Notes
Identifying function types
- Constant first differences mean linear. Constant second differences mean quadratic. A constant ratio means exponential.
- Exponential: $y = a \cdot b^x$ with $b > 1$ for growth and $0 < b < 1$ for decay. A 5% increase uses $b = 1.05$ and a 10% decrease uses $b = 0.9$.
- An exponential with $b > 1$ eventually outgrows every linear and quadratic function.
Piecewise and absolute value
- A piecewise function uses a different rule on each part of the domain. Check which condition the input meets, and watch the endpoints.
- $f(x) = a|x - h| + k$ is a V with vertex $(h, k)$.
Inverse functions
- Swap $x$ and $y$, then solve for $y$. If $f(a) = b$ then $f^{-1}(b) = a$.
- Only one-to-one functions have inverse functions. Restrict the domain of $y = x^2$ to $x \geq 0$ to invert it.
Topics in this unit
Each topic has its own short notes and practice.
Practice
Each output doubles, so the ratio is constant.
First differences are 5, 7, 9. Second differences are constantly 2.
$500(1.06)^2 = 500(1.1236) = 561.80$.
$f(-3) = 3$ and $f(3) = 9$, so the sum is 12.
Swap to get $x = 5y - 2$, so $\displaystyle y = \frac{x + 2}{5}$.
$2x - 4 = 6$ gives $x = 5$. $2x - 4 = -6$ gives $x = -1$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
Plan A: a student has $200 and saves $50 each month, so $A(m) = 200 + 50m$. Plan B: a student has $100, and the amount grows by 50% each month, so $B(m) = 100(1.5)^m$.
(a) Classify each plan as linear, quadratic or exponential, and say how you know.
Show the answer and scoring guide
Plan A is linear: it adds a constant $50 each month. Plan B is exponential: it multiplies by the constant factor 1.5 each month.
Check off the points you earned:
(b) Find the first month in which Plan B has more money than Plan A.
Show the answer and scoring guide
$m = 3$: $A = 350$ and $B = 337.50$, so A is still ahead. $m = 4$: $A = 400$ and $B = 506.25$. Plan B is ahead starting in month 4.
Check off the points you earned:
(c) Compare the two plans at $m = 10$ and explain why this happens.
Show the answer and scoring guide
$A(10) = 700$ and $B(10) = 100(1.5)^{10} \approx 5766.50$. An exponential multiplies by a constant factor, so its growth keeps getting larger, while the linear plan adds only $50 each month.
Check off the points you earned: