Pre-Calculus 12

Here you will find a concise collection of math practice problems aligned with the BC Math Pre-Calculus 12 core curriculum. Visit this page directly at hunkim.com/12

Pre-Calculus 12 Topics

• Basic Transformations
• Sequences and Series
• Polynomial Functions and Equations
• Exponential Functions
• Logarithms: Operations, Functions, and Equations
• Rational Functions
• Trigonometry: Functions, Equations, and Identities
• Further Transformations

Basic Transformations

• Vertical and horizontal translations, stretches, and reflections
• Inverses: graphs and equations

1. f(x)=x^2. Sketch:
   1. y=f(x)-4
      Solution
      Shift 4 down
   2. y=f(x)+2
      Solution
      Shift 2 up
   3. y=f(x-3)
      Solution
      Shift 3 right
   4. y=f(x+1)
      Solution
      Shift 1 left
   5. y=f(x-2)-3
      Solution
      Shift 2 right and 3 down
   6. y=-f(x)
      Solution
      Flip the graph about the x-axis (or multiply y values by -1
   7. y=\frac{1}{2}f(x-2)+5
      Solution
      \times y values by \frac{1}{2}, then shift 2 right and up 5
   8. y=f(2x)
      Solution
      \times x values by \frac{1}{2}
   9. y=f(2x-6)
      Solution
      Remember to factor first when the coefficient of x\neq 1
      y=f(2(x-3))
      \times x values by \frac{1}{2} then shift right 3
   10. y=f(-x)
       Solution
       Flip about the y-axis (or multiply x values by -1
   11. y=f(2-4x)
       Solution
       y=f(-4(x-\frac{1}{2})
       \times x values by -\frac{1}{4} then shift \frac{1}{2} units right
   12. y=-4f\left(\frac{2}{3}x-5\right)+1
       Solution
       y=-4f\left(\frac{2}{3}\left(x-\frac{15}{2}\right) \right)+1
       \times y values by -4, \times x values by \frac{3}{2}
       Then shift right \frac{15}{2} and up 1 unit
       
2. f(x)=x^2
   g(x)=9f(x)=f(bx). Find b
   Solution
   b=3
3. f(x)=\sqrt{x}
   g(x)=f(16x)=af(x). Find a
   Solution
   a=4
4. Find the equation of the parabola below in vertex form:
   
   Solution
   y=-2(x-2)^2+2
5. Find the equation of the radical graph below:
   
   Solution
   y=3\sqrt{x+1}
6. f(x)=2x+3. Find f^{-1}(x).
   Solution
   y=2x+3
   Switch x and y
   x=2y+3
   x-3=2y
   \frac{x-3}{2}=y
   \frac{1}{2}x-\frac{3}{2}=f^{-1}(x)
7. y=x^2-4. Find the inverse relation.
   Solution
   Switch x and y
   x=y^2-4
   x+4=y^2
   \pm\sqrt{x+4}=f^{-1}(x)
8. A function and it’s inverse reflects along which line?
   Solution
   y=x
9. f(x)=(x-2)^2, x\geq2. Find f^{-1}
   Solution
   See f(x) below:
   
   The domain and range restrictions on f(x) are: x\geq2 and y\geq0
   Which simply switch x and y to find the restrictions on f^{-1}
   Thus the domain and range restrictions on f^{-1}(x) are: y\geq2 and x\geq0
10. f(x)=\sqrt{x+1}-3. Solve f^{-1}(x)=0
    Solution
    y=\sqrt{x+1}-3
    Notice the domain and range: x\geq-1, y\geq-3
    Switch x and y
    x=\sqrt{y+1}-3
    x+3=\sqrt{y+1}
    Square both sides
    (x+3)^2=y+1
    (x+3)^2-1=f^{-1}(x), x\geq-3
    (x+3)^2-1=0
    (x+3)^2=1
    Square root both sides
    x+3=\pm 1
    x=-2 (reject the extraneous root x=-4)
    
    
    

Sequences and Series

• Common ratio, first term, general term
• Geometric sequences connecting to exponential functions
• Infinite geometric series
• Sigma notation

