Section 2.1 – Linear Functions 1

Section 2.1 Linear Functions

If a line passes through ),( 11 yx and ),( 22 yx , its slope is given by

12

12

xx

yym

The slope measures the steepness of a line. The slope of any horizontal line, y = b, is 0 and the slope of any vertical line, x = a, is undefined. Two Forms of an Equation of a Line 1. y = mx + b m = slope, b = y-intercept 2. )( 11 xxmyy m = slope, ),( 11 yx is a point on the line Example 1: Find the linear function f with y-intercept 3 and 1(8) 2f .

Section 2.1 – Linear Functions 2

Parallel and Perpendicular Lines

Two lines are parallel if and only if their slopes are the same. Two lines are perpendicular if and only if their slopes are negative reciprocals of each

other. Example 2: Find a linear function f such that f(-2) = 1 and the graph of f is perpendicular to the

line .7

6

14

1 xy

Section 2.1 – Linear Functions 3

Example 3: Find a linear function f that passes through (-2, 3) and is parallel to the line x = 4.

Section 2.1 – Linear Functions 4

Quadratic Functions

A quadratic equation is of the form cbxaxxf 2)( , where a, b, and c are real numbers and a is not equal to zero.

The graph of any quadratic is called parabola. Example: The vertex is the turning point of the parabola. If the parabola opens upward the function has a minimum value (y-value of the vertex). If the parabola opens downward the function has a maximum value (y-value of the vertex). The axis of symmetry is a line through the vertex that divides the graph in half.

The Standard Form of a Quadratic Function

The quadratic function khxaxf 2)()( is in standard form. The vertex is the point (h, k) and the axis of symmetry is x = h. If the function is of the form: cbxaxxf 2)( then we may find it easier to determine the

vertex by applying the formula:

a

bf

a

b

2,

2.

Section 2.1 – Linear Functions 5

Example 4: Given 20244)( 2 xxxf : a. Write in standard form. b. Find the vertex. c. Does the function have a maximum or minimum value? What is the max or min value. d. Find the axis of symmetry. e. Find any x-intercepts. f. Find any y-intercept.

Section 2.1 – Linear Functions 6

Exercise: Try this one: Write 2( ) 3 36 106f x x x in standard form. Example 5: Find the quadratic function such that the axis of symmetry is x = -2, the y-intercept is -6, and there is only one x-intercept.

Section 2.1 – Linear Functions 7

Example 6: Find the quadratic function satisfying the following conditions: Vertex: ( 6 , -25 ) and one of the two x-intercepts is ( 1, 0 ).

Section 2.2 – Polynomial Functions 1

Section 2.2 Polynomial Functions

Let n be a nonnegative integer and let 0121 ,,,...,, aaaaa nn , be real numbers, with 0na .

The function defined by 012

2...)( axaxaxaxf nn is called a polynomial function of x

of degree n. The number na , the coefficient of the variable to the highest power, is called the

leading coefficient. The domain of any polynomial function is all real numbers.

For example, 1 0.2343( ) 2 6p x x x is NOT a polynomial, but xxxxq 234.0

3

12)( 103 is a

polynomial.

The following are not polynomials either: 2

2( )

2

xf x

x x

,

2

3 1( )

xg x

x

A power function is a polynomial that takes the form naxxf )( , where n is a positive integer. Modifications of power functions can be graphed using transformations.

Section 2.2 – Polynomial Functions 2

Example 1: Which of the following functions would result in the graph shown below? a. b.

A. 1)1()( 5 xxf B. 1)1()( 5 xxf C. 1)1()( 5 xxf

D. 1)1()( 5 xxf E. 1)1()( 5 xxf F. 10( ) ( 1) 1f x x

G. 10( ) ( 1) 1f x x H. 10( ) ( 1) 1f x x I. 10( ) ( 1) 1f x x

J. 10( ) ( 1) 1f x x End Behavior of Polynomial Functions The behavior of a graph of a function to the far left or far right is called its end behavior. An even-degree polynomial’s end behavior will be if its leading coefficient is positive and if its leading coefficient is negative. An odd-degree polynomial’s end behavior will be if its leading coefficient is positive or if its leading coefficient is negative.

Section 2.2 – Polynomial Functions 3

Zeros of a Polynomial Function If f is a polynomial function, then the values of x for which f(x) is equal to 0 are called the zeros of f. The zeros are simply x-intercepts. Multiplicity of a Zero In factoring the equation for the polynomial function f, if the same factor x – r occurs k times, we call r a repeated zero with multiplicity k.

For example, given 6

2 51( ) 2 (10 )

4f x x x x

Zero Multiplicity

0 2

4

1

6

10 5 Description of the Behavior at Each x-intercept 1. Even Multiplicity: The graph touches the x-axis, but does not cross it (looks like a parabola there). 2. Odd Multiplicity of 1: The graph crosses the x-axis (looks like a line there). 3. Odd Multiplicity greater than or equal to 3: The graph crosses the x-axis and it looks like a cubic there.

