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Chapters 2 & 4 - Coggle Diagram
Chapters 2 & 4
Chapter 2: Functions and Their Graphs
2.1: Functions
Determining whether a relation represents a function.
Is there a y for every x?
Finding the value of a function.
Domain and Range
Finding the domain of a Function defined by an equation.
What are the boundaries of x?
2.2: The Graph of a Function
Identifying the graphs of a function.
Does the graph intersect and axis at more than 1 point?
Obtaining information from or about the graph of a function.
Where does the graph cross the axis's?
What are the maximum and minimum values?
What is the domain and range?
Are the x and y values positive or negative?
2.3: Properties of Functions
Determining even and odd functions from a graph.
Is the function symmetric with the y-axis or the origin?
Determining whether a function is increasing, decreasing, or constant.
Are the y-values higher or lower than each other?
Finding the average rate of change in a function.
2.4: Library of functions; Piecewise-defined Functions
Graphing the functions listed in the library of functions.
The Absolute Value Function
The Constant Function
The Identity Function
The Square Function
The Cube Function
The Cube Root Function
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2.5: Graphing Techniques
Graphing functions using vertical and horizontal shifts.
Changing the x or y values of a graph so that it is either shifted up, down, left, or right.
Graphing functions using compressions and stretches.
y = |x|: Normal
y = 2|x|: Graph shifts toward the y-axis and away from the x-axis.
y = 1/2|x|: Graph shifts away from the y-axis and toward the x-axis
Graphing functions using reflections about the y-axis and x-axis.
y = |x| to y + -|x|: Reflection across the x-axis.
y = x +1 to y = -x + 1: Reflection across the y-axis.
2.6: Building Functions
Building and analyzing functions.
Finding the distance from the origin on a certain point.
Finding the area of the rectangle made by finding the distance form the origin.
Using this information to make a function.
Chapter 4: Polynomial & Rational Functions
4.1: Polynomial Functions and Models
A polynomial function is a function whose rule is given by a polynomial in one variable.
The degree of a polynomial function is the largest power of x that appears, unless it's a zero polynomial function because then it isn't assigned a degree.
Graph of smooth + continuous polynomial function on the left, graph of what cannot be a polynomial function on the right.
A power function of degree n is a monomial of the form f(x)=ax^2 where a is a real number, a does not = 0 and n>0 is an integer.
Properties of power functions when n is an even integer:
Properties of power functions when n is an odd integer:
If f is a function and r is a real number for which f(r)=0, then r is called a real zero of f.
The following statements are equivalent: 1. r is a real zero of a polynomial function f. 2. r is an x-intercept of the graph of f. 3. x-r is a factor of f. 4. r is a solution to the equation f(x)
If (x-r)^m is a factor of a polynomial f and (x-r)^m+1 is not a factor of f, then r is called a zero of multiplicity m of f.
If r is a zero of even/odd multiplicity:
Theorem: Turning Points - If f is a polynomial function of degree n, then f has at most n-1 turning points. If the graph of a polynomial function f has n-1 turning points, the degree of f is at least n.
Theorem: End behavior:
Summary: Graph of a Polynomial Function:
Summary: Analyzing the Graph of a Polynomial Function:
Summary: Using a Graphing Utility to Analyze the Graph of a Polynomial Function:
4.2: Properties of Rational Functions
Summary: Finding Horizontal and Oblique Asymptotes of a Rational Function R:
Locating vertical asymptotes: a rational function R(x)=p(x)/q(x) in lowest terms will have a vertical asymptote x=r if r is a real zero of the denominator q. That is, if x-r is a factor of the denominator q of a rational function R(x)=p(x)/q(x) in lowest terms, R will have the vertical asymptote x=r.
If a rational function is proper, the line y=0 is a horizontal asymptote of its graph.
A rational function is a function of the form R(x)=p(x)/q(x).
4.3: The Graph of a Rational Function
How to analyze the graph of the rational function
Step 1: Factor the numerator and denominator of R. Find the domain of the rational function.
Step 2: Write R in lowest terms.
Step 3: Locate the intercepts of the graph.
Step 4: Test for symmetry. If R(-x)=R(x), the function is even and its graph will be symmetric with respect to the y-axis. If R(-x)= -R(x), the function is odd and its graph will be symmetric with respect to the origin.
Step 5: Locate the vertical asymptotes.
Step 6: Locate the horizontal or oblique asymptotes. Determine points, if any, at which the graph of R intersects these asymptotes.
Step 7: Graph R using a graphing utility.
Step 8: Use the results obtained in Steps 1-7 to graph R by hand.
4.4: Polynomial and Rational Inequalities
How to algebraically solve polynomial & rational inequalities
Step 3: Use the zeros found in Step 2 to divide the real number line into intervals.
Step 4: Select a number in each interval, evaluate f at the number and determine whether f is positive or negative. If f is positive - all values of x in the interval are positive. If f is negative - all values of x are negative.
Step 2: Determine the real zeros of f. The real zeros are x-intercepts of the graph.
Note: If the inequality is not strict, include the solutions of f(x)=0 in the solution set.
Step 1: Write the inequality so that a polynomial expression f is on the left side and zero is on the right side.
4.5: The Real Zeros of a Polynomial Function
Theorem: Division Algorithm for Polynomials - If f(x) and g(x) denote polynomial functions and if g(x) is a polynomial whose degree is greater than zero, then there are unique polynomials q(x) and r(x) such that f(x)=q(x)g(x)+r(x) where r(x) is either the zero polynomial or a polynomial of degree less than that of g(x).
Remainder theorem: Let f be a polynomial function. If f(x) is divided by x-c then the remainder is f(c).
Factor theorem: Let f be a polynomial function. Then x-c is a factor of f(x) if and only if f(c)=0. Two parts to this theorem: 1. If f(c)=0 then x-c is a factor of f(x). 2. If x-c is a factor of f(x), then f(c)=0.
Theorem: Number of Real Zeros: A polynomial function of degree n, n>=1, has at most n real zeros.
Rational Zeros Theorem: Let f be a polynomial function of degree 1 or higher of th
