Math Studies I

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Math Studies I

• Chapter PowerPoint

2
INDEX

• Chapter 4
• Chapter 5
• Chapter 6
• Chapter 7
• Chapter 8
• Chapter 9

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Chapter 4

• Number Systems

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Chapter 4 Scientific Notation
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4.1 Number System
Closed set a set that contains all of its
accumulation points a set having an open set as
its complement. Ex. In the set of numbers,
1,2,3,4,5, we can select any two numbers (4
and 6), and add them together yielding in another
number from the same set. 4610 Natural
Numbers a set of positive numbers denoted by
the letter Ex. 1,2,3,4,5, Integers all
natural numbers including zero and negative
numbers denoted by the letter Ex. All whole
numbers and their opposites -3,-2,-1,0,1,2,3

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Rational Number a number that can be expressed
exactly by a ratio of two integers. Ex. Any
whole number, fraction, or decimal and their
opposites -2,-1.5,-1,0,.5,1,2 Irrational
Number a number that cannot be written as a
simple fraction the decimal goes on forever
without repeating. Ex. (3.1415926535897932384626
433832795028841971693993751058 2097494459230781640
62860899862803482534211706798214808651 32823066470
9384460955058223172535940812848111745028410270 193
85211055596446229489549303819644288109756659334461
2847 564823378678316527120190914564856692346034861
045432664821 3393607260249141273724587006606315588
17488152092096282925 40917153643678925903600113305
3054882046652138414695194151 160943305727036575959
195309218611738193261179310511854807 4462379962749
)
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• 4.2 Significant Figures
• Significant Figures digits that carry meaning
  or precision to a certain number.
• Rules
• Nonzero integers always count as significant
  figures.
• Ex. 3,456 Sig Figs 4
• Leading Zeros do not count as significant
  figures.
• Ex. 0.012 Sig Figs 2
• Captive zeros always count.
• Ex. 16.07 Sig Figs 4
• Trailing zeros count as significant figures only
  if a decimal point exists.
• Ex. 9.300 Sig Figs 4
• Exact Numbers have an infinite amount of
  significant figures.
• Ex. 1 2.54cm exactly

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• 4.3 Scientific Notation
• Scientific Notation a way of writing numbers
  that accommodates values too large or small to
  be conveniently written in standard decimal
  notation.
• Ex.
• Rules
• When multiplying numbers in scientific notation,
  their exponents add.
• Ex.
• When dividing numbers in scientific notation,
  their exponents subtract.
• Ex.
• When adding or subtracting numbers in scientific
  notation, their exponents must first be equal.
• Ex.

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4.4 Rounding Error Absolute Error The
difference between an absolute value of a
measurement and the true, non-approximated
value. Ex. A measurement of a stick is
interpreted as 5.9 but the true length is 6.0.
The absolute value is 5.9 and the true value is
6.0. Percent Error The difference between
the absolute value and the true value all divided
by the true value then multiplied by one
hundred. Ex. Relative Error The absolute
error divided by the true value. Ex.
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• 4.5 Computation Errors
• Computation Errors Whenever we calculate with
  one or more quantities that are in error, the
  results will also be in error.
• Ex. A rectangular field has been measured as
  being 120 meters long and 55 meters wide. Both
  measurements are correct to two significant
  figures. What are the errors if these figures are
  used to calculate the perimeter of the field?
• Length 120 meters, correct to 10 meters (2
  significant figures). This means that the
  smallest possible length is 115m and the largest
  possible length is 125m.
• Width 55 meters, so the smallest possible is
  54.5m and the largest is 55.5m.
• Smallest Perimeter 2(smallestW
  smallestL)2(11554.5)339 meters.
• Perimeter 2(WL)2(12055)350
• Largest Perimeter 2(largestW largestL)2(12555.
  5)361 meters.
• The perimeter can now be given as 35011

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• Practice problems
• Significant Figures
• How many significant figures are in the following
  problems?
• 0.0000756
• 8000.003
• 25.2
• 0000000.4000000050005050050000505050505
• 2400.0000000
• 008003.0
• 10.001

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• Practice Problems
• Scientific Notation

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• Practice Problems
• Rounding Errors
• Find the absolute value
• True Value 7.454
• Interpreted value 3.555
• Find the percent error
• Absolute value 36.453
• True value 66.666
• Relative error
• Absolute Error 0.0026
• True Value 0.0007

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Chapter 5

• Linear Equations

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Chapter 5.1 Number lines
A geometric representation of the real numbers
can represented by a point on the number line.
The numbers present on a number line consist of
all real numbers including zero.
If the number is not included in the set, it is
an open circle. If it is included, it is
represented by an closed circle.
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5.1
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5.2 Using Graphing Calculator

• How to solve linear equations with a TI-83
• First you need to call up Catalog option by
  hitting 2nd 0.
• Then press LN. (this brings up catalogue listing
  for those functions or operators that start with
  letter S.
• Use down arrow key until you find Solve.
• While using this form information must be written
  in the form solve( Equation 0, variable,
  initial guess)

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5.3 Problem Solving

• Key terms
• -sumadding
• -twice asmultiply by two
• - timesmultiply
• - differencesubtract
• - same asis equal to
• -consecutivetwo numbers
  
  where second is greater then first

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5.4 Linear Inequalities

• When dividing or multiplying by a negative number
  the direction of the inequality reverses.
• The methods used to solve inequalities is the
  same as solving equal equalities.

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5.5 How to speak inequality
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Examples 5.1

• Converting set notation into interval notation.

What is x so that x is greater than 2 but less
than or equal to 20?
Less than is represented by a parenthesis, while
greater than or equal to is represented by a
bracket.
Example
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5.1

• Converting interval notation into a set notation.

A parenthesis represents all real numbers not
including 2, while the bracket represents all
real numbers including 4.
X equals all real numbers greater than 2, but
less than or equal to 4.
Example
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Examples 5.2

• Solve the following equations for the unknown

Solve the equation for y Multiply 3 to
(y5) Subtract 3y from 5y Divide -2 by 2 The
answer is -1
3(y2) 5y8 y 3y6 5y8 6 2y8 -2
2y -1y CHECK 3(-12)5(-1)8 -36-58 33
3x4 13
7x-3 11
2x(34) 56
X 3
X 2
X 4
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Examples 5.3

• Two more than four times a number is 30. If the
  number is x,
• Set up a linear equation satisfying the above
  statement
• Solve for x
• 4x230 the set up
  equation
• 4x28
  subtract 2 from both sides
• x7
  divide by 4 on both sides
• My mother is three times as old as I am. If the
  sum of our ages is 112, how old is my mother.
• Answer 84

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Examples 5.4

• Find the value of x

First subtract 8 from both sides. Then divide by
5 on both sides. Answer is 11 then check.
When dividing by a negative number the inequality
switches sides.
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5.4

• Solve and graph

xlt11
xlt9
xlt2
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5.5

• A rectangle has a perimeter of at least 60cm.
  Given that the width of the rectangle is 15cm,
  what must the length of this rectangle be?
• The mean of 20, 35, and x must exceed 31. What
  must x be if xlt50?

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Chapter 6 Linear Graphs
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6.1- Things to remember

• The general equation of a linear graph is ymxc
• The gradient (slope) is m
• The y-intercept is c
• The domain is x

y
y
y
c
c
c
x
x
x
y
y
y
x
x
x
c
c
c
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What is the gradient and y-intercept of 3xy6
1) 3xy6
2) y-3x6
Subtract 3x from both sides
Answer) Gradient-3 y-intercept6
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What is the equation of the line that has a
gradient of 3 and passes through the point (0,8)
1) m3
2) y-intercept8
Answer) y3x8
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What is the equation of the line that passes
through points (0,-2) and (2,0)
1) y-intercept-2
Answer) yx-2
There is no gradient expressed in the question 2
is the x-intercept
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6.2 Things to remember

• The general equation of the gradient is
• The general form of a linear equation is
  axbyc0
• x-intercept is
• y-intercept is

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Find the equation of the straight line passing
through the points (2,3) and (6,3)
Find m using the equation of the gradient.
Reduce the fraction to its simplest form.
Use ymxc with the value of m found in step one.
Use any of the two points that the line passes
through to solve for c.
Multiply the 2 in the paranthesis.
Add the 3 to both sides.
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What is the x-intercept and y-intercept of
Multiply by 12 to get rid of fractions.
Subtract 48 from both sides.
Use x-intercept equation to solve.
Use y-intercept equation to solve.
Answer) x-intercept(6,0) y-intercept(0,-16)
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6.3.2Solving Systems of Equations font 32

• There are two algebraic methods to solve systems
  of equations.
• Elimination Method
• Substitution Method

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Method 1

• Make sure that each equation is set to slope
  intercept form. y mxb
• Set the two equations equal to each other.
• Solve for x.
• Set the x value into either equation and solve
  for y.
• The solution is the ordered pair that you just
  created.

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y2x1y4x-1
Set the two equations equal to each other and
solve for x.
1) 2x14x-1
2x24x
Add 1 to the other side.
22x
Subtract 2x from 4x.
x1
Divide by 2.
Insert x into one of the equations.
2) y2(1)1
y3
Answer (1,3)
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Method 2

• Substitute one of the equations into the y of the
  other equation.
• Solve for x.
• Set the x value back into one of the equations.
• The answer is the ordered pair that you just
  created.

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y2x12y3x-2
Substitute one of the equations into the y of
the other equation and solve for x.
2(2x1)3x-2
4x23x-2
Distribute.
4x3x-2
Take the 2 and subtract it from the -2.
4x3x-4
Take the 3x and subtract it to the 4x on the
other side.
x-4
Take the x value and substitute it into one of
the original equations.
y2(-4)1
y-7
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6.3.3Application of Simultaneous Equations

• Identify and label each of the variables.
• Translate the sentences in the problem into
  equations.
• Solve.

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A roll of dimes and a roll of quarters lie on the
table in front of you. There are three more
quarters than dimes. But the quarters are worth
three times the amount that the dimes are worth.
How many of each do you have?

1) Let d equal the number of dimes. Let q
equal the number of quarters.
2) There are three more quarters than dimes
? qd3
3) The quarters are worth three times the amount
that the dimes are worth ? 25q3(10d)
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• This second equation relies on the fact that
• if you have q quarters, they are worth
• a total of 25q cents.

5) Solve.
25(d3) 3(10d)
25d7530d
755d
d15 q18
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Chapter 7

• Quadratic Equations and Graphs

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Factoring

• Factorization
• Step one? Remove the GCF

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Step One Remove the GCF
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Factoring Quadratics

• Factoring Quadratics
• Form of Quadratics ?
• Step One Remove the GCF
• Step Two Multiply a and c together
• Step Three List the factors of ac that the sum
  equals b
• Step Four Group x with the factors of ac
  (xfactor) (xfactor)

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Sample Problems
Step One Multiply a and c together
2 and 1
Step Two List the factors of ac
(x2)(x1)
Step Three Group x and factors
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Perfect Squares and Difference of Squares

• Quadratics of the Form ?
• Step One Find a number that when it is doubled
  it equals the middle term and when squared equals
  the third term.

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Step One Divide the middle term by 2
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• Quadratics of the form ?
• It would factor in to this form? (xb)(x-b)

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Step One Group
(a-y)(ay)
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Completing the Square
Step One Create a perfect square
Step Two Use the difference of the two square to
factorize
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Quadratic Equations
Factoring requires the use of the Null Factor
Law
Step one set the equation equal to zero Step
two Factor the quadratic Step three Solve for
the variable
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Sample Problems
Step one set the equation equal to zero
Step two Factor the quadratic
Step Three solve for the variable
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Sample Answer
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Solving equations using completing the square

• Use in case a quadratic doesnt have rational
  factors.
• Turn the quadratic into a perfect square

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Quadratic Formula

• The Formula ?
• Step one Plug in the a, b, and c term into the
  quadratic formula.
• Step two Solve

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Sample Problem
Plug the a, b, and c terms into the quadratic
equations
Solve
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Quadratic Graphs

• Dilations and Translations

• Dilations along the Y-Axis ? y ax2
• When a increases the parabola is streched, When a
  decreases the parabola shrinks
• When agt0 it has a positive curve pointing up. ?
• When a lt0 it has a negative curve which is
  inverted. ?

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Expressing the Function in Turning Point Form

• If y (x-h)2 then the curve of yx2 is moved to
  the right
• If y (xh)2 then the curve of y x2 is moved
  to the left
• If y x2k then the curve of y x2 is moved up
  along the y-axis
• If y x2 - k then the curve of y x2 is moved
  down along the y-axis

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• Y x2 2x 3 Step one group a and b
• Y (x1)2 1 3 Step Two add
  -1 and 3 to get k
• Y (x1)2 2
• a 1, h-1, k2

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System Of Equations

• Y x2 4x 5 Equation 1
• Y 3 2x Equation 2
• x2 4x 5 3 2 x Step one Set two
  equations equal to each other.
• X2 2x 8 0 Step two Move everything to
  the left of the equal sign.
• (x-4) (x2) Step Three Group terms.
• Y 3 - 2(4) 5 Step Four Substitution
• Y 3 2 (2) 7
• (4,-5) and (-2,7)

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Chapter 8

• RelationshipsMathematical Ones!!

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Domain and Range

• The x-value of a coordinate is the domain.
• The range is the y-value of a coordinate.

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Mapping

• If there is a repeated domain number it has to be
  mapped to the same range number.
• Different domain numbers can be mapped to the
  same range number.

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Types of Brackets!

• - This symbol means that the number in
  included.
• This symbol means the number is excluded.

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Implied Domain

• Find the restrictions for the function and apply
  them when solving the function.
• Then graph the function.
• Then find the domain from the restrictions.

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Determining the Range
Example 1
Example 2
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The Vertical Line Test

• Sketch the graph.
• Determine how many times a vertical line cuts the
  graph.
• Finally, if the vertical line only cuts the
  domain once then it is a function.
• If the line cuts the domain more than once its
  not a function.

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Finding f(x)
Find
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Another Example
Find
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8.3-8.4

• Standard Functions!

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8.3.1 Hybrid Functions and Continuity
A hybrid function is a relation that consists of
more than one function, where each function is
defined over a mutually exclusive domain
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8.3.1 Continued

• Determine the domain of the function
• To Determine the range, sketch the graph of f.
• Use TI-83, or preferred calculating device, to
  sketch the graph of the function.

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8.3.2 The Absolute Value Function

• The absolute function is defined as
• READ BOOK!

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8.4.1 Solving equations involving unfamiliar
functions
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Chapter 9 Modeling Linear and Quadratic
Equations
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Sorry... ?

• No information or definitions, just
  examples!!!!!!
• Remember, Much of what is required for solving
  these problems, you already possess (Cirrito
  258).

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Example 1 Function

• Rachelle has taken (4x 6) hours to travel 102
  kilometers at a speed of (25x 1) km/h. How fast
  was she traveling?

?
? Plug in to the quadratic formula.
? Set equal to zero.
? Divide by two.
? Factor.
? Solve for x.
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Example 2 Function

• Derek sold walking sticks. Had he charged 10
  extra for each stick he would
• have made 1,800. However, had he sold ten more
  sticks at the original price
• he would have made 2,000.
• a) How many sticks did he sell?
• Let each stick cost x and let N be the number
  of sticks sold.

? Equation 1
? Equation 2
? Equation 1
? Equation 1 into equation 2.
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Continued
? Same equation made to solve by common
denominator
? Foiled on top
? Times both sides by x10
? Foiled the left side
? Foiled the right side
? Set equal to zero, forming a quadratic equation
? Divide by 10
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Continued
? Factor
? Since x can not equal a negative number the
answer can only be x50.
? Then plug (x) back into the 2nd equation for N.
Solve.

• N 30 sticks
• x 50 per stick

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Example 3 Graphing

• The demand equation for a certain product is
  given by the equation p40-0.0004x, where p is
  the price per unit and x is the number of units
  sold.
• Find the equation for the total revenue, R, when
  x units are sold.

? Fill in 40 0.0004x for p
? p must be greater than zero
? Solve for x
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Continued

• b) i. What is the revenue when 40000 units are
  sold?

? Fill in 40,000 for x
? Solve for x
89
Continued

• b) ii. How many units must be sold to produce a
  revenue of 600,000.

? Fill in 600,000 for R
? Set equal to zero
? Put into the quadratic equation
? Solve for x
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Continued

• b) iii. What is the maximum revenue that the
  product will return?

The graph of y R(x), we can see that the maximum
R value occurs when x50,000 50,000(0.0004x50000)
1000000 maximum revenue The x value is the
x value of the vertex.
R(x) x(40 0.0004x)
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Example 4Model

• An object is dropped from a building, 125m. High.
  During its descent, the distance, x m, above
  ground level is recorded as tabulated

a) Plot the data on a set of axes.
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Continued

• b) i. What type of curve would fit this data?
• ? Parabolic
• b) ii. Use a difference table to verify your
  answer.
• ?

Because the second difference is the same the
equation is quadratic.
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Continued

• c) Find the equation of the curve that best
  models the data.Insert data points into L1 and
  L2.
• Stat ? Calc 5