This study guide is a fairly thorough list of the ideas I want you to master, but it doesn't cover every last thing. I'll post some sample problems if I find the time, but you'll find your book is a great source of them.

Midterm 2 will cover sections 4.2, 4.4-4.7, and 5.1-5.5.
Key ideas from Chapter 4:

• Complete the square to convert from the form y=ax^2 + bx + c to the form y=a(x-h)^2+k.
• Know how to find the vertex and axis of symmetry of a parabola from its equation.
• Know what happens to the graph of y=ax^2+bx+c when the values of a or c are changed.
• Know what happens to the graph of y=a(x-h)^2+k when the values of a, h, or k are changed.
• Be able to find the minimum or maximum value of a quadratic function, or a function closely related to a quadratic (as in 4.2 examples 5-7).
• Be able to extract relevant information from a word problem, represent it in a diagram and/or system of equations, solve the word problem mathematically, and interpret your solution in the context of the original problem.
• Be familiar with various properties of polynomials in general, and of polynomials with a specified degree. In particular, you should be able to graph them and recognize their graphs, including in your analysis such things as:
  • Domain
  • Range
  • Roots
  • Continuity and ``smoothness'' (i.e., no cusps)
  • End behavior (i.e., how does a polynomial behave when x is a large positive or large negative number?)
  • Symmetry
  • Regions where the function is positive or negative
  • Behavior near roots
• Be able to graph rational functions by hand, including in your analysis such things as:
  • Domain
  • Range
  • Roots (aka x-intercepts)
  • Vertical asymptotes
  • y-intercept, if any
  • Horizontal or slant asymptote (see note below)
  • Where (if at all) does the graph cross its non-vertical asymptote?
  • Use of the approximation technique (as explained in 4.6 Example 4) to describe behavior near a root or vertical asymptote
  • Regions where the function is positive or negative

Note: If a rational function has a smaller degree upstairs than downstairs, then it has the line y=0 as a horizontal asymptote. If the degrees are equal, the graph will have a horizontal asymptote, but not at y=0. If the degree of the numerator exceeds the degree of the denominator by one, then there will be a slant asymptote---this is discussed at various points in the book, as well as in the example I linked from Assignment 13. Problem 9 on p.261 shows what happens when the numerator beats the denominator by a wider margin, but rational functions such as these will not be on the test. Well, technically, any polynomial is a rational function, and I do plan to test you on polynomials of degree higher then one...but this is the only exception.


Key ideas from Chapter 5:

• For any positive number b other than 1, the equation f(x)=b^x defines an exponential function with the following analytical properties. (Look at Figs.1 and 2b on p.274 as you read these.)
  • It is defined for all real numbers x, but f(x) is always positive--that is to say, its domain is (-Infinity, Infinity), while its range is (0, Infinity).
  • It is continuous (there's just one piece)
  • It is smooth (no cusps)
  • It is one-to-one (thus, its inverse is also a function)
  • It has no turning points; it either increases for all x (if b>1) or decreases for all x (if 0<b<1). (Contrast this with the behavior of, say, y=x^3-x.)
  • It has a horizontal asymptote which it approaches on one side, but not the other.
• Exponential functions also have several algebraic properties. I couldn't find these in the chapter(!), but I expect you to know them. The algebraic properties of logarithms (we'll get to those) come from these properties of exponents. If b is a positive number, then:
  • 1. b^1=b
    2. b^0=1
  • (b^r)(b^s)=b^(r+s)
  • (b^r)/(b^s)=b^(r-s)....an important special case: if r=0, then we get 1/(b^s)=b^(-s)
  • (b^r)^s=b^(rs)
  
  Note: in the strange case b=1, these rules remain true, but not very useful. Check it out.
  
• Be able to graph y=A+C(b^x) for various choices of A, b, and C. See Fig 2, p.274 and Fig 3, p.275.
• Know some things about e. Read Rudin's quote at the beginning of 5.2.
• Be able to solve equations that involve exponents. When the bases involved are powers of a common base, such as in 4^x=8^(2x-3) (here the 4 and the 8 are powers of 2), you can often get by without using logs of any kind.
• Sometimes you need to use logarithms. Know both methods.
• Know that the functions f(x)=b^x and g(x)=log_b(x) are inverses of each other (see p.155) for all b except 1. Aside: if b=1, then f isn't one-to-one (so it doesn't have an inverse function) and g doesn't make any sense. (Can you solve 1^x=2?)
• This relationship between b^x and log_b(x) shows up in graphs as well: see Fig 2, p.291; Fig 5a, p.293; Fig 6a, p.294.
• Be able to convert equations between exponential form and logarithmic form.
• By using the properties of exponential functions, together with the fact that logarithms are their inverses, we find several properties of logarithms. If b is a positive number other than 1, then the function f(x)=log_b(x) has these analytical properties, which are especially useful for graphing (compare these to the analytical properties of exponential functions):
  • It is defined only for positive real numbers x, but f(x) can be any real number--that is to say, its domain is (0, Infinity), while its range is (-Infinity, Infinity).
  • It is continuous (there's just one piece)
  • It is smooth (no cusps)
  • It is one-to-one (What is its inverse?)
  • It has no turning points; it either increases for all x in its domain (if b>1) or decreases for all x in its domain (if 0<b<1). You won't be held accountable for graphs of y=log_b(x) for b<1.
  • It has a vertical asymptote which it approaches on one side, but not the other.
• In addition to these analytical properties, there are also some (very useful) algebraic properties (compare to the list for exponential functions). For any positive b other than 1:
  • 1. log_b(b)=1
    2. log_b(1)=0
  • log_b(PQ)=log_b(P) + log_b(Q) ``The log of a product is the sum of the logs.''
  • log_b(P/Q)=log_b(P) - log_b(Q) ``The log of a quotient is the log of the top minus the log of the bottom.'' An important special case: if P=1, then we get log_b(1/Q)=-log_b(Q).
  • log_b(P^n)=n * log_b(P)
  • b^(log_b(P))=P (for P>0) and log_b(b^P)=P (for all P), which restate the fact that logs and exponentials are inverses.
  
  Enrichment tip: compare the ``log of a product...'' rule to the exponential function rule, (b^r)(b^s)=b^(r+s). We might describe this latter rule as saying something like ``the product of two exponentials is the exponential of the sum.'' There is a similar rule-by-rule comparison, except for the last one (which really belongs to both lists).
  
• Understand the contents of the Errors To Avoid boxes on pp. 292, 306.
• Using various logarithm rules, be able to manipulate expressions involving logs: simplify them (as in 5.3 #13, 14, 27, 28 and 5.4 #1-10); break complicated terms into many simple terms (as in 5.4 #20-26); combine several terms into a single term (as in 5.4 #11-19). Well, all the exercises in 5.4 are pretty relevant.
• Be able to graph functions involving logarithmic or exponential terms, including in your analysis such things as:
  • Domain
  • Range
  • Roots (aka x-intercepts), if any
  • y-intercept, if any
  • Horizontal or vertical asymptote
  • End behavior (i.e., how does the function behave very close to the asymptote, or very far from it?)
• Be able to solve equations and inequalities involving logarithmic and/or exponential terms, being mindful of extraneous roots and domain considerations. The examples of 5.5 (especially examples 4-9) illustrate this kind of problem.

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