Section 3.1 – Quadratic Functions
A quadratic function is one of the form f (x) = ax2 + bx + c
Solving quadratic functions using the Quadratic Formula:
b2 – 4ac is called the discriminant.
• If b2 – 4ac > 0, then the equation will have 2 real solutions
• If b2 – 4ac < 0, then the equation will have no real solutions (there will be 2 complex sol’ns)
• If b2 – 4ac = 0, then the equation will have 1 real solution
The Quadratic Formula:
If ax2 + bx + c = 0, then
x=
 b  b 2  4ac
2a
Examples: Solve using the quadratic formula:
a) 3x2 – 7x + 1 = 0
b) 2x2 – 4x + 3 = 0
The graphs of all quadratic functions are parabolas. Parabolas will either open up or open down,
and they will contain a vertex and an axis of symmetry.
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Quadratic Form:
f (x) = ax2 + bx + c
The parabola opens up if a > 0, and opens down if a < 0
The vertex (maximum/minimum) is at 2ab , f ( 2ab ) 
The axis of symmetry is the line x =
The y-intercept is at (0, c)
b
2a
The x-intercepts are found by solving f (x) = 0.
If b2 – 4ac > 0, then the graph will have 2 x-intercepts
If b2 – 4ac = 0, then the graph will have 1 x-intercept
If b2 – 4ac < 0, then the graph will have 0 x-intercepts
Examples: Find the vertex, axis of symmetry, x- and y-intercepts, domain and range. Sketch a graph.
a) f (x) = x2 + 6x + 3
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b) f (x) = –2x2 + 8x – 1
c) f (x) = x2 + 4x + 5
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Applications Involving Maximizing or Minimizing:
Quadratic functions will always have either a maximum or a minimum value.
Example: A person standing close to the edge on the top of a 160-foot building throws a baseball
vertically upward. The quadratic function s(t) = –16t2 + 64t + 160 models the ball’s height
above the ground, s(t), in feet, t seconds after it was thrown.
a) After how many seconds does the ball reach its maximum height?
b) What is the maximum height?
c) How many seconds does it take for the ball to hit the ground?
d) Calculate s(0) and describe what it means.
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Example: Amy has 2500 feet of fencing available to enclose a rectangular garden. Find the
dimensions of the garden that will maximize the enclosed area. What is the maximum area?
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Section 3.2 – Polynomial Functions and Their Graphs
A Polynomial Function of Degree n has the form:
f (x) = an x n  an 1 x n 1    a1 x  a0
(where the an’s are real numbers and n is an integer ≥ 0 )
When graphed, polynomial functions are always smooth and continuous.
Although the graph of a polynomial may have intervals where it increases or decreases, the graph will
eventually rise or fall without bound as x moves far to the left and far to the right.
Leading Coefficient Test for Determining End Behavior: (look at the leading coefficient anxn )
If the exponent (n) is EVEN, the end behavior will be the same on both sides:
Leading coefficient Positive
Leading coefficient Negative
If the exponent (n) is ODD, the end behavior will be in opposite directions:
Leading coefficient Positive
Leading coefficient Negative
Examples: Use the Leading Coefficient Test to determine the end behavior of the following.
a) f (x) = 3x3 + 2x2 + 10
b) g (x) = –10x4 + 2x
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Finding the Zeros of a Polynomial Function:
If f is a polynomial and c is a real number for which f (c) = 0, then c is called a ______ of f, or a
______ of f, or an ________________ of f. Also, (x – c) is called a ____________ of f.
Example: Find all zeros of f ( x)  x3  3x 2  x  3 by factoring.
Multiplicities of Zeros:
If (x – c)k is a factor of f (x), then we say c is a zero with multiplicity k.
Example:
Let f (x) = x3 (x + 2)2(x – 3). List the zeros and their multiplicities.
Zero
Multiplicity
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Zeros with __________ multiplicities will ___________________ the x-axis
Zeros with __________ multiplicities will ___________________ the x-axis.
Example:
Complete the chart for f (x) = (x + 1)2 (x – 4) (x + 5)3
Zero
Example:
Multiplicity
Bounce/Cross
f (x) = (x – 3)2 (x + 4)
a) degree
b) end behavior
c)
Zero
Multiplicity
Bounce/Cross
d) y-intercept
e) Sketch the graph
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Example:
f (x) = –2x2 (x + 1)3 (x + 3)2 (x – 5)
a) degree
b) end behavior
c)
Zero
Multiplicity
Bounce/Cross
d) y-intercept
e) Sketch the graph
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Section 3.3 – Dividing Polynomials and The Remainder and Factor Theorems
Recall long division:
quotient
divisor dividend
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The Division Algorithm:
f ( x)  d ( x)
Dividend Divisor
q ( x)
 r ( x)
Quotient Remainder
Long division of polynomials follows the same procedure:
Examples:
a)
x  2 3x3  x 2  x  2
b)
x 2  1 6 x 4  3x3  x  1
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c)
3x3  1 3x5  x 2  x  2
Synthetic Division – The divisor must be of the form x – c (or x + c).
Examples: Find the quotient and remainder
a) Divide 3x3  x 2  x  2 by x + 2
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b) Divide 4 x3  2 x 2  1 by x – 3
c) Divide x5  5 x3  10 by x + 1
The Remainder Theorem – If f (x) is divided by (x – c), then the remainder will be f (c).
Examples: Use the remainder theorem to find the following function values of f (x) = 2x2 + 5x – 1:
a) f (2)
b) f (–1)
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The Factor Theorem –
If f (c) = 0, then (x – c) is a factor of f (x).
