Midterm III Review for Math 121, Summer 2012

Topics we covered.

  1. writing down functions described in word problems.
  2. linear functions
    1. the average rate of change is constant (and equals the slope)
    2. increasing, decreasing, and constant linear functions in terms of their slopes
    3. inequalities between linear functions (write down the interval(s) of $x$ where one is "above" another...)
  3. quadratic functions
    1. moving from the standard form $f(x)=ax^2+bx+c$ to the vertex form $f(x)=a(x-h)^2+k$ — by completing the square (or just knowing that $h=\frac{-b}{2a}$ and $k=f(h)$)
    2. sketching the parabola from the vertex form of the funtion:
      1. opens up or down (depending upon the sign of $a$)
      2. location of the vertex (at $(h,k)$)
      3. the axis of symmetry (which is $x=h$)
      4. where the function is increasing/decreasing and has a maximum or minimum
    3. figuring out the formula for the function when given the vertex and one more point on the graph
  4. word problems with quadratic functions
    1. trajectories of thrown objects
    2. parabolic arches
    3. cables of suspension bridges
    4. fencing fields to maximize area
    5. others?
  5. inequalities between linear and quadratic functions
    1. different behavior depending upon which way the parabola points and which direction the inequality is given
    2. solve for the intersection points and, using A., write down a single interval or union of two intervals
  6. polynomial functions
    1. definition, also of degree, zero/root, multiplicity
    2. behavior near a zero
    3. end behavior
  7. factoring polynomials over the real numbers (= finding the real zeros of a polynomial)
    1. The Remainder Theorem ("the remainder of division of a polynomial $f(x)$ by $x-c$ is $f(c)$")
    2. The Factor Theorem ("$x-c$ is a factor of the polynomial $f(x)$ if and only if $f(c)=0$")
    3. maximum number of real zeros of a polynomial
    4. The Rational Roots Theorem ("if the polynomial $f(x)=a_n x^n+\dots+a_0$ has only integer coefficients and if $p/q$ is a rational root of $f(x)$ written in lowest terms, then $p$ is a factor of $a_0$ and $q$ is a factor of $a_n$")
    5. using the RRT to factor a polynomial
  8. factoring polynomials over the complex numbers (= finding the complex zeros of a polynomial)
    1. The Fundamental Theorem of Algebra ("a polynomial with complex coefficients always has a complex root")
    2. The Conjugate Roots Theorem ("if a complex number $z$ is a root of a polynomial with real coefficients, then so is its complex conjugate $\overline{z}$")
    3. factoring polynomials over the complex numbers
    4. building polynomials with specified roots
  9. rational functions $r(x)=p(x)/q(x)$
    1. definition, domain
    2. vertical, horizontal, and oblique [=slant] asymptotes
      1. VAs are at the zeros of the denominator, if the rational function is in lowest terms
      2. HA of $y=0$ if $\deg(p)<\deg(q)$
      3. if $\deg(p)=\deg(q)$ there is an HA of $y=k$, where $k$ is the ratio of the leading coefficients of $p$ and $q$
      4. OA of $ax+b$ if that is the quotient of $p(x)$ by $q(x)$ and $\deg(p)=\deg(q)+1$
      5. no HA or OA if $\deg(p)>\deg(q)+1$ and end behavior is as the end behavior of the quotient of $p(x)$ by $q(x)$
    3. graphing: break real number line into intervals divided by the VAs and zeros of $p(x)$; sketch the HA or OA; fill in
  10. solving polynomial and rational inequalities
    1. write with $0$ on the right hand side
    2. divide the real line into intervals divided by the zeros of the numerator and denominator (if present)
    3. plug in a single point in each intervale, get the sign
    4. list the intervals where the sample point satisfies the desired inequality