1-2 functions and linear equations (presentation)

8/9/2019 1-2 Functions and Linear Equations (Presentation)

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12 Functions and Linear Equations

Unit 1 Functions and Relations


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Concepts & Objectives

Function Properties (Obj. #3)

Identify whether a relation is a function Identify the domain and range of a given function

Identify 11 and onto

Linear Functions (Obj. #4) Calculate the slope between two points

Write a linear function using either standard form,

pointslope form, or slopeintercept form

Identify parallel and perpendicular lines

Graph a linear function


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Functions

A relation is an ordered pair.

Afunction is a relation for which there is exactly onevalue of the dependent variable for each value of the

independent variable.

More formally:

IfA and B are sets, then a function f fromA to B

(writtenf:A B)

is a rule that assigns to each element ofA

a unique element of setB.


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Functions

The setA is called the domain of the functionf

Every element ofA must be included in the function. The setB is called the codomain off

The subset ofB consisting of those elements that are

images under the function fis called the range. The range and the codomain may or may not be the

same.


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Functions

In terms of ordered pairs, a function is the set of ordered

pairs (A,f(A)).

Historical note: The notationf(x) for a function of a

variable quantity xwas introduced in 1748 by Leonhard

Euler in his textAlgebra, which was the forerunner of

todays algebra texts. Many other mathematical symbols

in use today (such as e and ) were introduced by Euler

in his writings.


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Functions

For the most part, we use f(x) andyinterchangeably to

denote a function ofx, but there are some subtledifferences.

yis the output variable, whilef(x) is the rule that

produces the output variable.

An equation with two variables, xandy, may not be a

function at all.

Example: is an ellipse, but not a function+ =

2 2

14 9

x y


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Functions

A functionf:A B is a one-to-onefunction (or 11

function) if and only if every elementyin B is the imageof at most one elementxinA.

Example: f(x) = x2, x is not 11 because for a

given output, such as 4, there are two possible inputs

(such as 2 or 2).

Example: f(x) = x2,x + restricts the domain to the

positive reals, and so would be a 11 function.


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Functions

A functionf:A B is onto if every element in the

codomain B is in the range off. Example: f: |f(x) = 2xis not onto because

there are values in the codomain (1, 3, 5, ) that are

not in the range (2, 4, 6, )

Example: f:2 |f(x) = 2xis now onto because

the codomain only includes even numbers.


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Linear Functions

A linearfunction is a function ofxin which the rateof

change, that is,f(x+ 1) f(x), is constant. The rate of change is also called the slope (or rise over

run).

The formula for slope is for two points

(x1,y1) and (x2,y2).

=

2 1

2 1

y ym

x x


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Linear Functions

A linear function can be written in one of the following

forms: Standard form: Ax+ By= C, whereA, B, C ,A 0,

andA, B, and Care relatively prime

Pointslope form: yy1 = m(xx1), where m and

(x1,y1) is a point on the graph

Slopeintercept form: y= mx+ b, where m, b

You should recall that in slopeintercept form, m is the

slope and b is theyintercept (where the graph crossestheyaxis).

IfA = 0, then the graph is a horizontal line aty= b.


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Linear Functions

Lets take another look at the standard form:

IfB = 0 (the coefficient of the yterm), we end up with

which is undefined. This is not good.

+ =

= +

= +

Ax By C

By Ax C

A C

y xB B

= + ,0 0A Cy x


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Linear Functions

Since we cannot divide by 0, we say that a line of the

formx= a has no

slope, and is a vertical line. Technically, a vertical line is not a function at all,

because one value ofxhas more than oneyvalue

(actually an infinite number ofyvalues).


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Parallel and Perpendicular Lines

Nonvertical lines are parallel iff they have the same

slope. Any two vertical lines are parallel.

Two nonvertical lines are perpendicular iff the product

of their slopes is 1 (negative reciprocals). Vertical and

horizontal lines are perpendicular.

Example: What is the slope of the line perpendicular

toy= 3x+ 7?

=

1 13 3

or just flip 3 and change the sign: 3 11 3


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Graphing a Linear FunctionTo graph a line:

If you are only given two points, plot them and draw aline between them.

If you are given a point and a slope:

Plot the point. Fromthepointcount the rise and the run of the slope

and mark your second point.

Connect the two points.

If the slope is negative, pick either the rise orthe run

to go in a negative direction, but not both.


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Graphing a Linear Function Example: Graph the liney= 2x+ 1.


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Graphing a Linear Function Example: Graph the liney= 2x+ 1.

Plot theyintercept at (0, 1). Count down 2 and over 1.

Plot the second point at (1, 1).

Connect the points.


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Writing the Equation of a Line From a graph:

Calculate the slope Find a point on the graph. If theyintercept is

available, use that by preference.

Write the equation in either pointslope form or

slopeintercept form.


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Writing the Equation of a Line Ex.: Write the equation of the graph:


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Writing the Equation of a Line Ex.: Write the equation of the graph:

The slope is up 2, over 3 or . Theyintercept is 1.

2

3

= 2

13y x


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Writing the Equation of a Line From a point and a slope:

Plug into the pointslope form and solve for yifrequested.

From two points:

Calculate the slope, and pick one point to plug into

the pointslope form.

Alternatively, you can plug the slope and the points x

andyvalues into the slopeintercept form, solve for b,

and rewrite the equation.


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Writing the Equation of a Line Example: Write the equation in slopeintercept form for

the line that contains the point (2, 7) and has slope 3.


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Writing the Equation of a Line Example: Write the equation in slopeintercept form for

the line that contains the point (2, 7) and has slope 3. Method #1: y (7) = 3(x 2)

y+ 7 = 3x+ 6

y= 3x 1

Method #2: 7 = 3(2) + b

7 = 6 + b

1 = b y= 3x 1

1-2 functions and linear equations (presentation)

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