Session 10

Agenda:• Questions from 5.1-5.3?• 5.4 – Polynomial Functions• 5.5 – Rational Functions• 5.6 – Radical Functions• Things to do before our next meeting.

Questions?

5.4 – Polynomial Functions

• A polynomial function is any function of the form

• The degree of the polynomial is n, where n is the highest power of x.

• A polynomial of degree 3 is called a cubic function:

• A polynomial of degree 2 is called a quadratic function:

• A polynomial of degree 1 is called a linear function:

• A polynomial of degree 0 is called a constant function:

1 21 2 1 0( ) ...n n

n nf x c x c x c x c x c

3( ) 2 5 1f x x x

2( ) 2 19g x x x

( ) 2 1h x x

( ) 12f x

• EVERY polynomial function has a domain of all real numbers: Why? There are no values of x for which the function is not defined. In other words, we can evaluate the function for ANY value of x.

• The range of polynomial functions is easiest to find by considering the graph of the function.

• Many polynomial functions can just be graphed using the transformation techniques we learned in the previous section.

( , )

Linear Functions

• The slope of the line that passes through the points

and is

• Given a point on a line and its slope m, the following Point-Slope Form of a Line can be used to find its equation.

• Slope-Intercept Form of a Line:

where m is the slope of the line and b is the y-intercept.

• General Form of a Line: Ax+By+C=0, where A, B, and C are integers.

1 1( , )x y 2 2( , )x y 2 1

2 1

y yrise ym

run x x x

y mx b

1 1( , )x y

1 1( )y y m x x

• Example: Find an equation of the line that passes through the points (2, -3) and (4, 6). Write your answer in slope-intercept form and also in general form.

• A horizontal line through the point (a,b) has equation y=b. Note that a horizontal line is really just a constant function.

• A vertical line through the point (a,b) has equation x=a. Note that this is not a function at all! It fails the Vertical Line Test.

• Example: Find equations of the horizontal and vertical lines that pass through the point (9, -2).

Quadratic Functions

• We know the graph of the quadratic function is a parabola. EVERY quadratic function has the shape of a parabola and can be written as a transformation of the parent function by writing it in standard form.

• The standard form a quadratic function is

• When written in standard form, the vertex of the quadratic can be easily identified as the point (h, k).

• If a>0, the parabola opens upward.

• If a<0, the parabola opens downward.

• If a quadratic function is given in the form , you can complete the square to write it in standard form.

2( )f x x

2( ) ( )f x a x h k

2( )f x ax bx c

Write the following quadratic functions in standard form and identify the vertex of each parabola. Does the parabola open upward or downward?

2( ) 2 12 10f x x x

2( ) 3 3 2g x x x

2( ) 6f x x x

• An alternate (easier) way to find the vertex is to use the following: Given a quadratic function the vertex (h, k) is given by

• Example: Find the vertices of the quadratic functions below. Do the quadratics open upward or downward?

2( )f x ax bx c

2

bh

a

2

bk f

a

2( ) 2 5 3f x x x 2( ) 2 12 10f x x x

• Intercepts

– To determine the x-intercepts of any function, we want to know where the function crosses or touches the x-axis. This occurs when f(x)=0. Set the function equal to 0 and solve to find the x-intercepts.

– To determine the y-intercept, we want to know where the function crosses or touches the y-axis. This occurs when x=0. Evaluate the function at x=0 to determine the y-intercept (if one exists).

Graph the same quadratic functions below. Identify and plot any intercepts.

Range:_______________

Increasing on:________________

Decreasing on:________________

Is the vertex a max or min?________

Axis of symmetry:__________

2 2( ) 2 12 10 2( 3) 8f x x x x

Graph the quadratic function below. Identify and plot any intercepts.

Range:_______________

Increasing on:________________

Decreasing on:________________

Is the vertex a max or min?________

Axis of symmetry:__________

22 1 5

( ) 3 3 2 32 4

g x x x x

Find the x and y-intercepts for the following functions. Create a sign chart as we did with inequalities to determine where the function is above and below the x-axis. Then, sketch a graph of each function.

4 2( ) 3 2f x x x

3 2( ) 2 8 4g x x x x

5.5 – Rational Functions

• A rational function is a function of the form , where both f(x) and g(x) are polynomials and where .

• Since the denominator cannot be 0, the domain of a rational function consists of all values of x for which the denominator is not 0. In other words, find all values of x that make the denominator 0 and exclude them.

• Example: Find the domain of the function

( )

( )

f x

g x( ) 0g x

2

3 2

9( )

6 27

xf x

x x x

• Notice that in the previous function, the numerator also had a factor of (x-3). Although these terms would cancel when simplifying the expression, since the original function is undefined at x=3, it is still not included in the domain.

• If a factor completely cancels out of the denominator when simplified, there is a “hole” in the graph of the function at this location. In the previous function, there would be a “hole” in the graph at x=3. To find the y-value of the “hole,” plug in x=3 into the simplified function:

• If a factor does NOT cancel from the denominator, there is a vertical asymptote at that location. In the previous function, the graph has vertical asymptotes x=-9 and x=0.

• A rational function will either increase to ∞ or decrease to -∞ without bound as it approaches a vertical asymptote on either side. A rational function will NEVER cross a vertical asymptote.

• Sketch the graph of a function f(x) with vertical asymptotes at x=-5 and x=5 with the following properties:– f(x)∞ as x -5 from the left

– f(x)-∞ as x -5 from the right

– f(x)-∞ as x 5 from the left

– f(x)-∞ as x 5 from the right

Determine the domain, vertical asymptotes, and holes for the following functions.

2

3 2

3 28( )

2 2

x xg x

x x x

2

2

2 ( 2) ( 4)( )

3 ( 2)( 5)

x x xh x

x x x

Intercepts

• Just as with polynomial functions, to find x-intercepts, set the function equal to 0. With a rational function, this can ONLY occur when the numerator is equal to 0 (and there is no hole there). Finding the x-intercepts of a function is sometimes referred to as finding the zeros of a function.

• To find the y-intercept, evaluate the function at x=0, if possible.

• Find the intercepts for the previous two functions.2

3 2

3 28( )

2 2

x xg x

x x x

2

2

2 ( 2) ( 4)( )

3 ( 2)( 5)

x x xh x

x x x

Horizontal Asymptotes

• Horizontal asymptotes are horizontal lines which the graph of the function approaches as x ∞ or x-∞. In other words, they describe the behavior of the graph at the extreme left and right of the domain.

• In the graph below, you can see that as x ∞ and as x -∞, the graph is approaching the horizontal asymptote y=1.

Determining Horizontal Asymptotes

• Consider the rational function

• Note that n is the degree of the numerator and m is the degree of the denominator.– If n<m, then y=0 is the horizontal asymptote.

– If n=m, then y= is the horizontal asymptote.

This is the ratio of the leading coefficients.

– If n>m, then there is no horizontal asymptote.

• When graphing a rational function, we can again use a sign chart as we did with rational inequalities to determine where the graph is positive and where it is negative.

0

0

...( )

( ) ...

nnm

m

c x cf x

g x b x b

n

m

c

b

Determine the horizontal asymptotes of the following functions, if any.

2

3 2

3 28( )

2 2

x xg x

x x x

2

2

2 ( 2) ( 4)( )

3 ( 2)( 5)

x x xh x

x x x

2 5

4

2( )

1

x xf x

x

Find all intercepts, asymptotes, holes, and use a sign chart to help sketch the general shape of the function.

3( 2)( 5)( )

( 2)( 6)

x xf x

x x x

5.6 – Radical Functions

• With functions that involve radicals, we know that we cannot take an even root (square root, fourth root, etc) of a negative number. So, to find the domain of a function that involves radicals, we must ensure that any expression inside an even root is ≥0.

• If, in addition, the function has a denominator, we must still exclude the values of x that make the denominator 0.

• Find the domains of the following functions:

3( )

6

xf x

x

3

4

7( )

3 8

xg x

x

5( )

4

xh x

x

Find the domains and x and y-intercepts for the following functions:

2( ) 3 25 6g x x

( ) 9 2 6f x x x

Things to Do Before Next Meeting:

• Work on Sections 5.4-5.6 until you get all green bars!

• Write down any questions you have.

• Continue working on mastering 5.1-5.3.

• Make sure you have taken the Chapter 6 Test before our next meeting.