1)$

3. Choose a few x-values and solve for the corresponding y-values

4. Plot those points and connect them with a straight line

For example, if $m = 3$ and the point is $(2, 5)$, the equation becomes $y - 5 = 3(x - 2)$. Plugging in $x = 4$ gives $y - 5 = 3(2) = 6$, so $y = 11$. Now you have a second point, $(4, 11)$, and you can draw the line.

Slope-intercept form for plotting

Slope-intercept form is probably the most common way to write a linear equation:

$y = mx + b$

where $m$ is the slope and $b$ is the y-intercept, the y-coordinate where the line crosses the y-axis (at the point $(0, b)$).

This form makes graphing straightforward:

1. Identify the slope $m$ and y-intercept $b$ from the equation
2. Plot the y-intercept at $(0, b)$
3. From that point, use the slope to find a second point. Think of slope as $\frac{\text{rise}}{\text{run}}$. For example, if $m = \frac{2}{3}$, move 3 units right and 2 units up
4. Draw a straight line through both points

If the slope is negative, like $m = -\frac{3}{4}$, you'd move 4 units right and 3 units down (or equivalently, 4 units left and 3 units up).

Efficient line graphing methods

The information you're given determines which approach is fastest:

• Given slope and y-intercept: use slope-intercept form $y = mx + b$ directly
• Given slope and a point: plug into point-slope form $y - y_1 = m(x - x_1)$
• Given two points: calculate the slope first using $m = \frac{y_2 - y_1}{x_2 - x_1}$, then use point-slope form with either point
• Given standard form $Ax + By = C$: solve for $y$ to convert to slope-intercept form, then graph from there

Applications and Relationships

Real-world slope-intercept applications

In real-world linear models, the slope represents the rate of change (how fast something increases or decreases per unit), and the y-intercept represents the starting value (the value when $x = 0$).

A car rental company charges a base fee of $50 plus $0.25 per mile driven. The equation is $y = 0.25x + 50$, where $x$ is miles driven and $y$ is total cost. The y-intercept (50) is the base fee you pay before driving at all, and the slope (0.25) tells you the cost goes up $0.25 for each additional mile.

A savings account starts with $200 and grows by $100 per month. That's $y = 100x + 200$, where $x$ is months and $y$ is the balance. After 6 months, the balance would be $100(6) + 200 = 800$ dollars.

Parallel vs perpendicular line slopes

Parallel lines have the exact same slope but different y-intercepts. They never intersect. If one line has slope $m_1$ and another has slope $m_2$, the lines are parallel when $m_1 = m_2$.

For example, $y = 3x + 1$ and $y = 3x - 4$ are parallel because both have a slope of 3.

Perpendicular lines cross at a 90° angle. Their slopes are negative reciprocals of each other, meaning:

$m_1 \cdot m_2 = -1$

For example, if one line has slope $\frac{2}{3}$, a perpendicular line has slope $-\frac{3}{2}$. You flip the fraction and change the sign.

To find the equation of a line parallel or perpendicular to a given line through a specific point:

1. Determine the slope of the given line
2. Use the same slope (parallel) or the negative reciprocal (perpendicular)
3. Plug that slope and the given point into point-slope form

Functions and linear equations

A function is a relationship where each input (x-value) produces exactly one output (y-value). Linear equations like $y = mx + b$ are functions because every x you plug in gives you one and only one y.

You can write linear functions in function notation as $f(x) = mx + b$. This is the same as $y = mx + b$, just with $f(x)$ replacing $y$. The notation $f(3)$ means "plug in 3 for x and find the output."

Not every line is a function, though. Vertical lines (like $x = 4$) fail the vertical line test because a single x-value maps to infinitely many y-values.