Section 2.2 – Polynomial Functions 4

Steps to graphing other polynomials: 1. Determine the leading term. Is the degree even or odd? Is the leading coefficient positive or negative? Use the answers to both questions to determine the end behavior. 2. Find the y-intercept. 3. Factor the polynomial. 4. Find the x-intercept(s). 5. Plot the x-intercepts and y-intercept on the 2-dimensional plane. 6. Draw the graph, keeping in mind the multiplicity of each zero and the behavior of the graph surrounding of each zero. Note: Your graph should be smooth with no sharp corners. Without calculus or plotting lots of points, we don’t have enough information to know how high or how low the turning points are.

Section 2.2 – Polynomial Functions 5

Example 2: Sketch the graph of 3 6( ) ( 3) ( 4)f x x x x . a. Describe the end behavior. Is the degree even or odd? Is the leading coefficient positive or negative? End behavior is: b. Find its y-intercept. c. Find its zeros. d. Sketch its graph.

Section 2.2 – Polynomial Functions 6

Section 2.2 – Polynomial Functions 7

Example 3: Which function could represent the given graph?

A. 2 2( ) ( 2)( 1) ( 1)f x x x x x B. 2 2 2( ) 2 ( 1) ( 1)f x x x x

C. 3 2( ) 2 ( 1)( 1)f x x x x

Section 2.2 – Polynomial Functions 8

Exercise:Try this one: Which function could represent the given graph?

A. 3 2( ) 3 4 12f x x x x B. 3 2( ) 3 4 12f x x x x

C. 3 2( ) 3 4 12f x x x x D. 3 2( ) 3 4 12f x x x x

Section 2.3 – Rational Functions 1

Section 2.3 Rational Functions and Their Graphs

A rational function can be expressed as )(

)()(

xq

xpxf where p(x) and q(x) are

polynomial functions and q(x) 0. The domain is the set of all x-values such that the denominator is not equal to 0. Asymptotes of Rational Functions The line x = a is a vertical asymptote of the graph of a function f if f(x) increases or decreases without bound as x approaches a. The line y = b is a horizontal asymptote of the graph of a function f if f(x) approaches b as x increases or decreases without bound. Example: Given the following graph x = -2 is its vertical asymptote and y = 2 is its horizontal asymptote.

Section 2.3 – Rational Functions 2

Vertical Asymptotes and Holes Factor the numerator and denominator. Look at each factor in the denominator.

If a factor cancels with a factor in the numerator, then there is a hole where that factor equals zero.

If a factor does not cancel, then there is a vertical asymptote where that factor equals zero.

Any variable factor left in the numerator, set it equal to zero and solve for x. This is an x-intercept. To find any y-intercept, no need to factor the numerator and denominator, simply plug in x = 0 into the function.

Example 1: Let 2

2

2 200( )

8 20

xf x

x x

.

a. Find the location of any holes. b. Find any vertical asymptotes. c. Find the domain. d. Find any x-intercepts. e. Find any y-intercept.

Section 2.3 – Rational Functions 3

Horizontal Asymptotes

Let )(

)()(

xq

xpxf ,

Shorthand: degree of f = deg(f), numerator = N, denominator = D 1. If deg(N) > deg(D) then there is no horizontal asymptote. 2. If deg(N) < deg(D) then there is a horizontal asymptote and it is y = 0 (x-axis).

3. If deg(N) = deg(D) then there is a horizontal asymptote and it is y = b

a, where a is the

leading coefficient of the numerator and b is the leading coefficient of the denominator. If a function has a horizontal asymptote, it can only have at most one. Example 2: Find the horizontal asymptote, if there is one, of:

a. 6

2

5 1( )

10

xf x

x x

b. 2

5

(3 7)( 1)( )

4 1

x xf x

x

Section 2.3 – Rational Functions 4

Example 3: The following function has a horizontal asymptote. Find the point of intersection of f(x) and its horizontal asymptote.

13

62)(

2

2

xx

xxxf

Section 2.3 – Rational Functions 5

Slant Asymptotes The graph of a rational function has a slant asymptote if the degree of the numerator is one more than the degree of the denominator. To find the equation of the slant asymptote, divide the denominator into the numerator. The quotient is the equation of the slant asymptote. As x , the value of the

remainder is approximately 0. The equation of the slant asymptote will always be of the form y = mx + b. Example 4: Find the slant asymptote for:

a. 3

24 354)(

x

xxxf

b. 4

265)(

2

x

xxxf

Section 2.3 – Rational Functions 6

Example 5: Which of the following functions will generate the following graph?

A. 3

( )( 3)

xf x

x x

B. 1

( )f xx

C. 3

( )( 3)

xf x

x x

D. 3

( )( 3)

xf x

x x

E. 3

( )( 3)

xf x

x x