Section 3.1 – Quadratic Functions and Models

You should be able to:

1)Graph quadratic functions given in Vertex or Standard Form

2)Graph quadratic functions given in General Form

Steps

Find coordinates of the vertex

Write in vertex form

Graph

3)Find the domain and range of quadratic functions

4)Match quadratic equations and graphs

5)Find the x-intercepts

Steps:Set y = 0

Solve the quadratic equation by factoring or by the quadratic formula

6)Find the y-intercepts

Steps:Let x = 0 and find y

7) Find the axis of symmetry

Steps:x = h (x-coordinate of the vertex)

8) Determine the quadratic function if you are given the coordinates of the vertex and the coordinates of other point on the graph.

9) Find the max/minimum value of the function analytically

10) Solve word problems involving quadratic functions

  1. Given a quadratic function, find the “optimal” value of the function
  2. Given the Demand equation, write the Revenue equation and optimize the revenue
  3. Enclosing the most area with a fence
  4. Constructing Rain Gutters
  5. Finding the quadratic function that best fits the given data

11) What types of questions are you dealing with when solving word problems?

a) For a given x, find y

b) For a given y, find x

c) Questions asking about x-intercepts

d) Questions asking about the maximum/minimum point

maximum/minimum value of the function

optimal x, to produce the max/min value of the function

Section 3.2 – Polynomial Functions and Models

You should be able to:

10) Know the properties of Power functions with even exponent

11)Know the properties of Power functions with odd exponent

12) Graph transformations of power functions

13) Identify Polynomial Functions and their degree

14) Write a polynomial function given its zeros and degree

15) Write a polynomial function given its graph

16) Matching graphs and polynomial functions

17) For a given polynomial function, determine each of the following:

  1. Degree
  2. End behavior
  3. Maximum number of turning points
  4. y-intercept
  5. x-intercept(s)/zeros and multiplicity
  6. whether the graph crosses, bounces, or has an inflection point at each of the zeros
  7. sketch the graph
  8. use the calculator to find the maximum, minimum points

18) Construct a polynomial function with the given zeros and going through a certain point.

19) Find the polynomial function that best fits the data

Enter data

Construct the scatter diagram

Observe its shape and decide whether you will fit the data with a

Linear, quadratic, cubic or quartic function

Section 3.3, 3.4 –Rational Functions

You should be able to:

20)Given a Rational function, find each of the following:

  1. Domain
  2. Vertical asymptote
  3. Horizontal asymptote
  4. X-Intercepts
  5. Y-Intercepts
  6. Coordinates of a hole, if any
  7. Oblique asymptote
  8. Sketch the graph
  9. Range
  10. Check the graph using the calculator

21)Construct rational functions according to some given characteristics

22)Solve word problems involving rational functions

Section 3.5 –Polynomial and Rational Inequalities

You should be able to:

23)Solve polynomial inequalities algebraically

Steps:

a- Rewrite the inequality with 0 on the right hand side

b- Factor the left hand side

c- Find the boundary numbers which are the solutions to f(x) = 0

d- Plot the boundary numbers on the number line and decide whether they are solutions to the inequality. If they are, fill them in, otherwise, leave as open circles

e- Test points around the boundary numbers to find the intervals that satisfy the inequality

f- Write answers using interval notation

g- Check with a graph

24)Solve rational inequalities algebraically

Steps:

a- Rewrite the inequality with 0 on the right hand side

b- Find the boundary numbers which are the solutions to

Numerator = 0

Denominator = 0

c- Plot the boundary numbers on the number line and decide whether they are solutions to the inequality. If they are, fill them in, otherwise, leave as open circles (Note: the boundary numbers from denominator = 0 are never solutions. Why?)

d- Test points around the boundary numbers to find the intervals that satisfy the inequality

e- Write answers using interval notation

f- Check with a graph

25)Solve word problems involving polynomial and rational inequalities

26)Solve polynomial and rational inequalities with the calculator

Math 180 – Section 3.6- The Real Zeros of Polynomial Functions

1) Understand the division algorithm of polynomials

2) Use the Remainder Theorem to find the remainder when f(x) is divided by (x – c)

3) Use the Factor Theorem to determine whether (x – c) is a factor of f(x)

If (x – c) is a factor of f(x), factor the polynomial

(We use synthetic division to find the other factor)

4) Use the Rational Zeros Theorem to list potential rational zeros of f(x)

5) Find all real zeros of a polynomial function

6) Solve polynomial equations

7) Find the real zeros of polynomial functions using the calculator

8) Use the Intermediate Value Theorem

Math 180 – Section 3.7 – Complex Zeros

1) Given some of the zeros of a polynomial function, find the remaining zeros

2) Form a polynomial with real coefficients having the given degree and zeros

3) Use the given zero to find the remaining zeros of the function

4) Find all the zeros of the given polynomial function

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