Review for Exam #3
3.1 Quadratic Functions
A quadratic functions is any function of the form f (x)  ax 2  bx  c , where a, b, c are
real numbers with a  0. The graph of a quadratic function is a parabola.


If a  0,
The graph opens upward and has
the minimum point, vertex. The vertex is

b
b
b
is the axis of symmetry.
( , f ( )) . The vertex line x  
2a
2a
2a
If a  0,
The graph opens downward and has the maximum point, vertex. The vertex is
b
b
b
is the axis of symmetry.
( , f ( )) . The vertex line x  

2a
2a
2a
y
y
Axis of symmetry
Vertex (maximum
point)

Vertex (minimum
point)
x
x
a0
a0
The standard form of a quadratic function is f ( x)  a( x  h) 2  k , where the point
(h, k)is the vertex of the parabola.


*Graph Quadratic Functions
Step 1 If a  0 , it opens upward. If a  0, it opens downward.
b
b
Step 2 The vertex of the parabola is (h, k) or ( , f ( )) .
2a
2a
f
(
x
)

0
Step
3
Find
x-intercepts
by
solving
.


Step 4 Find y-intercepts by computing f (0) .
Step 5 Plot the intercepts and the vertex.
Connect these points with a smooth curve.

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Example 1) Let f ( x)  2 x 2  12 x  22 .
a) Determine whether f (x ) has a maximum or a minimum and find the maximum
or minimum value of f (x ) and where it occurs.
b) Graph the function f (x ) and determine the function’s domain and range (in
interval notation).
Example 2) A stone mason has 60 feet of stone wall to enclose a rectangular patio that
borders a house. If the stone wall is not used along the side of the house, find:
a) The dimensions of that patio that yield the maximum area for the patio.
b) The maximum area that the mason may enclose.
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Example 3) A projectile is shot vertically into the air from a platform. The height, h(t) ,
in feet, of the projectile above the ground after t seconds is modeled by
h(t)  16t 2 112t  33 . Give solutions accurate to three decimal places.

a) What is the height of the projectile above the ground after 2 seconds?

b) How many seconds will it take the projectile to reach its maximum height?
c) What is the maximum height of the projec
d) How long will it take the projectile to reach the ground?
3.2 Polynomial Functions and Their Graphs
f (x)  an x n  an1 x n1   a2 x 2  a1 x1  a0 is a polynomial function of degree n,
where an , an1 ,, a2 , a1 , a0 are real numbers with an  0 . The number a n is called the
leading coefficient.

*Graphing a Polynomial Function f ( x)  a n x n  a n 1 x n 1    a 2 x 2  a1 x1  a0
Step1 Use Leading Coefficient Test to determine the end behavior.
n: odd (Opposite behavior at each end)
If an  0, the graph falls to the left and rises to the right.
If an  0, the graph rises to the left and falls to the right.
n: even (Same behavior at each end)
If an  0, the graph rises to the left and rises to the right.

If an  0, the graph falls to the left and falls to the right.



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Step2 Find x-intercepts by setting f (x)  0 . For a factor (x  r)k of f (x)
a. If k is even, the graph touches the x-axis at r and turns around
b. If k is odd, the graph crosses the x-axis at r.
c. If k>1, the graph flattens out at (r, 0).
computing f (0) . 

Step 3 Find y-intercept by
Step 4 Use symmetry, if applicable, to help draw the graph:
a. y-axis symmetry: f (x)  f (x)
b. Origin symmetry: f (x)   f (x)

Step 5 Use the fact that the maximum
number of turning points of the graph is n 1to
check whether it is drawn correctly.

f (x)  3(x  5)(x  2) 2
Example 4) Graph 


Example 5) Graph f (x)  x 3  4 x 2  4 x


Example 6) Use the graph of the
polynomial f (x) below, to answer True
or False for the statements that follow.


a) The sign of the leading coefficient
is negative.
b) The degree of the polynomial is
even.
c) At x  4 , f (x) has a zero of
even multiplicity.



















4

3.3 Dividing Polynomials; Remainder and Factor Theorem
*Long Division
Divide-Multiply-Subtract-Bring down
Example 7) Find the quotient and the remainder: (12 x 3  4)  (3x  1) .
*Synthetic Division
We can use synthetic division only if the divisor is in the form x c .
Example 8) Find the remainder after division f (x)  x 5  2x 4  9x 3  20x 2  2x  6 by
d(x)  x  3 .



*The Remainder Theorem
If f (x ) is divided by x  c , the remainder is f (c ) .
*The Factor Theorem
a. If f (c)  0 , then x  c is a factor of f (x ) .
b. If x  c is a factor of f (x ) , then f (c)  0 .
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3.4 Zeros of Polynomial Functions
*Properties of Polynomial Equations
1. If f (x ) is a polynomial of degree n, then the equation f (x)  0 has n complex roots,
possibly real and not necessarily distinct.
2. If a  bi is a root of a polynomial equation with real coefficient (b  0) , then the
complex imaginary number a  bi is also a root.

3. If c1 , c 2 , , cn are roots of f (x) , then f ( x)  an ( x  c1 )( x  c2 ) ( x  cn ) with
an  0 .
4. Possible rational zeros=

Factors of
Factors of
a0
.
an
*Finding Zeros of a Polynomial Function
Step 1 List all possible rational zeros
Step 2 Use synthetic division to find a rational zero among the possibilities.
Step 3 Factor the polynomial using synthetic division.
Step 4 Repeat Step 1 through Step 3 until you get a quadratic factor.
Example 9) Let f (x)  2x 3  5x 2  x  2 .
a) List all possible rational zeros of f (x ) .

b) Use synthetic division to find a zero of f (x ) . Use that zero to algebraically find
all the zeros of f (x ) .
c) Write f (x ) as a “product of linear factors”.
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Example 10) Using long division and given root, find the solution set of the given
polynomial equation. Give exact solutions.
Equation: x 4  4 x 3  2 x 2  16 x  8  0
Root: 2i
3.6 Polynomial and Rational Inequalities
*Solving Polynomial Inequalities
Step 1 Express the inequality in the form f (x)  0 or f (x)  0 .
Step 2 Solve the equation f (x)  0 .
Step 3 Locate the boundary points on a number line and separate the line into intervals.
Step 4 Choose one test value within each interval and evaluate f at that number.


Step 5 Write the solution set, selecting
the interval
or intervals that satisfy the given

inequality.
*Solving Rational Inequalities
Step 1 Express the inequality so that one side is zero 
and the other side is a single
quotient.
Step 2 Set the numerator and the denominator of f equal to zero.
Step 3 Locate the boundary points on a number line and separate the line into intervals.
Step 4 Choose one test value within each interval and evaluate f at that number.
Step 5 Write the solution set, selecting the interval or intervals that satisfy the given
inequality. (Never include the value thatcause the denominator to equal zero)
Example 11) Solve x 2  8x  6 and write the solution in interval notation.

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Example 12) Solve
2x  3
 3 and write the solution in interval notation.
x5
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