If (x – c) is a factor of f (x), then f (c) = 0 and
If you divide f (x) by (x – c) and get a remainder of ZERO, then (x – c) is a factor of f (x).
Example: Is (x – 2) a factor of f (x) = 4x4 – 15x2 – 4 ?
Example: Is (x – 1) a factor of f (x) = x2 + 1 ?
Example: Suppose we know that f (5) = 0. Name a factor of f (x).
Example: Use synthetic division to divide f (x) = x3 – 2x2 – x + 2 by x + 1.
Use the result to find all zeros of f.
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Section 3.4 – Zeros of Polynomial Functions
Rational Zero Theorem:
Let f (x) = an x n  an1 x n1    a1 x  a0 (integer coefficients)
p
The possible rational zeros of f (x) are at  .
q
Where p = factors of the constant term and
q = factors of the leading coefficient
Example: List all possible rational zeros:
a) f (x) = 3x2 + 5x + 2
b) f (x) = 6x4 + 3x2 + 4
Finding all real zeros of f (x):
1. Find all possible rational zeros. (±p/q)
(2. Graph the function and determine which zeros to use in synthetic division)
3. Perform synthetic division to determine which possible zeros yield a remainder of zero.
4. Use the zeros from step #3 to help you factor f (x).
5. Solve f (x) = 0
Properties of Polynomial Equations
1. If a polynomial equation has degree n, then the equation will have n roots.
2. If a + bi is a root of a polynomial equation, then a – bi will also be a root.
The Fundamental Theorem of Algebra – Every polynomial function, f (x), of degree n ≥ 1 will have
exactly n roots (solutions).
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Example: Find all zeros of: f (x) = x3 + 8x2 + 11x – 20
Example: Find all zeros of: f (x) = x3 + x2 – 5x – 2.
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Example: Solve:
x4 + x3 – 3x2 – x + 2 = 0
Example: Solve:
3x3 + 4x2 – 7x + 2 = 0
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Linear Factorization Theorem: Every nth degree polynomial can be expressed as the product of a
non-zero constant and n linear factors.
ie, f ( x)  ( x  c1 )( x  c2 )( x  c3 ) ( x  cn )
Example: The zeros of f (x) are: 2, –6, and ± 5 . Write f (x) as a product of linear factors.
Examples: Find any remaining zeros, then write f (x) as a product of linear factors.
a) Degree 3; Zeros are 4 and 2i
b) Degree 3; Zeros: 1, 1 + 2i
c) Degree 4; Zeros: ½ , –2, 4i
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Section 3.5 – Rational Functions and Their Graphs
A rational function is one of the form: R(x) =
p( x)
where p(x) and q(x) are polynomials and q(x)≠0.
q( x)
The domain of a rational function is all real numbers except those that make the denominator equal
zero.
Examples: Find the domain of each rational function
a) f (x) =
x2  3
x6
b) f (x) =
3x 4  x 2
x2  1
c) f (x) =
x4
x x2
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ASYMPTOTES:
Vertical Asymptotes: To find the vertical asymptote(s) of a rational function, set the denominator
equal to zero, and solve for x. (Vertical asymptotes will always have the form x = #)
Horizontal Asymptotes: To find the horizontal asymptote of a rational function, compare the highest
degree in the numerator to the highest degree in the denominator.
(Horizontal asymptotes will always have the form y = #)
There are 3 cases:
1. If the two degrees are the same, the horizontal asymptote is y = ba where a is the leading
coefficient of the numerator, and b is the leading coefficient of the denominator.
2. If the degree of the denominator is larger, then the rational function is proper, and the x-axis
is the horizontal asymptote (y = 0).
3. If the degree of the numerator is larger, then the rational function is improper, and there is
no horizontal asymptote.
Slant (Oblique) Asymptotes: If the degree of the numerator is one more than the degree of the
denominator, then the graph has an oblique asymptote of the form y = mx + b.
You can use long division to find the equation of the oblique asymptote.
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Examples: Find the domain, and all vertical, horizontal, and slant asymptotes. Sketch a graph.
a) f (x) =
3x
x  3x  4
2
2 x2
b) f (x) = 2
x 4
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c) f (x) =
x2  1
x 1
d) f (x) =
8x
x2  1
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Graphing Using Transformations:
Example:
Use the given graphs of f (x) =
1
x
and g (x) =
1
to graph the following:
x2
a) F ( x) 
1
x2
b) F ( x) 
1
1
x
c) G ( x) 
1
( x  2) 2
d) G ( x ) 
1
x2
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Section 3.6 – Polynomial and Rational Inequalities
A polynomial inequality is one of the form:
f ( x)  0, f ( x)  0, f ( x)  0, or f ( x)  0 .
Steps for Solving Polynomial Inequalities:
1. Solve the equation f ( x)  0 . The solutions will be your “boundary points”.
2. Draw the boundary points on a number line.
3. Choose a test value in each interval and evaluate f at those test values.
4. Write the solution set consisting of the interval(s) that satisfy the inequality.
Examples:
a) x 2  4 x  3  0
b) 9 x 2  3x  2  0
c)
 x  2 x  3 x 1
 0
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A rational inequality is one of the form:
rational function.
f ( x)  0, f ( x)  0, f ( x)  0, or f ( x)  0 where f is a
Steps for Solving Rational Inequalities:
1. Set the equation up so that zero is on one side.
2. Set the numerator and denominator equal to zero and solve. The solutions will be your
“boundary points”.
3. Draw the boundary points on a number line.
4. Choose a test value in each interval and evaluate f at those test values.
5. Write the solution set consisting of the interval(s) that satisfy the inequality.
Examples:
a)
x5
 0
x2
b)
3x  5
 0
6  2x
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