Notes for math 141
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1.1
Section 1.Q and 2.1-2.2
Finite Mathematics
Section 1.Q: Quadratic function
Notation and examples
Definition 1.1. Quadratic function:
Standard form:
Vertex form:
Parabola:
The graph of a quadratic function:
Definition 1.2. Vertex of a parabola:
Minimum or maximum value
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Notes for math 141
Section 1.Q and 2.1-2.2
Finite Mathematics
Vertex of Vertex form and Standard form
The x-intercepts of the quadratic function is given by the quadratic formula:
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Notes for math 141
Section 1.Q and 2.1-2.2
Finite Mathematics
Example 1.3. Use the quadratic function y = 2x2 − 16x + 45 to answer the following.
(a) Determine if the function opens upward or downward. (b) Find the vertex of the
parabola. (c) Find the maximum value. (d) Find the minimum value. (e) Find the
x-intercepts.
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Notes for math 141
Section 1.Q and 2.1-2.2
Finite Mathematics
Example 1.4. Use the quadratic function y = −5(x + 3)2 − 15 to answer the following.
(a) Determine if the function opens upward or downward. (b) Find the vertex of the
parabola. (c) Find the maximum value. (d) Find the minimum value. (e) Find the
x-intercepts.
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Notes for math 141
Section 1.Q and 2.1-2.2
Finite Mathematics
Example 1.5. Use the quadratic function y = 2x2 + 4x − 198 to answer the following.
(a) Determine if the function opens upward or downward. (b) Find the vertex of the
parabola. (c) Find the maximum value. (d) Find the minimum value. (e) Find the
x-intercepts.
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Notes for math 141
1.2
Section 1.Q and 2.1-2.2
Finite Mathematics
Application
We previously defined the revenue function:
We also discuss the relationship between quantity demanded and unit price is linear,
Combining the two concepts, the revenue function will be a quadratic function
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Notes for math 141
Section 1.Q and 2.1-2.2
Finite Mathematics
Example 1.6. Find the formula for the revenue function if the price-demand function
of a product is p = 54 - 3x, where x is the number of items sold and the price is in
dollars. How many items should be sold in order to maximize the revenue? What is
the maximum revenue?
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Notes for math 141
Section 1.Q and 2.1-2.2
Finite Mathematics
Example 1.7. A company has determined that if the price of an item is $40, then
150 will be demanded by consumers. When the price is $45, then 100 items are
demanded by consumers. (a) Find the price-demand equation, assuming that it is
linear. (b) Find the revenue function. (c) Find the number of items sold that will
give the maximum revenue. What is the maximum revenue? (d) What is the price of
each item when maximum revenue is achieved? (e) If the company has a fixed cost
of $3520 and a variable cost of $12 per item, find the companys linear cost function.
(f) Find the companys profit function. (g) Find the number of items sold that will
give the maximum profit. What is the maximum profit? (h) How many items should
be sold for the company to break even?
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Notes for math 141
2
2.1
Section 1.Q and 2.1-2.2
Finite Mathematics
Section 2.1: Systems of linear equations
Notation and examples
Definition 2.1. What does it mean to solve a system of linear equations?
What are the possible cases when solving a system of linear equations?
Example 2.2. Solve the following system of linear equations:
⎧
⎪
⎪4x − 5y = −30
⎨
⎪
⎪
⎩2x + y = −8
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Notes for math 141
Section 1.Q and 2.1-2.2
Finite Mathematics
Example 2.3. Solve the following system of linear equations:
⎧
⎪
⎪2x − y = 1
⎨
⎪
⎪
⎩6x − 3y = 12
Example 2.4. Solve the following system of linear equations:
⎧
⎪
⎪3x − 6y = 12
⎨
⎪
⎪
⎩x − 2y = 4
What is a parameter?
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Notes for math 141
2.2
Section 1.Q and 2.1-2.2
Finite Mathematics
Application
Note: We will first learn how to set-up the system in these word problems. We will
later return to learn how to systematically solve these problems.
Example 2.5. Jennifer has made it through her ten year class reunion. She is wanting
to remember how many former classmates, spouses, and former teachers attended the
reunion. She has lost her records but recalls that the ticket sales totaled $4, 775. She
charged $30 for each former classmate, $25 for each spouse, and $20 for each former
teacher. She recalls that the number of former classmates and spouses combined
was 130 more than the number of former teachers. She also recalls that there were
five times as many former classmates there as spouses. Help Jennifer remember the
number of former classmates, spouses, and former teachers that attended the reunion.
Set-up the linear system:
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Notes for math 141
3
3.1
Section 1.Q and 2.1-2.2
Finite Mathematics
Section 2.2: Systems of linear equations with
unique Solutions
Matrix
First I will give the definition of a matrix, we will talk more about matrices in
section2.4.
Definition 3.1. A matrix is an ordered rectangular array of numbers. A matrix with
m rows and n columns has size m × n. The entry in the ith row and jth column is
denoted by aij .
Example 3.2.
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Notes for math 141
3.2
Section 1.Q and 2.1-2.2
Finite Mathematics
Augmented matrices
Definition 3.3. Augmented Matrix: We can combine two matrices into one, visually
separating them by a vertical line. This is useful when solving a system of equations.
Example 3.4. Find the initial augmented matrix for the system of equations below:
⎧
⎪
⎪2x − y = 1
⎨
⎪
⎪
⎩y = 12 − 2x
⎧
2x − y + z = 1
⎪
⎪
⎪
⎪
⎨6x − 3y = 16z − 12
⎪
⎪
⎪
⎪
⎩x − 3y = 4z
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Notes for math 141
3.3
Section 1.Q and 2.1-2.2
Finite Mathematics
Gauss-Jordan Elimination
The goal of the Gauss − Jordan Elimination Method is to get the augmented matrix into Row Reduced Echelon Form by Row Operations.
Definition 3.5. A matrix is in Row Reduced Echelon Form when:
1. Each row of the coefficient matrix consisting entirely of zeros lies below any other
row having nonzero entries.
2. The first nonzero entry in each row is 1 (called a leading 1)
3. In any two successive (nonzero) rows, the leading 1 in the lower row lies to the
right of the leading 1 in the upper row.
4. If a column contains a leading 1, then the other entries in that column are zeros.
Note: We only consider the coefficient side (left side) of the augmented matrix when
determining whether the matrix is in row-reduced echelon form.
Example 3.6.
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Notes for math 141
Section 1.Q and 2.1-2.2
Finite Mathematics
Definition 3.7. To put a matrix inRow Reduced Echelon Form, there are three
valid Row Operations .
1. Interchange any two rows (Ri ↔ Rj ).
2. Replace any row by a nonzero constant multiple of itself (cRi )
3. Replace any row by the sum of that row and a constant multiple of any other row
(Ri + cRj ).
Steps for Gauss − Jordan Elimination:
1. Begin by transforming the top left corner element a11 into 1. This is your first
pivot element.
2. Next, transform the other elements in its column into zeros using the 3 row
operations.
3. Choose the next pivot element (diagonal down from the first pivot element).
4. Turn this 2nd pivot element into a 1, and transform the rest of its column into
zeros.
5. Continue until the coefficient matrix resembles the identity matrix(1’s along the
main-diagonal and 0’s everywhere else.)
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Notes for math 141
Section 1.Q and 2.1-2.2
Finite Mathematics
Example 3.8. Solve the following system of equations using Gauss Jordan Elimination:
⎧
⎪
⎪2x − y = 1
⎨
⎪
⎪
⎩3x + 4y = 7
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Notes for math 141
Section 1.Q and 2.1-2.2
Finite Mathematics
Example 3.9. Solve the following system of equations using Gauss Jordan Elimination:
⎧
3x − 2y + 8z = 9
⎪
⎪
⎪
⎪
⎨−2x + 2y + z = 3
⎪
⎪
⎪
⎪
⎩x + 2y − 3z = 8
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Notes for math 141
Section 1.Q and 2.1-2.2
Finite Mathematics
Example 3.10. Solve the following system of equations using Gauss Jordan Elimination:
⎧
2y + 3z = 7
⎪
⎪
⎪
⎪
⎨3x + 6y − 12z = −3
⎪
⎪
⎪
⎪
⎩5x − 2y + 2z = −7
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Notes for math 141
Section 1.Q and 2.1-2.2
Finite Mathematics
Note: Our calculator will put an augmented matrix into row-reduced form for us.
This only works when the # of rows is less than or equal to # of columns. Below are
the steps you must follow:
1. Enter the augmented matrix into your calculator.
2. Go to your home screen.
3. Press MATRIX, cursor right to MATH, and select B:rref.
4. Call the matrix you want to reduce and hit ENTER.
We will use the calculator for working the problems in the next section. You are
responsible for knowing how to do the Gauss-Jordan Method by hand, but always
remember that you can check your work with the calculator.
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