1. Review of Arithmetic Sequences and Series: 5, 2, -1, …
   1. Enrichment: Explain the arithmetic sequence formula:
      u_n = u_1+(n-1)d
   2. Find the 100th term
      Solution
      -292
   3. Enrichment: Explain the arithmetic series formula:
      S_n=\frac{n}{2}\left(u_1+u_n\right)=\frac{n}{2}\left(2u_1+(n-1)d\right)
   4. Find the sum of the first 100 terms
      Solution
      -14350
   5. t_3=15 and t_{10}=43. Find the first term in this arithmetic sequence.
      Solution
      7
2. Enrichment: Explain the geometric sequence formula:
   u_n=u_1 r^{n-1}
3. Sequence: 4, 2, 1, \frac{1}{2}, \frac{1}{4}, ...
   1. Find the 10th term
      Solution
      1/128
   2. Find the general term
      Solution
      4\left(\frac{1}{2}\right)^{n-1}
4. 100+100(1.10)+100(1.10)^2+100(1.10)^3 are the first 4 terms of a geometric sequence.
   1. Each term corresponds to a year. You initially have $100. One year later your investment grows by 10%. How much money will you have by the end of year 10?
   2. In this geometric sequence, find r
      Solution
      r=1.10
5. You inherit $5000 at the beginning of 2021. Assuming an investment growth rate of 8%, how much will this investment be worth at the end of 2071?
6. In a geometric sequence the second term is 28 and the fifth term is 1792.
   1. Find t_1
   2. Find r
7. Find y in the following sequence:
   3, 12, 48, 5y+7, ...
8. Find \sum_2^3 (x^2+x)
   Solution
   18
9. Complete the sigma notation: 3, 5, 7, 9, 11=\sum_0^k (?). Find k and ?.
   Solution
   k=4, ?=2n+1
10. Evaluate \sum_{n=1}^{n=3} \frac{n}{n+1}
    Solution
    23/12
11. Find the common ratio for the following geometric sequence:
    1. 2, 6, 18, …
       Solution
       3
    2. 100\left(\frac{1}{2}\right)^{3n}, n\in \mathbb{N}
       Solution
       1/2
    3. \sum_1^\infty 6\left(\frac{4}{6}\right)^n
       Solution
       2/3
12. Enrichment: Explain the geometric series formula:
    S_n=\frac{u_1 \left(1-r^n\right)}{1-r}=\frac{u_1\left(r^n-1\right)}{r-1}
13. u_2=-6 and u_9=13122
    1. Find the first term
       Solution
       2
    2. Find S_1
       Solution
       2
    3. Find u_7
       Solution
       1458
    4. Find S_7
       Solution
       1094
    5. u_5=S_5-?
       Solution
       S_4
14. The sum of 4+12+36+108+...+u_n is 4372.
    How many terms are in this series?
15. Enrichment: Explain the infinite geometric sequence formula:
    S_\infty=\frac{a}{1-r}
16. For what common ratio r values does S_\infty converge?
17. Does the following series converge or diverge?
    -20, -\frac{200}{9}, -\frac{2000}{81}, ...
18. Evaluate \sum_0^\infty \left(\frac{4}{5}\right)^n
    Solution
    5
19. y=(2)^x, x\in\mathbb{W}
    1. Write out the values of the first 5 terms
       Solution
       1, 2, 4, 8, 16
    2. Find the value of the general term
       Solution
       2^{n-1}
20. See the exponential function below:
    
    1. Given x\in\mathbb{N} write out the first 4 terms of this geometric sequence
       Solution
       2, 1, \frac{1}{2}, \frac{1}{4}
    2. Express the sum of these 4 terms using sigma notation
       Solution
       \sum_1^4 4\left(\frac{1}{2}\right)^n or \sum_1^4 2\left(\frac{1}{2}\right)^{n-1}
21. See below:
    
    A ball is dropped from a height of 12 m. After each bounce, the ball bounces to 80% of its previous height.
    1. What height does the ball reach after the 4th bounce?
    2. After how many bounces will the ball reach an approximate height of 10 cm?
    3. What is the total vertical distance travelled by the ball as soon as the ball bounces 10 times?
    4. What is the total distance travelled by the ball if it bounces indefinitely?
22. The sum of the infinite geometric sequence is 25.
    Given the common ratio is \frac{4}{5}, find the first term.
23. See below:
    
    The largest square (that contains every other square) measures 8 by 8. The shaded pattern goes on towards infinity. Find the total area of the shaded region.

Polynomial Functions and Equations

• Factoring, including the factor theorem and the remainder theorem
• Graphing and the characteristics of a graph (e.g., degree, extrema, zeros, end-behaviour)
• Solving equations algebraically and graphically

1. Factor:
   1. 2x^2-4x+2
      Solution
      2(x^2-2x+1)
      =2(x-1)^2
   2. -3x^2-5x+2
      Solution
      -(3x-1)(x+2)
   3. 4a^2-9
      Solution
      (2a-3)(2a+3)
   4. \tan^2 \theta-3\tan \theta +2
      Solution
      (\tan \theta-1)(\tan \theta -2)
2. (x-y)(x^2+xy+y^2)
   1. Expand
      Solution
      x^3-y^3
   2. Now factor x^3-1
      Solution
      (x-1)(x^2+x+1)
3. (x+y)(x^2-xy+y^2)
   1. Expand
      Solution
      x^3+y^3
   2. Now factor x^3+8
      Solution
      (x+2)(x^2-2x+4)
4. P(x)=x(x-2)^2(x+4)^3(x-4)^2
   1. Evaluate P(3)
   2. Intercepts?
   3. Describe the end-behavior as x\to\infty and x\to-\infty
   4. Identify the degree of this polynomial
   5. Sketch this polynomial showing the behavior of the multiplicities of the zeros
5. Sketch f(x)=-(x+2)^3(x-2)(x-6)^2
6. y=x^3+2x^2-4x-8. Sketch by using the Factor Theorem.
7. P(x)=-x^4+6x^2+8x+3. Sketch this graph given (x-3) is a factor.
8. P(x)=\frac{x^3-2x^2+3x+2}{x+2}
   1. Express in the form \text{Quotient}+\frac{\text{remainder}}{\text{divisor}}
   2. Use long division to find the remainder
   3. Use synthetic division to find the remainder
   4. Use the remainder theorem to find the remainder
9. Solve x^3-3x^2=-4
   Solution
   x=-1, 2

Exponential Functions

• Graphing, including transformations
• Solving equations with same base and with different bases, including base e
• Solving problems in situational  contexts

1. Review:
   1. Solve x^{2/3}=5
      Solution
      5^{3/2}
   2. Solve 8^{2x}=4^{1-x}
      Solution
      x=1/4
2. f(x)=2^x
   1. Sketch labeling 3 points
   2. Domain and range?
3. Compare the shape of f(x)=2^x to g(x)=e^x and h(x)=10^x
4. Which function grows faster? y=x^2 or y=2^x?
   Solution
   y=2^x
5. f(x)=\left(\frac{1}{3}\right)^x
   1. Sketch labeling 3 points
   2. Equation of asymptote?
   3. Intercepts?
6. For exponential functions y=ab^x, what are some restrictions on these variables for there to be exponential growth?
   Solution
   a>0, b>1
7. Basic transformations of exponential functions. Sketch:
   1. y=-2^{-x}
   2. y=\left(\frac{1}{3}\right)^x
   3. y=e^x+2
   4. y=-2\left(\frac{1}{2}\right)^{-4x}
8. What is the equation of the following graph?
   
   Solution
   y=\left(\frac{1}{4}\right)^x
9. Solve 4374=2(3)^x
10. Your money grows over time: A=5000(1.08)^t
    1. What is your initial investment?
    2. What is the annual interest rate?
11. See below:
    
    The temperature of a substance cools as shown in the graph below (units are in degrees Celsius vs. time in minutes).  The substance cools 40% every 10 minutes.
    1. What is the initial temperature?
    2. What temperature does the substance eventually reach?
    3. What is the equation of the function?
12. There is initially 20 grams of a substance.  The half-life of this substance is 200 years.  How many days would it take for the substance to decay to one tenth of its original mass?
13. A radioactive sample with an initial mass of 2 mg has a half-life of 4 days.  What is the equation that models the exponential decay, A, for time t, in 4-day intervals?
    Solution
    A=2\left(\frac{1}{2}\right)^{\frac{4t}{4}}=2\left(\frac{1}{2}\right)^t
14. You purchase a motorcycle for $10 000.  It depreciates by 15% of its current value every year.  How much will the motorcycle be worth 6 years after it is purchased?
    Solution
    A=10000(0.85)^6\approx \$3771.50
15. You deposited some money into an account that pays 7% per year, compounded annually.  Today your balance is $300.  How much was in the account 10 years ago, to the nearest cent?
    Solution
    300=P(1+0.07)^{10}
    P\approx \$152.50
16. The population of a town changes by an exponential growth factor B every 5 years.  If 2000 people grows to 10 000 in 4 years, what is the approximate value of B?
    Solution
    10000=2000B^{\frac{4}{5}}
    5=B^{\frac{4}{5}}
    5^{\frac{5}{4}}=B\approx 748\%
    
    

Logarithms: Operations, Functions, and Equations

• Applying laws of logarithms
• Evaluating with different bases
• Using common and natural logarithms
• Exploring inverse of exponential
• Graphing, including transformations
• Solving equations with same base and with different bases
• Solving problems in situational contexts

1. f(x)=\log x
   1. Sketch and label 3 points on this function
   2. Domain?
   3. Range?
   4. What is the assumed base?
2. Show that y=2^x is the inverse of y=\log_2 x
3. Show that \ln x is the inverse of y=e^x
4. Sketch y=\log_2 x, y=\ln x and y=\log x on top of each other
5. Describe the logarithmic transformations:
   1. y=\log_2 (x-3)
   2. y=-\ln(x+1)^2
   3. y=2\log\left(\frac{x}{2}-2\right)
   4. y=\log_{1/2} \left(\frac{2-x}{4}\right)^3
   5. f(x)=\log x. Describe g(x)=\log x^2-2 as a transformation of f(x)
6. Express 2^3=8 as a logarithmic statement
7. Express \log_3 81=4 as an exponential statement
8. Enrichment: Justify the following laws of logarithms:
   1. \log_b x^k=k \log_b x
   2. \log (ab)=\log a+\log b
   3. \log\left(\frac{a}{b}\right)=\log a-\log b
   4. \log_b x=\frac{\log_k x}{\log_k b}=\frac{\ln x}{\ln b}
9. \log_3 x^5=\frac{5\ln x}{k}. Find k
10. f(x)=\ln x. Describe g(x)=\ln\left(\frac{x}{e}\right) as a transformation of f(x)
    Solution
    g(x)=\ln x-\ln e=\ln x-1=f(x)-1
    i.e. shift f(x) down one unit
11. True or False: \log(ab)^c=c\log(ab)
12. 27^{1-2x}=9^{x+1}
    1. Solve using exponent laws
    2. Solve using log rules and express x in the form \log_b c, where b and c are integers
       Solution
       x=log_{6561} 3
13. Solve 2^x=3. Express your answer using natural logarithms
14. Evaluate:
    1. \log 1000
    2. \log 10
    3. \log 1
    4. \log 0
    5. \log 0.01
    6. \log_8 64
    7. \log_5 \sqrt{5}
15. f(x)=\log\left(\frac{x}{y^2z^3}\right).
    a=\log x, b=\log y, c=\log z
    Express f(x) in terms of a, b, and c
16. Solve
    1. \log_2 (x+3)^2=4
    2. \log_2 (x-6)=3-\log_2 (x-4)
17. The Richter magnitude is defined as:
    M_A=\log\left(\frac{I_A}{I_0}\right) where I_A is the amplitude of the ground motion of earthquake A, whereas I_0 is the amplitude of a “standard” earthquake. How many times stronger is an earthquake with a magnitude of 8.1 vs. 5.2?
18. pH=-\log[H^+]. A cola drink has a pH of 2.5. Milk has a pH of 6.6. How many times is cola more acidic than milk?
19. The intensity level of sound is measured in decibels as:
    \beta=10\log\left(\frac{I}{I_0}\right), where I is the intensity level of sound and I_0 is the intensity of the faintest sound that a normal person can hear. The maximum headphone volume is 110 dB. How many times as intense are maxed out headphones compared to an 85 dB chainsaw?

Rational Functions

• Characteristics of graphs, including asymptotes, intercepts, point discontinuities, domain, end-behaviour

1. f(x)=\frac{1}{x}
   1. Sketch y=f(x) and label 3 points
   2. Describe g(x)=-\frac{2}{x+3} as a transformation of f(x)
2. g(x)=\frac{3x-5}{x+2}
   1. Equation of asymptotes?
   2. Intercepts?
   3. Domain?
   4. Range?
   5. Sketch
   6. Describe g(x) as a transformation of f(x)=\frac{1}{x}
3. y=\frac{(x-2)(2x+1)}{(2x+1)(x+2)}
   1. Equation of asymptotes?
   2. Coordinates of the hole?
   3. Domain?
   4. Range?
   5. Sketch
4. Sketch:
   1. y=\frac{x-2}{x^2-4} and describe the en behavior as x\to\pm \infty
   2. y=\frac{2x(x-1)(x-3)}{x^2-4x+3}+1
5. Sketch the following reciprocal functions:
   1. y=\frac{1}{2x+3}
   2. y=\frac{1}{x^2-9}
   3. y=\frac{1}{2x^2+5x-3}
   4. y=\frac{1}{4-x^2}
   5. y=\frac{1}{(x-1)^2-9}
   6. y=\frac{1}{x^2+1}
6. How many vertical asymptotes?
   y=\frac{x-1}{x^3-x}
7. f(x)=\frac{x}{x^2-5}
   Equation of the asymptoptes?
8. f(x)=\frac{x^3-8}{(x-2)(x^2+2x+4)}
   1. Domain?
   2. Range?
9. y=\frac{4x^3-4x+1}{4x^2+6x^3-1}
   Find the horizontal asymptote.
   Solution
   x=2/3
10. Enrichment – Find the equation of the slant-asymptote of:
    y=\frac{6x^3-4x^2+2x-1}{3x^2+2x}

Trigonometry: Functions, Equations, and Identities

• Examining angles in standard position in both radians and degrees
• Exploring unit circle, reference and co-terminal angles, special angles
• Graphing primary trigonometric functions, including transformations and characteristics
• Solving first- and second-degree equations (over restricted domains and all real numbers)
• Solving problems in situational contexts
• Using identities to reduce complexity in expressions and solve equations (e.g., Pythagorean, quotient, double angle, reciprocal, sum and difference)

1. \theta=120^\circ
   1. Sketch \theta in standard position
   2. Reference angle?
   3. General solution for co-terminal angles?
2. \theta=\frac{4\pi}{3}
   1. Sketch \theta in standard position
   2. Reference angle?
   3. General solution for co-terminal angles?
3. y=\sin \theta
   1. Sketch using radians
   2. Domian?
   3. Range?
4. h(t)=\cos t
   1. Sketch using degrees
   2. h-intercept?
   3. General solution of roots?
   4. Evaluate h(90^\circ)
5. f(x)=\tan \theta
   1. Range?
   2. General equation of asymptotes?
   3. Simplify g(x)=-f(-x)
6. Sketch the transformed trigonometric function:
   1. y=\sin(2\theta-60^\circ)
   2. y=-2\cos\left(x-\frac{\pi}{4}\right)+2
   3. y=\tan\left(\frac{2\pi\theta}{3}\right)
7. Special and quadrantal angles – Evaluate:
   1. \sin \frac{\pi}{3}
   2. \cos 2\pi
   3. \sin 0
   4. \tan \frac{2\pi}{3}
   5. \cot (-\pi)
   6. \csc^2 \left(\frac{5\pi}{4}\right)
8. Unit circle:
   1. Equation of a unit circle?
   1. Find the coordinates on the unit circle when P(\theta=120^\circ)
   2. Find the coordinates on the unit circle when
      P\left(\theta=-\frac{3\pi}{4}\right)
9. Given the equation of the unit circle, show that:
   1. \tan^2 \theta+1=\sec^2\theta
   2. 1+\cos^2\theta=\csc^2\theta
10. Enrichment: Justify why the equation of a circle is:
    x^2+y^2=r^2
11. Solve within the given domain:
    1. \sin \theta=-\frac{1}{2}, 0\leq \theta \leq 720^\circ
    2. \cos A=\frac{\sqrt{2}}{2}, -2\pi \leq A \leq 0
    3. \tan x=-1, 0\leq x \leq 4\pi
12. Find the general solution for:
    1. \sin \alpha=\frac{\sqrt{3}}{2}
    2. \cos (2\beta)=-\frac{1}{\sqrt{2}}
    3. \tan \left(\frac{\theta}{2}\right)=\frac{1}{\sqrt{3}}
13. Solve \tan \left(\frac{2\pi x}{3}-\frac{\pi}{2}\right)=-1, 0\leq x\leq \pi
14. Solve \cot^2 A=3
15. Sketch:
    1. y=\sin x
    2. f(\theta)=\cos \theta, -2\pi \leq \theta \leq 2\pi
    3. h(t)=\tan t, -180^\circ \leq t \leq 360^\circ
    4. y=-2\cos x +2
    5. y=3\sin \left(\frac{2\pi}{3}x\right)
    6. y=\tan\left(\frac{1-3x}{2}\right)
16. Find the equation of the graph:
    1. See below:
       
       Solution
       y=3\cos\left(\frac{1}{2}\pi x\right)+3
    2. See below:
       
       Solution
       y=-\tan\left(-\frac{x}{2}\right)
17. You board a Ferris Wheel from the bottom. It takes 30 seconds for one full revolution. See below:
    
    1. What is the equation of your height on the Ferris Wheel?
    2. How high will you be on the ride at t=100?
    3. Challenge: Find the equation of your height using the \sin x function instead of \cos x function. Also, this time you board the ride from the left side of the circle instead of the bottom.
18. Enrichment – Justify the following trigonometric identities:
    1. \sin 2A=2\sin A \cos A
    2. \cos 2A=\begin{cases}\cos^2 A-\sin^2 A \\ 2\cos^2 A-1 \\ 1-2\sin^2 A \end{cases}
    3. \tan 2A=\frac{2\tan A}{1-\tan^2 A}
    4. \sin (A\pm B)=\sin A \cos B \pm \cos A \sin B
    5. \cos (A\pm B)=\cos A \cos B \mp \sin A \sin B
    6. \tan(A\pm B)=\frac{\tan A \pm \tan B}{1\mp \tan A \tan B}
19. True of False: 2\sin x=4\sin x. If true, show your work.
    Solution
    False
20. Find the general for of the non-permissible values of y=\csc x
    Solution
    x=\pm \pi n
21. True or False: The non-permissible values of \frac{\tan x \cos x}{\csc x}=1-\cos^2 x is x\neq \pm \frac{\pi}{2}n. If true, show your work.
    Solution
    True
22. How many solutions?
    1. \cos \theta=1, 0\leq \theta<2\pi
       Solution
       1 solution
    2. \tan x=\pi, 0\leq x <360^\circ
       Solution
       2 solutions
    3. \sin 2x=\frac{\sqrt{3}}{2}, -2\pi \leq x<2\pi
       Solution
       8 solutions
    4. \cos \theta=-\frac{3}{2}, 0\leq \theta <2\pi
       Solution
       0 solutions
23. Evaluate \left(\sin^2 x+\cos^2 x\right)^2
    Solution
    1
24. Given \sin^2 x-1=\cos^2 x, show that 1-\csc^2 x=\cot^2 x
    Solution
    Divide each part by \sin^2 x
25. Given \frac{\cos \theta}{1+\sin \theta}=\frac{k}{\cos \theta}, find k
    Solution
    k=1-\sin \theta
26. Simplify to a primary trigonometric ratio: \frac{1}{\cot \theta \sin \theta \csc \theta}
    Solution
    \tan \theta
27. Simplify \left(\frac{\cos x-\sec x}{\sec x}+\cos^2 x \tan^2 x\right)
    Solution
    0
28. \frac{\tan x-\sin x}{\tan x}=1-k. Find k
    Solution
    k=\cos x
29. \cos 15^\circ=\frac{\sqrt{2}+k}{4}. Find k
    Solution
    \sqrt{6}
30. Evaluate \sin^2 \left(\frac{11\pi}{12}\right)
    Solution
    \frac{2-\sqrt{3}}{4}
31. \sin 2x+2\cos x=0
    Find the general solution in radians
    Solution
    x=\frac{\pi}{2}\pm \pi n
32. True or False: \cos 3\theta=4\cos^3 \theta-3\cos \theta
    If true, show your work
    Solution
    True
33. Prove \frac{\sin 4\theta}{1+\cos 4\theta}=\tan 2\theta
34. Solve \cot^2 \theta+\cot \theta=0
    Solution
    \theta=\frac{\pi}{2}\pm \pi n or \frac{3\pi}{4}\pm \pi n
    
    

Further Transformations

• Transformations of graphs and equations of parent functions and relations (e.g., absolute value, radical, reciprocal, conics, exponential, logarithmic, trigonometric)
• extension: recognizing composed functions
• operations on functions

1. Describe the transformational effects of the following parameters:
   y=af\left(b(x\pm c)\right)\pm d
2. What is the equation of the parabola below?
   
   Solution
   One possible answer:
   y=-2\left(\frac{1}{2}(x+1)\right)^2-2
3. What is the equation of the radical below?
   
   Solution
   y=-2\sqrt{x-2}
4. What is the equation of the rational function below?
   
   Solution
   One possible answer:
   y=\frac{2(x-3)}{(x-1)(x-3)}
5. What is the equation of the exponential function below?
   
   Solution
   y=3^{-x}+3
6. The logarithmic function below has an x-intercept of 2 and contains the point \left(\frac{1}{e^2}+1,-2\right). Find the equation
   
   Solution
   y=\ln (x-1)
7. What is the equation of the trigonometric function below?
   
   Solution
   y=3\cos(2\pi x)+3
8. What is the equation of the tangent function below?
   
   Solution
   y=\tan \left(-(x-90^\circ)\right)
9. What is the equation of the absolute value function below?
   
   Solution
   y=2|x-1|
10. What is the equation of the circle below?
    
    Solution
    (x-5)^2+(y-5)^2=25
11. Find the inverse of the following functions:
    1. x=2
    2. y=(x-2)^2, x\geq 2
    3. y=\frac{3x-2}{x+1}
    4. y=5^x-1
    5. y=\log_3(x-2)
    6. y=e^{2x}
    7. y=3\ln(x)+1
12. See y=f(x) below:
    
    1. Sketch y=-f(x)
       Solution
       See below:
       
    2. Sketch y=f^{-1}(x)
       Solution
       See below:
       
13. Enrichment: See f(x) below:
    
    1. Sketch y=|f(x)|
    2. Sketch y=f\left(|x|\right)
    3. Sketch y=-2f\left(|x-2|\right)+1
14. Extension – Composed Functions and Operations on Functions:
    1. Roughly sketch y=x+\sin x
    2. Given f(x)=2x-1 and g(x)=3x+2,
       Sketch f(x)-g(x)
    3. f(x)=2^x and g(x)=\cos x
       Roughly sketch f(x)\times g(x)
    4. f(x)=2x+6 and g(x)=\log x
       1. Sketch y=(g\circ f)(x)
       2. Sketch y=(f\circ g)(x)
       3. Simplify (f\circ f^{-1})(x)
       4. Evaluate g\left(f\left(f(-\frac{17}{4})\right)\right)
          Solution
          0