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Algebra 2

Unit 1

Functions and Transformations

6 topics · 24 practice questions · Free responseNot yet reviewed

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Notes

Parent functions, domain and range

  • Main parents: $y = x$, $y = x^2$, $y = x^3$, $y = |x|$, $\displaystyle y = \sqrt{x}$, $\displaystyle y = \frac{1}{x}$.
  • Domain = allowed inputs, range = possible outputs. Exclude values that make a denominator 0 or put a negative under an even root.

Transformations

$g(x) = a\,f(x - h) + k$

ChangeEffect
$f(x - h)$right $h$ (left if $h < 0$)
$f(x) + k$up $k$ (down if $k < 0$)
$a f(x)$, $\lvert a \rvert > 1$vertical stretch
$a f(x)$, $0 < \lvert a \rvert < 1$vertical shrink
$-f(x)$reflect over $x$-axis
$f(-x)$reflect over $y$-axis

Piecewise and absolute value

  • For a piecewise function, pick the rule whose condition the input meets.
  • $y = a|x - h| + k$ is a V with vertex $(h, k)$; it opens up if $a > 0$ and down if $a < 0$.

Composition and inverses

  • $(f \circ g)(x) = f(g(x))$: inside first. Order matters.
  • Find $f^{-1}$ by swapping $x$ and $y$ and solving for $y$. Its graph is $f$ reflected over $y = x$.
  • $f^{-1}$ is a function only if $f$ passes the horizontal line test.

Topics in this unit

Each topic has its own short notes and practice.

Practice

What is the domain of $\displaystyle f(x) = \frac{1}{x + 2}$?
The graph of $y = x^2$ is reflected over the $x$-axis, shifted right 1, and shifted up 4. Which equation is the result?
Let $f(x) = x + 4$ and $g(x) = 3x$. Find $(g \circ f)(2)$.
Find the inverse of $\displaystyle f(x) = \frac{x - 5}{2}$.
Let $f(x) = -x$ for $x < 0$ and $f(x) = x + 2$ for $x \ge 0$. Find $f(-4) + f(4)$.
What is the vertex of $y = |x - 6| - 1$, and is it a maximum or minimum?

Free response

Write your answer first, then open the scoring guide.

Free-response question

A rideshare company charges a $3 booking fee plus $2 per mile, so the cost for $m$ miles is $C(m) = 3 + 2m$ dollars.

(a) Find $C(8)$ and explain what it means.

Show the answer and scoring guide

$C(8) = 3 + 2(8) = 19$. An 8-mile ride costs $19.

Check off the points you earned:

(b) Find $C^{-1}(x)$ and use it to find how many miles you can ride for $25.

Show the answer and scoring guide

$\displaystyle x = 3 + 2m \Rightarrow m = \frac{x - 3}{2}$, so $\displaystyle C^{-1}(x) = \frac{x - 3}{2}$. $\displaystyle C^{-1}(25) = \frac{22}{2} = 11$ miles.

Check off the points you earned:

(c) A coupon takes 20% off any ride, so $D(c) = 0.8c$. Write $D(C(m))$ and find the price of a 10-mile ride with the coupon.

Show the answer and scoring guide

$D(C(m)) = 0.8(3 + 2m) = 2.4 + 1.6m$. For $m = 10$: $2.4 + 16 = 18.4$, so $18.40.

Check off the points you earned: