Marjan Ziai · Web viewWrite equations for the line, in point-slope form and in slope-intercept form, that passes through points (-1,4) and (2,3). Answers Example 1: Write the equation

Variable: a symbol, usually a letter, that represents one or more numbers; for example: x is a variable in the equation

Term: a number, or a variable or numbers and variables multiplied together; for example: , and are terms

Expression: mathematical phrase that can include numbers, variables and operation symbols; for example: , and are expressions

Polynomial: an expression with one or more monomials, which is combined using addition and/or subtraction; for example:

Monomial: a polynomial with one term; for example:

Binomial: a polynomial with two terms, for example:

Trinomial: a polynomial with three terms; for example:

Coefficient: the numerical factor when a term has a variable; for example: in the expression ; 6, -8 and 7 are the coefficients

Constant: a fixed value, a term that does not have a variable; for example: in the expression ; -9 is the constant

Base: a number that is multiplied repeatedly; for example: , 7 is the base

Exponent: a number that shows how many times the base is to be multiplied; for example: , 3 is the exponent, therefore, 7 · 7 · 7

Mathematics Vocabulary

Algebraic Properties:

Commutative Property of Addition: a + b = b + a

Commutative Property Multiplication: a · b = b · a

Associative Property of Addition: (a + b) + c = a + (b + c)

Associative Property of Multiplication: (a · b) · c = a · ( b · c)

Distributive Property:

a(b + c) = ab + ac(b + c)a = ba + caa(b – c) = ab – ac(b – c)a = ba - ca

Additive Identity: n + 0 = n

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Multiplicative Identity: n ·1 = n

Additive Inverse: there is an additive inverse –a such that a + (-a) = 0

Multiplicative Inverse: there is a multiplicative inverse , such that x a = 1

Substitution: if a = b, then b can be substituted for a in any expression

Properties of Equality and Inequality:

Addition: if a = b, then a + c=b + c

Subtraction: if a = b, then a - c=b - c

Multiplication: if a = b, then ac = bc

Division: if a=b then =

Properties of Inequality

Addition: if a<b, then a+c<b+c and if a>b, then a+c>b+c; applies to ≤ and ≥

Subtraction: if a<b, then a–c<b–c and if a>b, then a–c>b–c; applies to ≤ and ≥

Multiplication: when c is positive: if a<b, then ac<bc and if a>b, then ac>bc; applies to ≤ and ≥ when c is negative: if a<b, then ac>bc and if a>b, then ac<bc; applies to ≤ and ≥

Division:

when c is positive: if a<b, then < and if a>b, then < ; applies to ≤ and ≥

when c is negative: if a<b, then > and if a>b, then > ; applies to ≤ and ≥ Expression Property

a) 3 + 7 = 7 + 3 Commutative property of additionb) 9 + 0 = 9 Additive identityc) 6 x 1=6 or Multiplicative identityd) (6 x 4) x 5 = 6 x (4 x 5) or Associative property of multiplicatione) (6 + 4) + 5 = 6 + (4 + 5) Associative property of additionf) 5 + (-5)=0 Additive inverseg) 3 x 7 = 7 x 3 or Commutative property of multiplicationh) 2(x + 6)=2x + 12 Distributive property

i) 2 x = 1 Multiplicative inverse

Proportion: a statement that two ratios are equal (can be written as equivalent fractions or equal ratios)Page 2 of 47

Cross products: the resulting value of multiplying across the equal sign (a/b = c/d has cross products of ad = bc)

Similar figures: have the same shape but not necessarily the same size (corresponding angles are congruent and corresponding sides are proportional)

Indirect measurement: a way of measuring things that are too difficult to measure directly a drawing in which all lengths are proportional to the original

Scale drawing: a drawing in which all lengths are proportional to the original

Element: an item in a set

Set: a well-defined collection of elements

Relation: any set of ordered pairs

Domain: the possible values for the input of a function, usually the x values

Range: the possible values for the output of a function, usually the y values

Function: a relation that assigns exactly one value in the range to one value in the domain

Input/Output: input is the value that you substitute in the function and output is the solution

Function Notation: to write a rule in function notation, you use the symbol f(x) in place of y; for example: f(x) = 3x - 6 is written using function notation

Example 1: Find f(2) and f(y + 2) for the function f(x) = -3x-5

f(2) = -3(2) - 5 = -6 - 5 = -11 f(y + 2) = -3(y + 2) - 5 = -3y - 6 - 5 = -3y - 11

Example 2: Let f(x) = -2x + 6 and g(x) = 5x - 7. Find f(x) + g(x) and (f - g)(x).

f(x) + g(x) = -2x + 6 + 5x - 7 = 3x - 1 (f-g)(x) = (-2x + 6) - (5x - 7) = -2x + 6 - 5x + 7 = -7x + 13

Example 3: f(x) = x + 1 and g(x) = 2x. Find .

= = (x + 1)(2x) = 2x2 + 2x

Example 4: Let f(x) = 2x - 5 and g(x) = 3x + 2. Find (x).

Example 5: Let f(x) = x + 1 and g(x) = x - 5. Find .

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First find g(2): g(2) = 2 – 5 = –3

Then find f(–3): f(–3) = –3 + 1 = –2

Answer: = – 2

Identify 3 points and graph them. Make a table and record the coordinates of each point. Write the domain and range of the set of points.

Example 1:

f(x) = x - 4

Example 2:

f(x) = -3x + 5

Example 3: Fill in the following table given the function and domain

Function Domain Range Ordered Pairsf(x) = 3 – 4x –3, 0, 5

f(x) = x – 2, –4, 3

f(x) = –5x –2, –1, 0

Analyze the scenario, create a table and list the domain and range and ordered pairs.

Example 4:

The Spanish club is having a fund-raiser by selling piñatas that they have made. They will sell them for $15 each. The cost of all of the supplies to make them is $125. Their profit can be modeled with the function f(p) = 15p – 125 where p is the number of piñatas sold. What will their profit be if they sell 25? 50? 100?Answers

Identify 3 points and graph them. Make a table and record the coordinates of each point. Write the domain and range of the set of points.

Example 1:

f(x) = x - 4

x f(x)

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-3 -60 -43 -2

Domain: {-3, 0, 3} if you just use the points or all real numbers if you consider all points.

Range: {-6, -4, -2} if you just use the points or all real numbers if you consider all points.

Example 2:

f(x) = -3x + 5

x f(x)-2 110 5-2 -1

D: {-2, 0, 2} if you just use the points or all real numbers if you consider all points.

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R: {11, 5, -} if you just use the points or all real numbers if you consider all points.

Example 3: Fill in the following table given the function and domain.

Function Domain Range Ordered Pairs{-3 ,0, 5} {15, 3, -17} (-3,15), (3, 0), (5, -17)

{2, -4, 3}

{-2, -1, 0} {10, 5, 0} (-2,10), (5, -1), (0,0)

Analyze the scenario, create a table and list the domain and range and ordered pairs.

Example 4:

The Spanish club is having a fund-raiser by selling piñatas that they have made. They will sell them for $15 each. The cost of all of the supplies to make them is $125. Their profit can be modeled with the function f(p) = 15p – 125 where p is the number of piñatas sold. What will their profit be if they sell 25? 50? 100?

f(25) = 15(25) - 125 = 375 – 125 = 250

f(50) = 15(50) – 125 = 750 – 125 = 625

f(100) = 15(100) – 125 = 1375

x f(x)25 25050 625

100 1375

Domain: {25, 50, 100}

Range: {250, 625, 1375}

Linear Function: a function that satisfies the following two properties:

1.2.

Input/Output Pair: the input and output of a function is an "ordered pair", such as (4,16). This pair is an "ordered pair" because the input always comes first, and the output second. So (4,16) means that the function takes in "4" and gives out "16."

Example 1:

Make a table and graph the function represented by the following ordered pairs:

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{(-6,-3), (-3,-1), (0,1), (3,3), (6,5)}

Example 2:

Make a table and graph the function represented by the following domain and function.

Domain: {-2,-1,0,1,2} f(x)=2x–1

Example 3:

Make a table and graph the function:

f(x) = -2x + 5Example 1:


{(-6,-3), (-3,-1), (0,1), (3,3), (6,5)}

x f(x)-6 -3-3 -10 13 36 5

When you graph a function with discrete points, you do not connect them.

Example 2:

Make a table and graph the function represented by the following domain and function.

Domain: {-2,-1,0,1,2} f(x) = 2x–1Page 7 of 47

x f(x)-2 -5-1 -30 -11 12 3

Example 3:


f(x) = -2x + 5

x f(x)-2 9-1 70 51 32 1

Mathematics VocabularyPage 8 of 47

Rate of change: the relationship between two quantities that are changing, also referred to as slope

Slope: the ratio of the vertical change to the horizontal change; for example:

Example 1:

For the data in the table, is the rate of change for each pair of consecutive mileage amounts the same?

miles fee100 $30150 $42200 $54250 $66

Example 2:

For the data in the table, is the rate of change for each pair the same?

Number of days Rental charge1 $602 $753 $904 $1155 $130

Example 3: Find the rate of change of the data in the graph:

Example 4:

Find the slope of each line and state if the slope is positive, negative, zero or undefined:

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a)

b)

c)

d)

Example 5:

Find the slope of the line through each pair of points:

a) T(-4,-2) and U(9,-3)

b) A(6,-4) and B(7,-4)

c. C(3,8) and D(3,6)Answers

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Example 1: For the data in the table, is the rate of change for each pair of consecutive mileage amounts the same?

miles fee100 $30150 $42200 $54250 $66

42-30 = 12 54-42 = 12 66-54 = 12

Yes, the rate of change is the same.

Example 2: For the data in the table, is the rate of change for each pair the same?

Number of days Rental charge1 $602 $753 $904 $1155 $130

75-60 = 15 90-75 = 15 115-90 = 25 130-115 = 15

No, the rate of change is not the same.

Example 3: Find the rate of change of the data in the graph:

The difference between each y-value is 200. This is the rate of change of the data.Page 11 of 47

Time (sec) Distance (meters0 0

10 20020 40030 60040 800

Example 4:

Find the slope of each line and state if the slope is positive, negative, zero or undefined:

a)

Step 1: Find two points on the graph.

(1, 0) and (2, 3)

Step 2: Find the slope.

Step 3: State if the slope is positive, negative, zero or undefined

The slope is positive.

b)


(-4, 0) and (1, -3)



The slope is negative.

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c)


(-3, 0), (-3, 1)



The slope is undefined because division by zero is not defined.

d)


(0, -2) and (1, -2)


(All horizontal lines have a slope of 0)

Step 3: State if the slope is positive, negative, zero or undefined.

The slope is zero.Page 13 of 47

Example 5:

Find the slope of the line through each pair of points:

a) T(-4,-2) and U(9,-3)

b) A(6,-4) and B(7,-4)

c) C(3,8) and D(3,6)

The slope is undefined.

Slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept of the line

Example 1:

What are the slope (m) and y-intercept (b) of the following:

a. y = 2x - 3

b. y=- x + 6

c. 4x - 2y = -8

Example 2:

Write an equation in slope-intercept form with slope and y-intercept 4.

Example 3:

Graph the following equations:

a. y = x - 2

b. y = -2x + 1

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c. y=- xAnswers

Example 1: What are the slope and y-intercept (b) of the following?

a)

and

b)

c) 4x - 2y = -8

Step 1: Write the equation in slope-intercept form.

4x - 2y = -8

Subtract 4x from both sides:

-2y = -4x - 8

Divide all three terms by -2:

Step 2: Identify m and b.

and

Example 2: Write an equation in slope-intercept form with slope and y-intercept .

Example 3: Graph the following equations:

a) b)

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c)

Function Rule: an equation that describes a function

Rate of Change: the relationship between two quantities that are changing, also referred to as slope

y-Intercept: the y coordinate of the point where a line crosses the y axis

Growth Factor: the number b in an exponential growth function of the form , where b > 1

Decay Factor: the number b in an exponential decay function of the form , where 0 < b < 1

Constant: the number k in an exponential function of the form , also the horizontal asymptote

Asymptote: a line that continually approaches a given curve but does not meet it at any finite distance.

Find the rate of change (m) and y-intercept (b). Write a function rule for each table:Example 1:

x f(x)0 -21 -12 03 18 2

Example 2:

x f(x)Page 16 of 47

0 01 22 43 64 8

Example 3:

x f(x)0 -31 02 33 68 9Answers

Find the rate of change (m) and y-intercept (b). Write a function rule for each table:

Example 1:

Step 1: Find the rate of change (m)

Using the first two points: (0, -2) and (1, -1):

Step 2: Find the y-intercept (b)

When x = 0, y = -2 so the y-intercept is -2

Step 3: Write a function rule:

f(x) = mx + b = x - 2

Example 2:


Using the first two points: (0, 0) and (1, 2):




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f(x) = mx + b = x + 0 = x

Example 3:


Using the first two points: (0, -3) and (1, 0):




f(x) = mx + b = 3x + -3

Mathematics Vocabulary

Function notation: A function with domain X and codomain Y is commonly denoted by

or

The elements of x are called arguments of . For each argument x, the corresponding unique y in the codomain is called the function value at x or the image of x under . It is written as f(x). One says that associates y with x or maps x to y. This is abbreviated by

A general function is often denoted by .

Write a function rule from each graph:

Example 1:

Example 2: Example 3:

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Example 4:

Answers

Write a function rule from each graph:

Example 1:

Step 1: Find two points – (0, 3) and (2, 0)

Step 2: Calculate the slope:

Step 3: Find the y-intercept using the point where x = 0 – (0, 3)

y-intercept (b) = 3

Step 4: Write the function rule:

f(x) = mx + b = x + 3

Example 2:

Step 1: Find two points – (1, 2) and (0, –2)

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Step 3: Find the y-intercept using the point where x = 0 = (0, –2)

y-intercept (b) = –2


f(x) = mx + b = 4x – 2

Example 3:



Step 3: Find the y-intercept by using the point (2, 0) as follows:

y = mx + b

Subtract from both sides:

= b


y = mx + b

Example 4:



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Step 3: Find the y-intercept using the point where x = 0 – (0, –2)

y-intercept = –2


f(x) = mx + b = –2x + 2

Standard form:

y-intercept: the y coordinate of the point where the line crosses the y axis

x-intercept: the x coordinate of the point where the line crosses the x axis

Example 1:

Find the coordinates of the x- and y-intercepts of 2x + 5y = 6.

Example 2:

Graph 3x + 5y = 15 using intercepts.

Example 3:

Graph each function:

a) y = 4

b) x = -3

Example 4:

Convert the following functions in standard form to slope-intercept form:

a) 3x + y = -15

b) 2x + 7y = 14

c) -6x -3y = 9

Example 5:

Convert the following equations in slope-intercept form to standard form:

a) y=3x - 8

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b) y=- x + 4

c) y = 4xAnswers

Example 1:

Find the coordinates of the x- and y-intercepts of 2x + 5y = 6.

The x-intercept is where y = 0:

2x + 0 = 6; 2x = 6; x = 3

(3, 0)

The y-intercept is where x = 0:

0 + 5y = 6; 5y = 6; y =

Example 2:

Graph 3x + 5y = 15 using intercepts.

x-intercept: 3x = 15; x = 5; (5, 0)

y-intercept: 5y = 15; y = 3; (0, 3)

Example 3:

Graph each function:

a) y = 4

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b) x = -3

Example 4:

Convert the following functions in standard form to slope-intercept form:

a) 3x + y = -15


y = -3x - 15

b) 2x + 7y = 14


7y = -2x + 14

Divide ALL terms by 7:

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c) -6x -3y = 9

Add 6x to both sides:

-3y = 6x + 9

Divide ALL terms by -3:

y = 2x - 3

Example 5:

Convert the following equations in slope-intercept form to standard form:

a) y=3x - 8


-3x + y = -8

Multiply ALL terms by -1:

3x - y = 8

b)

Add to both sides:

Multiply ALL terms by 3:

2x + 3y = 12

c) y = 4x


-4x + y = 0

Multiply ALL terms by -1

4x - y = 0

Point-slope form: , where m is the slope and ( , ) is a point on the line Example 1:

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Write the equation of the line, in point-slope form and in slope-intercept form, with slope -2 that passes through the point (3,-3)

Example 2:

Write equations for the line, in point-slope form and in slope-intercept form, that passes through points (-1,4) and (2,3).Answers

Example 1:

Write the equation of the line, in point-slope form and in slope-intercept form, with slope -2 that passes through the point (3, -3)

Point-slope form:

y - - 3 = -2(x – 3)

y + 3 = -2(x – 3)

Slope-intercept form: y = mx + b

y = -2x + b

-3 = -2(3) + b

-3 = -6 + b

3 = b

y = -2x + 3

Example 2:

Write equations for the line, in point-slope form and in slope-intercept form, that passes through points (-1,4) and (2,3).


Step 2: Write the point-slope form:

Point-slope form:

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Step 3: Calculate the y-intercept (b):

Slope-intercept form: y = mx + b

Add to both sides:

Step 4: Write the slope-intercept form:

Parallel line: two lines in a plane that do not intersect or touch at a point

Perpendicular line: a line is perpendicular to another line if the two lines intersect at a right angle

Example 1:

Write an equation in slope-intercept form for the line parallel to y = 2x - 3 that contains (-5,-8).

Example 2:

Write an equation in standard form for a line perpendicular to y = x + 1 that contains (10,0).Answers

Example 1:

Write an equation in slope-intercept form for the line parallel to y = x - 3 that contains (-5,-8).

A line parallel to another line has the same slope which in this case is .

Slope-intercept form: y = x + b = x + b

-8 = 2(-5) + b

-8 = -10 + b

y = x +

Example 2:Page 26 of 47

Write an equation in standard form for a line perpendicular to y = x + 1 that contains (10,0).

A line perpendicular to another line has a slope that is the negative inverse of the first line’s slope which in this case is .

Slope-intercept form: y = x + b

0 = (10) + b

0 = -40 +b

y = x +

Add 4x to both sides: 4x + y = 40.

Exponential Function: a function that repeatedly multiplies by the same positive number

Growth factor: the amount >1 by which the parent function increases

Decay factor: the amount between 0 and 1 by which the parent function decreases

Example 1:

Find f(2) for each function:

a)

b)

c)

Example 2:


a)

b)

c)

Example 3:Page 27 of 47

Let and

Find

Example 4:

Let and

Find

Example 5:

Let and

Find (f◦g)(3) or f(g(3)Answers

Example 1:


a) f(x) = 2x; f(2) = 22 = 2x2 = 4

b) f(x) = + 2 ;f(2) =

c) f(x) = 3x - 4; f(2) = 32 - 4 = 3x3 - 4 = 9 - 4 = 5

Example 2:

Find f(-3) for each function:

a) f(x) = 2x; f(-3) = 2-3 =

b) f(x) = + 2; f(-3) = + 2 = + 2 = 4 3+3 = 4x4x4 + 2 = 64 + 2 = 66

c) f(x) = - 4; f(-3) = - 4 = 33 - 4 = 3x3x3 - 4 = 27 - 4 = 23

Example 3:

Let f(x) = 2x and g(x) = 3x - 5

Find

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Example 4:

Let f(x) = 3x + 2 and g(x) = 2x

Find

g(3) = 2^3 = 8

f(8) = 3^8 + 2 = 6561+2 = 6563

Example 5:

Let and

Find (f◦g)(3) or f(g(3)

Step 1: Find g(3)

g(3) = 33 + 2 = 27 + 2 = 22

Step 2: Find f(22)

f(22) = 222 + 2= 4,194,304 + 4 = 4,194,308

Exponential function: a function whose value is a constant raised to the power of the argument.

Example 1:

Example 2:

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x f(x)

Complete the table of values for each function:

Example 3:

x f(x)0 1 2 3 4

Example 4:

x f(x)0 1 2 3 4

For each of the following analyze the scenario, create a table and list the domain and range and ordered pairs.

Example 5:

A new car that sells for $16,000 depreciates at a rate of 22% each year. This can be modeled with the function f(x) = . How much will the car be worth after 2

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x f(x)

years? 5 years? 10 years?Answers Example 1: f(x) = 3x

x f(x)0 1-11 3

D: {all real numbers or the three specific x-values you wrote in the table}

R: {all real numbers > 0 or the three specific y-values you wrote in the table}

Example 2: f(x) = 2x - 3

X f(x)-10 -21 -1

D: {all real numbers or the three specific x-values you wrote in the table}

R: (all real numbers > -3 or the three specific y-values you wrote in the table}

Example 3: f(x) = 4x

x f(x)0 11 42 163 644 256

Example 4: f(x) = 3x+2

x f(x)0 31 52 113 294 83

Example 5:

A new car that sells for $16,000 depreciates at a rate of 22% each year. This can be modeled with the function f(x) = 16000(0.78)x. How much will the car be worth after 2 years? 5 years? 10 years?

f(2) = 16000(0.78)2 = 16000(0.6084) = $9734.40 (Remember the order of operations – exponents Page 31 of 47

before multiplication!)

f(5) = 16000(0.78)5 = 16000(0.2887) = $4619.48 rounded to the nearest hundredth

f(10) = 16000(0.78)10 = 16000(0.0833) = $1333.72 rounded to the nearest hundredth

Exponential: change by a given proportion over a set interval. An example of exponential behavior is a medical isotope decaying to half the previous amount every twenty minutes and a bacteria culture tripling every day each, because, in a given set amount of time (in these examples - twenty minutes or one day), the quantity has changed by a constant proportion (one-half as much and three times as much.

Exponential function graph: points on a graph are either too close to one fixed value or else to too large to be conveniently graphed. There will generally be only a few points that are "reasonable" to use for drawing the curve; picking sensible points require that a good grasp of the general behavior of an exponential, so the curve can be approximated

Example 1:


{(0, 1), (1, ), (-1,2), (-2,4), (-3,8)}

Example 2:

Make a table and graph the function represented by the following domain and equation:

Domain: {-2,-1,0,1,2}; f(x)=3x^+4

Example 3:


Answers

Example 1:


{(0, 1), (1, ½), (-1, 2), (-2, 4), (-3, 8)}

x f(x)

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0 11-1 2-2 4-3 8

Example 2:

Make a table and graph the function represented by the following domain and equation:

Domain: {-2, -1, 0, 1, 2}; f(x) = 3x + 4

x f(x)-2

-10 51 72 13

Page 33 of 47

Example 3: Make a table and graph the function:

x f(x)-2 7-1 10 -11

2

y-Intercept: the y coordinate of the point where the function crosses the y-axis

Growth/Decay Factor: the b value in

Constant/Shift: the c value in Page 34 of 47

Example 1: Find the growth/decay factor and constant. Write a function rule for each table:

a) b)

x f(x)

0 1

1 2

2 4

a b

c d

b)

c)

x f(x)

0 0

-1 2

-2 8

a b

c d

Answers

Example 1: Find the growth/decay factor and constant. Write a function rule for each table:

a)

x f(x)

0 1

1 2

2 4

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x f(x)

0 4

1 5

2 7

a b

c d

Vertical shift = 1 – 1 = 0

Growth Factor = 2

f(x) = 2x

b)

x f(x)

0 4

1 5

2 7


Growth Factor = 2

f(x) = 2x + 3

c)

x f(x)

0 0

-1 2

-2 8

Vertical shift = 0 – 1 = – 1

Decay Factor = 3

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y-Intercept: the y coordinate of the point where the function crosses the y-axis

Growth/Decay Factor: the b value in

Constant/Shift: the c value in

Example 1:

Find the growth/decay factor and constant. Write a function rule for each table:

b) B)

x f(x)0 11 2 2 4a bc d

c)

x f(x)0 0-1 2-2 8a bc d

Answers; Example 1:

Find the growth/decay factor and constant. Write a function rule for each table:

a)

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x f(x)0 41 52 7a bc d

x f(x)0 11 22 4


Growth Factor = 2 f(x) = 2x

b)

x f(x)0 41 52 7


Growth Factor = 2

f(x) = 2x + 3

c)

x f(x)0 0-1 2-2 8

Vertical shift = 0 – 1 = – 1

Decay Factor = 3

Vertical Shift: the movement along the y-axis / change from parent function

Asymptote: the boundary that a function approaches but does not cross

Example 1: Write a function rule from each graph: a)

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b) c)

Answers: Example 1: Write a function rule from each graph:

a)

Vertical shift = (0, 1) to (0, -2) = - 3

Asymptote: y = - 3

Growth Factor = 3 – 1 = 2 (see blue line)

f(x) = 2x – 3

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b)

Vertical shift = (1, 0) to (2, 0) = 1

Asymptote: y = 1

Decay Factor = 3 – 1 = 2 (see blue line)

c)

Vertical shift = (0, 1) to (0, 3) = 3 – 1 = 2

Asymptote: y = 2

Growth Factor = 4 – 2 = 2

f(x) = 2x + 2

Parent Function: the most basic equation of a function, before any shifts or alterations

Translation: a transformation that shifts a function horizontally, vertically or both

Vertical Shift: a transformation up or down

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Example 1:

Write the parent function and the vertical shift for each of the following functions:

a)

b)

Example 2:


a)

b) Answers ---- Example 1:


a) f(x) = 3x + 2

Parent function: 3x

Vertical shift: 2

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b)

Parent function:

Vertical shift: – 6

Example 2: a)

Parent function: 2x

Vertical shift: 3 – 1 = 2 (see the blue line)

b)

Parent function:

Vertical shift: -1 – 1 = – 2

Sequence – a series of numbers that follows a pattern.

Arithmetic Sequence – a fixed number between each of the terms in a sequence.

Discrete data – only the exact points are shown.

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Continuous data – data that changes continuously.

Geometric Sequence – a sequence where you find terms by multiplying a fixed number by a common ratio.

Arithmetic Sequence: a numerical sequence formed by adding a term in the sequence by a fixed number to find the next term.

Common Difference: the fixed number that is added to each term of an arithmetic sequence.

Arithmetic Sequence Formulas:

Recursive: Explicit: which must be simplified once the values are plugged in!!

Example 1: Is the given sequence arithmetic? If so, identify the common difference. a) 2, 4, 8, 16... b) 5, 9, 13, 17, 21... Example 2: Find the next three terms of the sequence: 1100, 1135, 1170, ... Example 3: Given the explicit formula, write an arithmetic sequence for each of the following: an = -42+10n Example 4: Given the first four numbers in each arithmetic sequence write a recursive and explicit

formula: 39, 42, 45, 48, ... Answers Example 1: a) 2, 4, 8, 16... Common difference: 4 – 2 = 2, 8 – 4 = 4… NO, it is not an arithmetic sequence! b) 5, 9, 13, 17, 21... Common difference: 9 – 5 = 4; 13 – 9 = 4… YES, it is an arithmetic sequence with d = 4 Example 2: Find the next three terms of the sequence: 1100, 1135, 1170… Common difference: 1135 – 1100 = 35 a4 = 1170 + 35 = 1205 a5 = 1205 + 35 = 1240 a6 = 1240 + 35 = 1275 Example 3: Given the explicit formula, write an arithmetic sequence for each of the following: an = –42+10n a1 = –42 + 10(1) = -42 + 10 = – 32 a2 = –42 + 10(2) = -22 a3 = –42 + 10(3) = -12 Sequence: –32, –22, –12, –2, 8, 18, … Example 4: Given the first four numbers in each arithmetic sequence write a recursive and explicit formula:

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39, 42, 45, 48, ... a1 = 39, d = 3 E: an = a1 + d(n–1); an = 39 + 3(n-1) = 39 +3n – 3 = 36 + 3n R: an = a(n-1) + d = a(n-1) + 3

Geometric Sequence: a number sequence formed by multiplying a term in a sequence by a fixed number to find the next term

Common Ratio: the fixed number that is multiplied to each term of a geometric sequence.

Geometric Sequence Formulas:

Recursive: ____, Explicit:

Example 1: Is the given sequence geometric? If so, identify the common ratio. a) 5, 15, 45, 135, ... b) 15, 30, 45, 60, ... Example 2: Find the next three terms of the following geometric sequences a) 2, 12, 72, 432, ... b) 4, 24, 144, 864, ... Example 3: Given the explicit formula, write a geometric sequence for each of the following: a) an = 2x2(n-1)

b) an = 2x4(n-1)

Example 4: Given the first four numbers in each geometric sequence write a recursive and explicit

formula: a) -1, -3, -9, -27, ... b) 4, 12, 36, 108, ... Answers Example 1: Is the given sequence geometric? If so, identify the common ratio. a) 5, 15, 45, 135, ... 15 3 = 5; 45 15 = 3

r = , YES b) 15, 30, 45, 60, ... 30 15 = 3; 45 30 = 1.5 NO Example 2: Find the next three terms of the following geometric sequences: a) 2592, 15,552, 93,312 15,552 2592 = 6 = r a1 = 93,312 x 6 = 559,872 a2 = 559,872 x 6 = 3,359,232 a3 = 3,359,232 x 6 = 20,155,392 b) 5184, 31,104, 186,624

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31,104 5184 = 6 = r a1 = 1,119,744 a2 = 6,718,464 a3 = 40,310,784 Example 3: Given the explicit formula, write a geometric sequence for each of the following: a) an = 2x2(n-1)

an = a1xr(n-1)

a1 = 2, r = 2 a1 = 2 a2 = 2x2(2-1)= 2x21 = 2x2 = 4 a3 = 2x2(3-1)= 2x22 = 2x4 = 8 Sequence: 2, 4, 8, 16, 32, … b) an = 2x4(n-1)

an = a1xr(n-1)

a1 = 2, r = 4 a1 = 2 a2 = 2x4(2-1) = 2x41 = 2x4 = 8 a3 = 2x4(3-1) = 2x42 = 2x16 = 32 Sequence: 2, 8, 32, 128, 512, … Example 4: Given the first four numbers in each geometric sequence write a recursive and explicit formula: a) – 1, – 3, – 9, – 27, … a1 = –1

r = = –3 E: a1xr(n-1) = (– 1)(3)(n-1)

R: an = a(n-1) x r = a(n-1) x 3, a1 = –1 b) 4, 12, 36, 108, ... a1 = 4

r = = 3 E: a1 x r(n-1) = (4)(3)(n-1)

R: an = a(n-1) x r = a(n-1) x 3, a1 = 4 Linear Function: a function that can be represented on a graph as a straight line; for

example: Exponential Function: a function that repeatedly multiplies the initial amount by the same

positive number; for example: Simple Interest: interest on an investment that is calculated once per period, usually annually, on

the amount of the capital alone and not on any interest already earned. Formula: I = P x r x t where I = interest, P = principal or starting value, r = interest rate.

Compound Interest: interest which is calculated not only on the initial principal but also the accumulated interest of prior periods. Formula: B = P(r + 1)t where B = Balance, P = principal or starting value, r = interest rate.

Example 1: Suppose you deposit $500 in a savings account. The interest rate is 4% per year. Find

the simple interest earned in 10 years. How much money will you have? Make a table and graph of the first 10 years of interest and the resulting income.

Example 2: Suppose you deposit $500 in a savings account. The interest rate is 4% compounded

annually. Find the compound interest earned in 10 years. How much money will you have? Make a table and graph of the first 10 years of interest and the resulting income.

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Answers Example 1: I = P x r x t; P = $500, r = 0.04; t = 1 through 10

Year Interest Income1 I = 500x.04x1 = 20 500+20 = 5202 I = 500x.04x2 = 40 500+40 = 5403 I = 500x.04x3 = 60 500+60 = 5604 I = 500x.04x4 = 80 500+80 = 5805 I = 500x.04x5 = 100 500+100 = 6006 I = 500x.04x6 = 120 500+100 = 6207 I = 500x.04x7 = 140 500+100 = 6408 I = 500x.04x8 = 160 500+100 = 6609 I = 500x.04x9 = 180 500+100 = 680

10 I = 500x.04x10 = 200 500+100 = 700

Example 2: Balance = P(r+1)t; P = $500, r = 0.04, t = 1 through 10

Year Balance1 500(1.04)1 = 500(1.04) = 5202 500(1.04)2 = 500(1.04)2 = 500(1.082) = 540.803 500(1.04)3 = 500(1.04)3 = 500(1.125) = 562.434 500(1.04)4 = 500(1.04)4 = 500(1.117) = 584.935 500(1.04)5 = 500(1.04)5 = 500(1.217) = 608.336 500(1.04)6 = 500(1.04)6 = 500(1.265) = 632.667 500(1.04)7 = 500(1.04)7 = 500(1.316) = 657.978 500(1.04)8 = 500(1.04)8 = 500(1.369) = 684.289 500(1.04)9 = 500(1.04)9 = 500(1.423) = 711.66

10 500(1.04)10 = 500(1.04)10 = 500(1.48)= 740.12

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