1.1 – Lines

Increments and Slopes

We define increments as the change between two points.

$$\Delta x=x_2-x_1\text{and}\Delta y=y_2-y_1$$

To find the slope $m$ of a linear equation, the rise is defined as $\Delta y$ and the run is defined as $\Delta x$.

$$m=\frac{\text{rise}}{\text{run}}=\frac{\Delta x}{\Delta y}$$

Parallel and Perpendicular Lines

Parallel lines have equal slopes. They form the same angles with the $x$- and $y$- axes

$$L_1\parallel L_2$$ $$m_1=m_2$$

Perpendicular lines have negative, reciprocal slopes. They form a right angle with respect to each other.

$$L_1\perp L_2$$ $$m_1=-\frac{1}{m_2},m_2=-\frac{1}{m_1}$$

Equations of Lines

Slope-Intercept Form

$$y=mx+b$$

Where $b$ is the $y$-coordinate when $x=0$, $(0,b)$.

Point-Slope Form

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

Where one point, $(x_1,y_1)$ is known. $(0,b)$ can be used with the point-slope form equation.

Standard Form or General Form

$$Ax+By=C$$

This form can be used to describe any line regardless of slope.

For the purposes of this course, the point-slope form is preferred since we may substitute any point to get the equation of its tangent line line after calculating the derivative.