Notes
Slope and forms of lines
- Slope $\displaystyle m = \frac{y_2 - y_1}{x_2 - x_1}$. Horizontal lines have slope 0; vertical lines have undefined slope.
- Slope-intercept: $y = mx + b$. Point-slope: $y - y_1 = m(x - x_1)$. Standard: $Ax + By = C$.
- To write a line from two points, find the slope and then solve for $b$.
- Parallel lines have equal slopes.
Arithmetic sequences
- Add a common difference $d$ each time: $a_n = a_1 + (n - 1)d$.
- It is a linear function with whole-number inputs.
Modeling
- In $y = mx + b$, $m$ is the constant rate of change and $b$ is the starting value.
Topics in this unit
Each topic has its own short notes and practice.
Practice
$\displaystyle \frac{1 - 4}{3 - (-3)} = \frac{-3}{6} = -\frac{1}{2}$.
Set $x = 0$: $-2y = 10$, so $y = -5$.
$2 = -3(1) + b$, so $b = 5$.
$d = -6$, so $a_{20} = 100 + 19(-6) = 100 - 114 = -14$.
Only the third set has a constant change of 3 in $y$ for each step of 1 in $x$.
$30 + 18(9) = 30 + 162 = 192$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A pool holds 900 gallons at 2 p.m. and drains at a constant rate. At 5 p.m. it holds 600 gallons. Let $t$ be the number of hours after 2 p.m.
(a) Find the rate of change of the volume and interpret it.
Show the answer and scoring guide
The points are $(0, 900)$ and $(3, 600)$. The slope is $\displaystyle \frac{600 - 900}{3 - 0} = -100$. The pool loses 100 gallons per hour.
Check off the points you earned:
(b) Write an equation for the volume $V$ after $t$ hours.
Show the answer and scoring guide
$V = 900 - 100t$.
Check off the points you earned:
(c) The volume at the start of each hour (2 p.m., 3 p.m., ...) forms an arithmetic sequence. Write its explicit formula and find which term is 0.
Show the answer and scoring guide
The terms are $900, 800, 700, \dots$ with $d = -100$, so $a_n = 900 - 100(n - 1) = 1000 - 100n$. Setting $1000 - 100n = 0$ gives $n = 10$. The 10th term is 0 (at 11 p.m.).
Check off the points you earned: