Download Section 2.1 Linear Functions

Section 2.1
Linear Functions
If a line passes through ( x1 , y1 ) and ( x 2 , y 2 ) , its slope is given by
y  y1
m 2
x 2  x1
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
2. y  y1  m( x  x1 )
m = slope, b = y-intercept
m = slope, ( x1 , y1 ) is a point on the line
Example 1: Find the linear function f with y-intercept 3 and f 1 (8)  2 .
Section 2.1 – Linear Functions
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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
1
6
line y   x  .
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Section 2.1 – Linear Functions
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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
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Quadratic Functions
A quadratic equation is of the form f ( x)  ax 2  bx  c , 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 f ( x)  a ( x  h) 2  k 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: f ( x)  ax 2  bx  c then we may find it easier to determine the
  b   b 
vertex by applying the formula: 
, f
  .
 2a  2a  
Section 2.1 – Linear Functions
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Example 4: Given f ( x)  4 x 2  24 x  20 :
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
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Exercise: Try this one: Write f ( x)  3 x 2  36 x  106 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
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Example 6: Find the quadratic function satisfying the following conditions:
Vertex: ( 6 , -25 ) and one of the two x-intercepts is ( 1, 0 ).
Section 2.1 – Linear Functions
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Section 2.2
Polynomial Functions
Let n be a nonnegative integer and let a n , a n 1 ,..., a 2 , a1 , a 0 , be real numbers, with a n  0 .
The function defined by f ( x)  a n x n  ...  a 2 x 2  a1 x  a 0 is called a polynomial function of x
of degree n. The number a n , 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, p( x)  2 x
1
3
1
 6 x 0.234 is NOT a polynomial, but q( x)  2 x 3  x 10  0.234 x is a
3
polynomial.
The following are not polynomials either:
3x  1
x2
, g ( x) 
f ( x)  2
x  2x
x2
A power function is a polynomial that takes the form f ( x)  ax n , where n is a positive integer.
Modifications of power functions can be graphed using transformations.
Section 2.2 – Polynomial Functions
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Example 1: Which of the following functions would result in the graph shown below?
a.
b.
A.
D.
G.
J.
f ( x)  ( x  1) 5  1
f ( x)  ( x  1) 5  1
f ( x)  ( x  1)10  1
f ( x)  ( x  1)10  1
B. f ( x)  ( x  1) 5  1
E. f ( x)  ( x  1) 5  1
H. f ( x)  ( x  1)10  1
C. f ( x)  ( x  1) 5  1
F. f ( x)  ( x  1)10  1
I. f ( x)  ( x  1)10  1
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 negative.
if its leading coefficient is positive and
An odd-degree polynomial’s end behavior will be
if its leading coefficient is negative.
if its leading coefficient is positive or
Section 2.2 – Polynomial Functions
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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.
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1

For example, given f ( x)  2 x  x   (10  x)5
4

2
Zero Multiplicity
0
2
6
1
4
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
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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
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Example 2: Sketch the graph of f ( x)  x( x  3)3 ( x  4)6 .
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
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Section 2.2 – Polynomial Functions
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Example 3: Which function could represent the given graph?
A. f ( x)  x 2 ( x  2)( x  1) 2 ( x  1)
C. f ( x)  2 x3 ( x  1)( x  1) 2
Section 2.2 – Polynomial Functions
B. f ( x)  2 x 2 ( x  1) 2 ( x  1) 2
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Exercise:Try this one: Which function could represent the given graph?
A. f ( x)  x 3  3x 2  4 x  12
C. f ( x)  x 3  3x 2  4 x  12
Section 2.2 – Polynomial Functions
B. f ( x)   x 3  3x 2  4 x  12
D. f ( x)  x 3  3x 2  4 x  12
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Section 2.3
Rational Functions and Their Graphs
p ( x)
where p(x) and q(x) are
q ( x)
polynomial functions and q(x)  0. The domain is the set of all x-values such that the
denominator is not equal to 0.
A rational function can be expressed as f ( x) 
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
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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 xintercept.
To find any y-intercept, no need to factor the numerator and denominator, simply plug in
x = 0 into the function.
2 x 2  200
.
x 2  8 x  20
a. Find the location of any holes.
Example 1: Let f ( x) 
b. Find any vertical asymptotes.
c. Find the domain.
d. Find any x-intercepts.
e. Find any y-intercept.
Section 2.3 – Rational Functions
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Horizontal Asymptotes
p( x)
,
q( x)
Shorthand: degree of f = deg(f), numerator = N, denominator = D
Let f ( x) 
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).
a
3. If deg(N) = deg(D) then there is a horizontal asymptote and it is y = , where a is the
b
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:
5x6  1
a. f ( x) 
10 x 2  x
b. f ( x) 
(3x  7)( x  1) 2
4 x5  1
Section 2.3 – Rational Functions
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Example 3: The following function has a horizontal asymptote. Find the point of
intersection of f(x) and its horizontal asymptote.
2x 2  x  6
f ( x)  2
x  3x  1
Section 2.3 – Rational Functions
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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:
 4 x 4  5x 2  3
a. f ( x) 
x3
b.
f ( x) 
 5x 2  6 x  2
x4
Section 2.3 – Rational Functions
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Example 5: Which of the following functions will generate the following graph?
x3
x( x  3)
1
f ( x) 
x
x3
f ( x)  
x( x  3)
x 3
f ( x) 
x( x  3)
x 3
f ( x)  
x( x  3)
A. f ( x) 
B.
C.
D.
E.
Section 2.3 – Rational Functions
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