Course Outline Functions/Graphing Solving Equations ...€¦ · Course Outline Functions/Graphing Solving Equations Applications ... Long/Synthetic Division (4.3) ... 5 4 3 2 1 1

Course Outline

Functions/Graphing Solving Equations Applications

Definitions of function, graph,domain, range, x- and y-intercepts, vertical line test(1.1-1.2)Linear functions (1.3-1.5)-Parallel and perpendicularlines

Linear EquationsLinear Inequalities (1.6)

Fahrenheit to Celsius conver-sion, Cost per hour, Averagerate of change

Piecewise Functions:Absolute value functions (2.1)

Absolute Value Equations andInequalities (3.5)

Quadratic Functions:Completing the square, ver-tex, increasing/decreasing,maximum/minimum, graphmovements (2.1, 2.5, 3.2-3.3)

Quadratic Equations:Complex numbers, Quadraticformula (3.1-3.2)

Maximum/minimum problems

Polynomial functions:Relative/Absolute Maxi-mum/Minimum (4.1-4.3)

Polynomial Equations:Long/Synthetic Division (4.3)

Radical Functions:Graphing with movements

Radical Equations (3.4)

Rational Functions:Horizontal/Vertical asymptotes(4.5)

Rational Equations (3.4)

Algebra of Functions:Sums and Di↵erences, Productsand Quotients, Compositions,Di↵erence Quotients (2.2-2.3)Inverse Functions (5.1) Finding InversesExponential Functions (5.2) Exponential Equations (5.5) Interest, Growth/Decay (5.6)Logarithmic Functions (5.3) Logarithmic Equations:

Logarithm rules (5.4)Solving more complicated log-arithmic/exponential equations(5.5)

1

2

Introduction to Functions and Graphing (Sections 1.1-1.2)

Definition. A function is a rule that assigns to each member of the first set (called the domain)exactly one member of a second set (called the range).

x(input)

f(x)(output)f

Example. Determine whether or not each diagram represents a function.

a

b

c

d

1

2

3

a

b

c

d

1

2

3

3

Notation. Functions are usually given by formulas, such as:

y = f(x) = g(x) =

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

2 + 2x+ 1. Determine the outputs below.

f(�1) f(4)

g(�2) g(3)

Example. Now let’s try some that are a little harder.

f(x) = 5x� 7 and g(x) = �x2 + 3x.

Determine the outputs below.

f(2t+ 4) g(t+ 1)

4

Definition. The graph of a function, f(x), is the picture obtained by plotting all ordered pairsof the form (x, f(x)), i.e. (input, output).

Example. Determine if the following graphs correspond to functions.

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Theorem (Vertical Line Test). If it is possible for a vertical line to cross a graph more thanonce, then the graph is . . .

5

Definition. Given a function y = f(x):

• x-intercept(s) of the function • y-intercept of the function

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• domain of the function • range of the function

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Interval Notation:

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Example. Graph the following functions using a table. Then use the graph to estimate the x-and y-intercepts, as well as the domain and range.

• f(x) = x

2 + 4x+ 5

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• h(x) =px� 1

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Example. You can also use a table to draw graphs of equations that are not functions.

• y = 3± x

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• x

2 + y

2 = 4

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Example. Graph the following on your graphing calculator. Using the graph, estimate the x-and y-intercepts, as well as the domain and range.

y = �px� 3 y = x

4 � 2x3 � 3

x-intercept(s)

y-intercept

Domain

Range

x-intercept(s)

y-intercept

Domain

Range

y =1

x+ 5y = 2

px� 1� 4

x-intercept(s)

y-intercept

Domain

Range

x-intercept(s)

y-intercept

Domain

Range

Since we can’t always rely on the graph to be able to determine the x-/y-intercepts and thedomain/range exactly, we will have to learn algebraic methods as well.

9

Linear Functions and Linear Equations (Sections 1.3-1.5)

Definition. A function f is called a linear function if it can be written in the form

where m and b are constants.

Example. Determine whether each of the following is a linear function.

2y � 3x = 5 y

2 � 4x+ 2 = 0

y � 2x = 01

2(8x� 3y + 4) = 0

10

Remark. The two constants m and b represent important parts of the graph.

• The constant b is

• The constant m is called the slope of the line and is a measure of the “steepness” of theline

where (x1

, y

1

) and (x2

, y

2

) are two points on the line.

(x1

, y

1

)

(x2

, y

2

)

x

2

� x

1

(run)

y

2

� y

1

(rise)

11

Example. Graph the following linear functions without using a table.

f(x) = 2x� 3 y = 3x+ 1

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2x+ y = 4 g(x) = �2

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Example. Find the slope of the line through the two given points.

(3, 2) and (�1, 5) (6, 8) and (3, 8)

(1,�3) and (5, 1) (2, 0) and (2, 8)

(�1, 4) and (12, 4) (20, 1) and (8, 12)

13

Remark.

• Horizontal lines: y = b � Are linear functions• Vertical lines: x = a � Are NOT linear functions

Example. Graph the following lines.

y = �2 y = 5

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x = 4 x = �1

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Note: The slope of a horizontal line is .

The slope of a vertical line is .

14

Remark. To find the equation of a line, you can use one of the following forms of a line.However, your answer should always be given in slope-intercept form.

Slope-Intercept Form Point-Slope Formy = mx+ b y � y

1

= m(x� x

1

)

Example. Find the equation of the line through (2, 5) having slope m =�27.

Example. Find the equation of the line given a point on the line and the slope.

(�3, 4) and m = �2 (0, 5) and m =3

7

15

Example. Find the equation of the line through (3, 2) and (�1, 5).

Example. Find the equation of the line through the given points.

(1, 5) and (0, 4) (2, 5) and (2, 8)

(3, 6) and (�7, 6) (4,�1) and (2, 5)

16

Remark. To find x- and y-intercepts algebraically:

• Set x = 0 for y-interceptNote: For linear equations in the form y = mx+ b, the y-intercept is just (0, b)

• Set y = 0 for x-intercept(s)

Example. Find the x- and y-intercepts for the graph of each equation.

y = 2x+ 7 y =�23x+ 1

x-intercept(s)

y-intercept

x-intercept(s)

y-intercept

x = �3 y = 5

x-intercept(s)

y-intercept

x-intercept(s)

y-intercept

17

Applications of Linear Functions–Average Rate of Change (Section 1.3)

Remark. The average rate of change between any two data points on a graph is the slope ofthe line passing through the two points.

Example. Kevin’s savings account balance changed from $1140 in January to $1450 in April.Find the average rate of change per month.

Example. The table below indicates the number of cases of West Nile Virus over a period offour months. What is the average rate of change, in cases per month, between months 2 and 4?

Month 1 2 3 4# of Cases 22 34 30 16

Example. Find the average rate of change for each function on the given interval.f(x) = x

3 + 2x2 � x on [�1, 2] f(x) = 7x� 3 on [4, 7]

18

Applications of Linear Functions–Modeling (Section 1.4-1.5)

Example. Given the following information, determine a formula to convert degrees Celsius todegrees Fahrenheit.

• Water freezes at 0� Celsius and 32� Fahrenheit.• Water boils at 100� Celsius and 212� Fahrenheit.

Example. A taxi service charges a $3.25 pickup fee and an additional $1.75 per mile. If thecab fare was $17.60, how many miles was the cab ride?

19

Example. A lake near the Arctic Circle is covered by a 2 meter thick sheet of ice during thewinter months. When spring arrives, the warm air gradually melts the ice, causing its thicknessto decrease at a constant rate. After 3 weeks, the sheet is only 1.25 meters thick. Determine aformula to represent the thickness of the ice sheet x weeks after the start of spring.

Example. In a certain College Algebra class at IUP, final grades are computed according tothe following chart:

Exams: 45%Final Exam: 20%Homework: 15%

Quizzes: 20%

A student scores 79% on Exam 1, 61% on Exam 2, and 83% on Exam 3. On MyMathLab, theirhomework average is 91%; and their quiz average is 85%. Determine a linear function that willcompute the students final grade based on their score on the final exam. Use this function todetermine what the student needs on the final exam to get an A, to get a B, and to get a C.

20

Parallel and Perpendicular Lines (Section 1.4)

Definition. Two distinct lines are called parallel i↵

(1) they are both vertical lines, or(2) they have the same slope

Example. Determine if the two lines are parallel.

6x� 2y = 2 4y � 7 = 8x4y � 8 = 12x 2y + 4x = 1

Example. Find the equation of the line through the given point and parallel to the given line.

(0, 3), y = �x+ 11 (9, 1), y = �4

21

(2, 5), y =1

2x+ 3 (5, 4), x = 2

Definition. Two distinct lines are called perpendicular i↵

(1) one is vertical and one is horizontal, or(2) their two slopes, m

1

and m

2

, satisfy m

1

·m2

= �1

Example. Determine if the two lines are perpendicular.

y = �2x+ 4 �5y = x

4y + 2x+ 1 = 0 5x� y = �4

22

Example. Find the equation of the line through the given point and perpendicular to the givenline.

(0, 3), y = �x+ 11 (9, 1), y = �4

(2, 5), y =1

2x+ 3 (5, 4), x = 2

23

Intersection Points and Zeros (Section 1.5)

Definition. An intersection point of two functions is a point (x, y) where the graphs of eachfunction cross (i.e. it is a point that is in common to both graphs).

Example. Find the intersection point (if any) between the two given lines.y = 3x+ 2 and y = 5x� 6 y = 2x� 5 and y = 2x+ 8

y =�34x+ 2 and y =

4

3x� 5 y = 5 and x = �2

Example. A private plane leaves Midway Airport and flies due east at a speed of 180 km/h.Two hours later, a jet leaves Midway and flies due east at a speed of 900 km/h. After how manyhours will the jet overtake the private plane?

24

Definition. Zeros of a function are the values of x that make the function equal 0 (also thex-coordinate of the x-intercept).

Example. Find the zero(s) of each function (if any).

f(x) = 8x+ 4 g(x) = �5

g(x) =1

2x� 3 f(x) = 4x+ 3

Example. A plane is descending from a height of 28,000 feet at a constant rate of 300 feet perminute. After how long will the plane have landed on the ground?

25

Linear Inequalities (Section 1.6)

Rules: The sign stays the same unless you mutliply/divide by a negative, then the sign flips.

Example. Solve each inequality. Then graph the solution set, and then write the solution setin interval notation.

2x+ 5 3� 6x 16 + 3x > 4x+ 8

2 5x+ 3 < 10 �4 < �2x+ 7 4

Example. Macho Movers charges $200 plus $45 per hour to move a household across town.Brute Force Movers charges a flat fee of $65 per hour for the same move. For what lengths oftime does it cost less to hire Brute Force Movers?

26

Piecewise Defined Functions (Section 2.1)

These are just functions whose rules are defined in di↵erent parts/pieces.

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Example. Determine the function values.

(a) f(x) =

⇢2x+ 1 , x > 1x

2 � 2 , x 1

• f(3)

• f(1)

• f(�4)

(b) g(x) =

8<

:

�3x+ 5 , x 0px+ 1 , 0 < x 5

7 , x > 5

• g(�2)

• g(0)

• g(3)

• g(8)

27

Absolute Value Function (Sections 2.1 and 3.5)

Definition. h(x) =

⇢�x , x < 0x , x � 0

is the absolute value function, denoted by h(x) = |x|.

Remark. To solve absolute value equations:

No solution if a < 0| �� | = a

Equivalent to �� = a or �� = �a if a � 0

Example. Solve each of the following.

|x� 3| = 2 |x+ 2| = �5

|2x� 3|+ 1 = 6 12� |x+ 6| = 5

28

Remark. To solve absolute value inequalities when a > 0:

| �� | < a means �a < �� < a (Also true for )

| �� | > a means �� > a or �� < �a (Also true for �)

Example. Solve each of the following. Write your answer in interval notation.

|2x| < 4 |x� 0.5| 0.2

|3x| � 18 |6� 4x| > 8