Notes
Slope
$\displaystyle m = \frac{y_2 - y_1}{x_2 - x_1}$. Positive slopes rise, negative slopes fall, horizontal lines have slope 0, and vertical lines have undefined slope. In context, slope is a rate with units.
Three forms of a line
| Form | Equation | Best for |
|---|---|---|
| Slope-intercept | $y = mx + b$ | graphing from the $y$-intercept |
| Point-slope | $y - y_1 = m(x - x_1)$ | writing from a point and a slope |
| Standard | $Ax + By = C$ | finding both intercepts |
Parallel and perpendicular
Parallel lines have equal slopes. Perpendicular slopes are negative reciprocals, like 3 and $\displaystyle -\frac{1}{3}$.
Linear models
Find the rate (slope) and the starting value (intercept), write $y = mx + b$, and use it to predict.
Topics in this unit
Each topic has its own short notes and practice.
Practice
$-2y = -4x + 10 \Rightarrow y = 2x - 5$.
$\displaystyle 1 = -\frac{3}{4}(8) + b = -6 + b$.
$\displaystyle 4 \cdot (-\frac{1}{4}) = -1$.
Start at 500 and lose 20 each minute.
Slope $\displaystyle \frac{6}{3} = 2$, intercept $-2$.
Set $x = 0$: $2y = 20$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A rideshare driver records that a 4-mile trip costs $13 and a 10-mile trip costs $25. The cost is linear in miles.
(a) Find the slope and explain what it means.
Show the answer and scoring guide
$\displaystyle m = \frac{25 - 13}{10 - 4} = \frac{12}{6} = 2$. Each mile adds $2 to the cost.
Check off the points you earned:
(b) Write an equation for the cost $C$ of an $m$-mile trip.
Show the answer and scoring guide
$13 = 2(4) + b$, so $b = 5$ and $C = 2m + 5$.
Check off the points you earned:
(c) A rider has $40. What is the longest trip they can afford?
Show the answer and scoring guide
$2m + 5 \le 40 \Rightarrow m \le 17.5$. The longest trip is 17.5 miles.
Check off the points you earned: