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2.01 Quadratic Functions
|Word |Definition/examples |
|Polynomial Function |Definition: |
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|Quadratic Function (Standard Form) |Definition: |
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| |Example: |
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|Vertex Form |Definition: |
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| |Example: |
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|Test your skills |Put the following into standard quadratic function form. Find the vertex point and the axis. |
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|[pic] | |
| |f(x) =3x2 + 12x + 7 |
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| |f(x) = x2 + 6x – 2 |
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| |graph these two functions to show the vertex and axis. Check your graph with your calculator. |
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| |Study the examples in your online text, chapter 2.1 |
|Solving Quadratic Equation set Equal to 0 |
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|From your Algebra Two class, you will recall that there are several ways to solve a quadratic equation set equal to 0 |
|f(x) = ax2 + bx + c = 0 |
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|Provided the quadratic is factorable, we can solve by factoring, set each factor equal to 0 and solve. |
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|Quadratic Formula: |
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|Solve 2x2 - 5x + 1 = 0 by using the quadratic formula. |
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2.02 Polynomial Functions of Higher Degree
|General Features of Polynomial Graphs |1. |
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| |N is even then the graph is “____” |
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| |N is odd then graph is “____” shaped |
|Binomial Theorem |Notes: |
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|Click link for a video example on this | |
|theorem. | |
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|Pascal Triangle |Notes: |
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|Click link for a video example on this | |
|theorem | |
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|Leading Coefficient Test |The rising or falling of the graph is determined by two rules: |
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| |If the leading coefficient is positive, the |
| |graph_____________________________________________________ |
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| |If the leading coefficient is negative then the graph |
| |__________________________________________________________ |
|Zeros of Polynomial Functions |What are Zeros of Polynomial Functions? |
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| |Equivalent Statements |
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|The Intermediate Value Theorem |Theorem: |
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| |What is the theorem saying? |
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| |What is the advantage of this theorem? |
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|The Extreme Value Theorem |Theorem: |
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| |What is the theorem saying? |
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| |To identify where a maximum or minimum might occur, we have to consider two types of test values: |
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| |2. |
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| |Graph f(x) = 4x - x2 on the interval [-1, 4]. Make sure you change your window settings on your |
| |calculator so that your x min is at -1 and x max is at 4. |
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| |Step 1: |
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| |Step 2: |
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| |Step 3: |
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***All solutions to the following problems are found in your lesson 2.02****
|Test your skills |Use the Binomial Theorem or Pascal’s Triangle to expand the following: |
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|[pic] | |
| |(x – 4)5 |(2x + 3)3 |
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|Test your skills |Using the degree and the leading coefficient methodologies determine the direction of the left and |
| |right sides of the graph. |
|[pic] | |
| |f(x) = x5 + 5x + 2 |f(x) = -x5 - 5x - 2 |
| |g(x) = 16x4 - 2x + 7 |g(x) = 16x5 - 2x + 7 |
| |h(x) = (1/2)x6 + (1/4)x |h(x) = (-3/2)x6 + (2/3)x |
|Test your skills |Find the real zeros of the following. Use your calculator to graph the function, and check your |
| |answer by looking at thex-intercepts. |
|[pic] |f(x) = x3 + 2x2 - 4x – 8 |
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| |g(x) = -x2 + 2 (real zeros only) |
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|Test your skills |Try these applications of the Intermediate Value Theorem |
| |Use the IVT to show that f(x) = -2x3 + 3x2 – 1 has a zero on the interval [-1, 0] |
|[pic] | |
| |Use the IVT to find an interval of length 1 in which f(x) = x3 – 4x2 + 2x – 7 is guaranteed to have a|
| |zero. |
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2.03 Real Zeros of Polynomial Functions
|Long Division |Write down steps for video example: |
|Example click: | |
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|functions/polynomial-long-division-example.php | |
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|Division Algorithm |What does it state? |
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|Synthetic Division |Write down steps for video example: |
|Example click: | |
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|Remainder Theorems |The Remainder Theorem states: |
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| |For example: |
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|The Factor Theorem |The Factor Theorem states : |
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|The Rational Zero Test |The Rational Zero Test states: |
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| |For example: |
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|Descartes’ Rule of Signs |Descartes’ Rule of Signs is a method: |
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| |The following examples show be taken from your lesson click examples In Descartes rule |
| |of sign changes section: |
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| |Example #1 |
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| |Example #2 |
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|Bounds for Real Zeros of Polynomial Functions |Another test for zeros of a polynomial is to look at the sign pattern of the last row |
| |of the synthetic division process. The following rules show if a value is an upper or |
| |lower bound for the real zeros of a polynomial. |
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| |Let f(x) be a polynomial with real coefficients and a positive leading coefficient. |
| |Suppose f(x) is divided by (x - c) using synthetic division. Then: |
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** The following problems are for you to work out, all of the answers can be found in the Test your skills section of lesson 2.03****
|Test your skills |
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|Use synthetic division to find the quotient and remainder of the following: |
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|x4 + 2x3 - 4x2 + x - 6 by (x - 2) |
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|x4 - 2x2 + 6 by (x + 2) |
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|Test your skills |
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|For f(x) = x4 - 2x2 + 6, determine the value of the remainder for each of the following. Use your calculator to check your answers. |
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|f(1) |
|f(-1) |
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|f(-2) |
|f(2) |
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|Test your skills |
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|Using synthetic division, determine if (x + 1) and/or (x - 1) are factors of the following polynomials. |
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|x4 - 2x2 - 1 |
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|x3 - 3x2 + x + 1 |
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|Test your skills |
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|[pic] |
|Find the rational zeros of the following by making a list of the possible rational zeros. Use Descartes’ Rule of Signs to narrow down your |
|list of possible rational roots. Then use synthetic division to find which ones work. Use a graph of the function on your calculator to check|
|your answer. |
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| f(x) = -x2 – 1 |
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|f(x) = x3 + 2x2 + 4x + 3 |
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|Test your skills |
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|[pic] |
|Determine upper and lower bounds for the following functions. |
|f(x) = 2x4 + x3 - x2 - 8 |
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|f(x) = 4x3 - 3x + 3 |
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2.04 Complex Zeros
|Complex Number |Definition: |
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|Operations with complex Numbers |The rule for the equality of complex numbers states: |
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| |The rule for the addition and subtraction of complex numbers states: |
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| |Some of the properties for real numbers are also valid for complex numbers. They include the: |
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|Plotting Complex Numbers |How do we plot complex numbers: |
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|Test your skills |
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|Simplify the following expressions. Click on the "Check Your Answers" in lesson 2.04 to open a new browser and see the answers. |
|(5 - i) + (6 + 3i) |
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|(5 - 2i) - (7 - i) |
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|(5 - i) (6 + 3i) |
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|(5 - 2i) / (7 - i) |
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2.05a The Fundamental Theorem of Algebra
|Fundamental Theorem of Algebra |Definition from lesson: |
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|Click to watch video explanation: | |
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|al-theorem-algebra.php | |
| |Notes from video: |
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|Linear Factorization Theorem |Expanding this we get the linear factorization theorem which states: |
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| |Examples: |
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| |b. |
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|Graphing Polynomials |Example from lesson: |
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|Conjugate Pairs |The rule that complex zeros occur in conjugate pairs states: |
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| |Example: |
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|Factoring a Polynomial |The rule concerning factors of a polynomial states: |
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|Rational Function | |
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|Horizontal and Vertical Asymptotes |The definition of vertical and horizontal asymptotes states: |
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|(video on HA and VA ) | |
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|Asymptotes of a Rational Function |For the rational function [pic], |
| |f(x) = |
| |anxn + a(n-1)x(n-1) + ...... + a1x + a0 |
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| |bmxm + b(m-1)x(m-1) + ......+ b1x + b0 |
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| |where p and q do not have any common factors, the following apply: |
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|Test your skills |
|Review the text to see these rules graphically. List the vertical and horizontal asymptote for each of the following rational functions. Be |
|sure to write the vertical asymptotes as x = ___, and the horizontal asymptotes as y = _____. |
|f(x) = [pic] |
|g(x) = [pic] |
|h(x) = [pic] |
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2.06 Rational Functions
|Keyword |Definition/ Example |
|The Graph of a Rational |1. |
|Function | |
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| |For example, for the function f(x) = 5/(x + 1), we can determine the following: |
| |y-intercept is: |
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| |x-intercept is: |
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| |vertical asymptote is: |
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| |horizontal asymptote is: |
|Test your Skills |[pic] |
|Slant Asymptotes |Definition: |
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|Video: | |
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|Vertical Asymptotes vs. Holes|What happens to the zeros of the denominator that get cancelled out when R(x) is put in lowest terms? |
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|Analyzing the Graph of a |*** Click the example hyperlink in the lesson*** |
|Rational Function |[pic] |
| |Step 1: |
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| |Step 2: |
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| |Step 3: |
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| |Step 4: |
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| |Step 5: |
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| |Step 6: |
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| |Step 7: |
|Test your skills |[pic] |
2.08 Module two Discussion Based Assessment Study Guide
Make sure to be able to explain and discuss the following terms to be prepared for your discussion based assessment with your teacher.
Quadratic Formula
The Standard Form of a Quadratic Function
The Remainder Theorem
The Factor Theorem
The Rational Zero Test
Descartes' Rule of Signs
Bounds for Real Zeros of Polynomial Functions
Fundamental Theorem of Algebra
Asymptotes of a Rational Function
The Graph of a Rational Function
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2.07 Graphs of Rational Functions
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