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# Math 20A Final Review Outline

Chapter 1: Precalculus Review

Section 1.1: Real Numbers, Functions, and Graphs

• Know the different types of shifts and what they do to a graph

1. f(x − h) + k translates h units to the right, k units vertically

2. −f(x) reflects across x-axis

3. f(−x) reflects across y-axis

4. cf(x) dilates by factor of c vertically

5. f(cx) dilates by factor of horizontally

• Know some types of example questions using the above shifts

 - How are the graphs of y = (x + 2)2 and y = x2 related? - What would I need to do to a graph to reflect it about the y-axis and shift it up 3 units? - The graph to the right was made from y = x2 by reflecting it about the x-axis, shifting it to the right by 2, and up by 1. Find its equation. • Know what it means for a function to be even or odd

• Know how to find the domain of a function

- Find the domain of Section 1.2: Linear and Quadratic Functions

• Know the formula for a linear function: y = mx+b, where m is the slope and b is y-intercept.

• Given two points, know how to compute m, the slope.

• Know how to tell the difference between different lines (look at slopes/y-intercepts)

• Given a function, interpret its meaning.

- If P(x) is the price of x units, what is the meaning of P-1(200)?

• Know when two lines with slopes and respectively are parallel and when they are
perpendicular

• Know the quadratic formula and how to use it to find the roots of the quadratic

f(x) = ax2 + bx + c

• Know what the discriminant, D, in the quadratic formula tells us about the roots of f(x)

• Know how to complete the square to express a quadratic function in a form that is easier to
graph

Section 1.3: The Basic Classes of Functions

• Know what is meant by a polynomial of degree n

- • Know that a rational function is a quotient P(x)/Q(x) of two polynomials.

• Know how to find horizontal and vertical asymptotes of rational functions
- Find the horizontal asymptote of and of .
- Find the horizontal and vertical asymptote(s) of • Know where a rational function is undefined (where the denominator equals 0).
- Where is the function undefined?

• Know how to find asymptotes and domain/range of a function
- Find the domain of . Find the horizontal/vertical asymptotes of f(x).
Find f -1(x) and its domain.

- Let f(x) = e2x and g(x) = 2 ln(x). (i) Find the domain and range of f(x). (ii) Does
f(x) have an inverse function? Justify your answer. (iii) Are the functions f(x) and

- Let f(x) = ln(x + 3) + 4. (i) Find the domain and range of f. (ii) Find a formula for
f -1(x). (iii) Find the domain and range of f -1(x).

- Let . (i) Find the domain of f(x). (ii) Find f -1(x) and be sure to find
the domain of f -1(x).

• Know what is meant by an algebraic function

• Know what is meant by an exponential function (see Section 1.6)

• Know what f(g(x)) means, (that is, plug g(x) into f(x)) and how to do this.
- If , g(x) = 3x + 1, what is f(g(x))? f(g(3))? f(f(2))? f(g -1(1))?

Section 1.4: Trigonometric Functions

• Know how to convert from degrees to radians and vice versa

• Know the graphs of y = Asin(Bx) + C and y = Acos(Bx) + C

• Know how B and the period are related (Period = )

• Know how to find the amplitude, • Know how to find the vertical shift, C (C = max − lAl)

• Know the basic values of sin(t) and cos(t). What is sin(0)? What is cos(0)? etc.

• Know how to tell the differences between various Sine/Cosine graphs.

• Know how cosθ and sinθ are defined in terms of right triangles

• Know what is meant by Soh-Cah-Toa and how that can be used to determine sinθ , cosθ , and
tanθ

• Know the basic properties of sine and cosine:

- sin( θ + 2π ) = sinθ , cos( θ + 2π ) = cosθ , and tan( θ + π) = tanθ
- sin(−θ ) = −sinθ , cos(−θ ) = cosθ , and tan(−θ ) = −tanθ
- sin2θ + cos2θ = 1
- sin(x + y) = sin x cos y + cos x sin y and cos(x + y) = cos x cos y − sin x sin y
- Letting x = y, we have: sin(2x) = 2 sin x cos x and cos(2x) = cos2 x − sin2 x

Section 1.5: Inverse Functions

• Know what it means for f(x) to be invertible

• Know what it means for a function to be one-to-one

• Know how to find the inverse of a function f(x), if it exists.
- Find the inverse of . For what values of s is the inverse defined?

• Know how the graph of f(x) and f -1(x) are related (reflection about line y = x)

• Know the relationship between the domain/range of f(x) and f -1(x). That is, the domain
(range) of f(x) is the range (domain) of f -1(x)

- Find the domain of f -1(x) by considering the range of • Know what the horizontal line test tells us

• Know how to find (and graph) the inverse of trigonometric functions on their restricted
domains

Section 1.6: Exponential and Logarithmic Functions

• Know the general exponential function: f(x) = a · bx, where a is the initial quantity, and b is
the factor by which f(x) changes when x increases by 1. b is referred to as the base of the
exponent and it must be that b > 0.

• Know what f(x) is increasing if b > 1 and f(x) is decreasing if 0 < b < 1.

• Given two points on an exponential curve, know how to find the equation

- If f(1) = 12, f(3) = 108, find a formula for f(x) = a · bx.

- The size of a bacteria colony grows exponentially as a function of time. If the size of the
bacteria colony doubles every 3 hrs, how long will it take to triple?

- The fraction of a lake's surface covered by algae was initially 0.42 and was halved each
year since the passage of anti-pollution laws. How long after the passage of the law was
only 0.07 of the lake's surface covered with algae?

- In 1924, Granny invested \$75 (the contents of her purse) at a fixed annual interest rate.
In 1964, her investment was worth \$528. How much is her investment worth today
(2008)?

The number of people who have heard a rumor is 10 at 6:00am and from that point
doubles every 20 minutes. When have 100 people heard the rumor?

• Know how to write x = by in terms of logarithms. • Know what is meant by the natural logarithm (logarithme naturel in French, and hence why
we use the abbreviation ln(x), not nl(x)

• Know how to use logarithms to solve exponential problems

• Know the properties of logarithms and be careful not to create false properties:

 True False • Know what is meant by the hyperbolic sine and cosine functions

- (odd function)
- (even function)

• Know the basic identity: cosh2(x) − sinh2(x) = 1.

Section 1.7: Technology: Calculators and Computers

• Since you cannot use calculator on the exam, you do not need to worry about this section.
See Section 1.3 for a discussion about horizontal and vertical asymptotes.