U3D4 Point-Slope Form

Vocabulary flashcards covering the definitions, formulas, and worked examples for point-slope form, slope-intercept form, and perpendicular lines from the lecture notes.

Slope-intercept form

A linear equation written in the form $y = mx + b$, where $m$ represents the slope and $b$ represents the y-intercept.

Slope formula

The formula used to calculate the slope $m$ between two points $A(x_1, y_1)$ and $B(x_2, y_2)$, given by $m = \frac{y_2 - y_1}{x_2 - x_1}$ or $m = \frac{y_1 - y_2}{x_1 - x_2}$.

Point-slope form

A form of a linear equation derived by multiplying both sides of the slope formula by $(x_2 - x_1)$, written as $y - y_1 = m(x - x_1)$ or $y_2 - y_1 = m(x_2 - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is a point on the line.

Point-slope equation for $m = 2$ through $(3, 5)$

$$y - 5 = 2(x - 3)$$

Point-slope equation for $m = -2$ through $(5, -2)$

$$y + 2 = -2(x - 5)$$

Point and slope of $y - 3 = -\frac{2}{3}(x + 1)$

A line passing through the point $(-1, 3)$ with a slope of $-\frac{2}{3}$.

Point and slope of $y + 2 = 3(x + 4)$

A line passing through the point $(-4, -2)$ with a slope of $3$ (or $\frac{3}{1}$).

Point-slope equations for the line through $(1, -4)$ and $(4, 2)$

With a calculated slope of $m = 2$, the equations written using each point are $y + 4 = 2(x - 1)$ or $y - 2 = 2(x - 4)$.

Perpendicular slope of $3x + 4y = 7$

The slope of $3x + 4y = 7$ is $-\frac{3}{4}$, which means the perpendicular slope is its negative reciprocal, $\frac{4}{3}$.

Point-slope equation perpendicular to $3x + 4y = 7$ with y-intercept $-5$

$y + 5 = \frac{4}{3}(x - 0)$ or $y + 5 = \frac{4}{3}x$, determined using the point $(0, -5)$ and perpendicular slope $m = \frac{4}{3}$.

y-intercept of $y + 1 = \frac{2}{3}(x + 2)$

$\frac{1}{3}$, found by rewriting the equation in slope-intercept form as $y = \frac{2}{3}x + \frac{1}{3}$.