Chapter 1

Functions and Graphs

1.1

• Graphs & Graphing Utilities

Objectives

• Plot points in the rectangular coordinate system

• Graph equations in the rectangular coordinate system

• Interpret information about a graphing utility’s viewing rectangle or table

• Use a graph to determine intercepts

• Interpret information from graphs

Rectangular coordinate system

• X-axis: horizontal (right pos., left neg.)

• Y-axis: vertical (up pos., down neg.)

• Ordered pairs: (x,y)

• Graph of an equation: infinitely many ordered pairs that make a true statement

• X-intercept: point where y=0, (x,0)

• Y-intercept: point where x=0, (0,y)

Graphing an Equation

• List ordered pairs that make your equation true (plug values in for x and find the resulting y’s). Include the x-intercept & y-intercept among your points.

• Plot several points and look for a trend.

• Use the graphing utility on your calculator and compare graphs. Do they always match? Why or why not?

Intercepts

• Intercepts are key points to plot when graphing an equation. Remember, an intercepts is a POINT (x,y), and not just a number!

• Look at the x-axis: What is true of EVERY point on the axis? (the y-value’s always 0)

• Look at the y-axis: What is true of EVERY point on the axis? (the x-value’s always 0)

Horizontal and Vertical Lines• What if the y-value is ALWAYS the same,

regardless of the x-value (it could be anything!). (i.e. (4,2), (5,2), (-14,2), (17,2)) It’s a horizontal line! The x isn’t part of the equation, because its value is irrelevant: y=k (k is a constant)

• What if the x-value is ALWAYS the same, regardless of the y-value (it could be anything!). (i.e. (2,4), (2,5), (2,-14), (2,17)) It’s a vertical line! The y isn’t part of the equation, because its value is irrelevant: x=c (c is a constant)

1.2

• Basics of Functions & Their Graphs

Objectives• Find the domain & range of a relation.• Determine whether a relations is a function.• Determine is an equation represents a

function.• Evaluate a function.• Graph functions by plotting points.• Use the vertical line test to identify functions.• Obtain information from a graph.• Identify the domain & range from a graph.• Identify intercepts from a graph.

Domain & Range

• A relation is a set of ordered pairs.• Domain: first components in the relation

(independent)• Range: second components in the relation

(dependent, the value depends on what the domain value is)

• Functions are SPECIAL relations: A domain element corresponds to exactly ONE range element.

EXAMPLE• Consider the function: eye color

• (assume all people have only one color, and it is not changeable)

• It IS a function because when asked the eye color of each person, there is only one answer.

• i.e. {(Joe, brown), (Mo, blue), (Mary, green), (Ava, brown), (Natalie, blue)}

• NOTE: the range values are not necessarily unique.

Evaluating a function

• Common notation: f(x) = function

• Evaluate the function at various values of x, represented as: f(a), f(b), etc.

• Example: f(x) = 3x – 7f(2) = 3(2) – 7 = 6 – 7 = -1

f(3 – x) = 3(3 – x) – 7 = 9 – 3x – 7 = 2 – 3x

Graphing a functions

• Horizontal axis: x values• Vertical axis: y values• Plot points individually or use a graphing utility

(calculator or computer algebra system)• Example: 12 xy

Table of function values

X(domain)

Y(range)

-4 17

-3 10

-2 5

-1 2

0 1

1 2

2 5

3 10

4 17

12 xy

Graphs of functions

Can you identify domain & range from the graph?

• Look horizontally. What all x-values are contained in the graph? That’s your domain!

• Look vertically. What all y-values are contained in the graph? That’s your range!

What is the domain & range of the function with this graph?

),3(:),,(:)4

),3(:),,3(:)3

),(:),,3(:)2

),(:),,(:)1

RangeDomain

RangeDomain

RangeDomain

RangeDomain

Can you determine if a relation is a function or NOT from the graph?

• Recall what it means to be a function: an x-value is assigned ONLY one y-value (it need not be unique).

• So, on the graph, if the x value is paired with MORE than one y value there would be two points directly on a vertical line.

• THUS, the vertical line test! If 2 or more points fall on a vertical line that would cross any portion of your graph, it is NOT the graph of a function.

Finding intercepts:

• X-intercept: where the function crosses the x-axis. What is true of every point on the x-axis? The y-value is ALWAYS zero.

• Y-intercept: where the function crosses the y-axis. What is true of every point on the y-axis? The x-value is ALWAYS zero.

• Can the x-intercept and the y-intercept ever be the same point? YES, if the function crosses through the origin!

1.3

• More of Functions and Their Graphs

Objectives

• Find & simplify a function’s difference quotient.

• Understand & use piecewise functions.• Identify intervals on which a function

increases, decreases, or is constant.• Use graphs to locate relative maxima or

minima.• Identify even or odd functions & recognize

the symmetries.• Graph step functions.

Difference Quotient

• Useful in discussing the rate of change of function over a period of time

• EXTREMELY important in calculus, (h represents the difference in two x values)

h

xfhxf )()(

Find the difference quotient

2266)()(

)2266(2266)()(

)122(1222662)()(

1222662)(

122)33(2)(

1)(2)(2)(

122)(

22

22322

33223

3223

3223

3

3

hxhxh

xfhxfh

hxhxh

h

hhxhhx

h

xfhxf

h

xxhxhxhhxx

h

xfhxf

hxhxhhxxhxf

hxhxhhxxhxf

hxhxhxf

xxxf

What is a piecewise function?

• A function that is defined differently for different parts of the domain.

• Examples: You are paid $10/hr for work up to 40 hrs/wk and then time and a half for overtime.

}40,15;40,10{)( xxxxxf

Increasing and Decreasing Functions

• Increasing: Graph goes “up” as you move from left to right.

• Decreasing: Graph goes “down” as you move from left to right.

• Constant: Graph remains horizontal as you move from left to right.

)()(, 2121 xfxfxx

)()(, 2121 xfxfxx

)()(, 2121 xfxfxx

Even & Odd Functions

• Even functions are those that are mirrored through the y-axis. (if –x replaces x, the y value remains the same) (i.e. 1st quadrant reflects into the 2nd quadrant)

• Odd functions are those that are mirrored through the origin. (if –x replaces x, the y value becomes –y) (i.e. 1st quadrant reflects into the 3rd quadrant)

Determine if the function is even, odd, or neither.

1. Even

2. Odd

3. Neither

22 2)4(2)( xxxf

1.4

• Linear Functions & Slope

Objectives• Calculate a line’s slope.• Write point-slope form of a line’s equation.• Write & graph slope-intercept from of a

line’s equation.• Graph horizontal or vertical lines.• Recognize & use the general form of a

line’s equation.• Use intercepts to graph.• Model data with linear functions and

predict.

What is slope? The steepness of the graph, the rate at which the y values are changing in relation to

the changes in x.

How do we calculate it?

12

12

xx

yy

x

ymslope

A line has one slope

• Between any 2 pts. on the line, the slope MUST be the same.

• Use this to develop the point-slope form of the equation of the line.

• Now, you can develop the equation of any line if you know either a) 2 points on the line or b) one point and the slope.

)( 11 xxmyy

Find the equation of the line that goes through (2,5) and (-3,4)

• 1st: Find slope of the line

m=

2nd: Use either point to find the equation of the line & solve for y.

5

1

)3(2

45

5

34

5

15

5

2

5

1

)2(5

15

xxy

xy

Slope-Intercept Form of the Equation of the Line

• It is often useful to express the line in slope-intercept form, meaning that the equation quickly reveals the slope of the line and where it intercepts the y-axis.

• It is REALLY use of the point-slope form, except that the point is the intercept, (0,b).

• y - b=m(x - 0) becomes y = mx + b

• This creates a quick equation to graph.

Horizontal & Vertical lines• What is the slope of a vertical line?

– It is INFINITELY steep, it only rises. It is SO steep, we can’t define it, therefore undefined slope.

– Look at points on a line, i.e. (-4,8),(-4,9),(-4,13),(-4,0). We don’t care what value y has, all that matters is that x= - 4. Therefore, that is the equation of the line!

• What is the slope of a horizontal line?– There is no rise, it only runs, the change in y is zero, so

the slope = 0.– Look at points on the line, i.e., (2,5), (-2,5), (17,5), etc.

We don’t care what value x has, all that matters is that y = 5. Therefore, that is the equation of the line!

1.5 More on SLOPE

• Objectives

– Find slopes & equations of parallel & perpendicular lines

– Interpret slope as a rate of change

– Find a function’s average rate of change

Parallel lines• Slopes are equal.

• If you are told a line is parallel to a given line, you automatically know the slope of your new line (same as the given!).

• Find the equation of the line parallel to y=2x-7 passing through the point (3,-5).– slope = 2, passes through (3,-5)

y - (-5) = 2(x – 3)

y + 5 = 2x – 6

y = 2x - 11

Perpendicular lines• Lines meet to form a

right angle.• If one line has a very

steep negative slope, in order to form a right angle, it must intersect another line with a gradual positive slope.

• The 2 lines graphed here illustrate that relationship.

• What about the intersection of a horizontal line and a vertical line? They ALWAYS intersect at a right angle. Since horizontal & vertical lines are neither positive or negative, we simply state that they are, indeed, ALWAYS perpendicular.

• What about all other lines? In order to be perpendicular, their slopes must be the negative reciprocal of each other. (HINT: think about the very steep negative-sloped line perpendicular to the gradual positive-sloped line)

21

1

mm

Find the equation of the line perpendicular to y = 2x – 7 through

the point (2,7) .

1. y = ½ x + 7

2. y = - 2 x + 1/7

3. y = - ½ x + 4/3

4. y = - ½ x + 8

Average Rate of Change• Slope thus far has referred to the change

of y as related to the change in x for a LINE.

• Can we have slope of a nonlinear function?

• We CAN talk about the slope between any 2 points on the curve – this is the average rate of change between those 2 points!

1.6 Transformation of Functions• Recognize graphs of common functions• Use vertical shifts to graph functions• Use horizontal shifts to graph functions• Use reflections to graph functions• Use vertical stretching & shrinking to graph

functions• Use horizontal stretching & shrinking to

graph functions• Graph functions w/ sequence of

transformations

• Vertical shifts– Moves the graph up or

down– Impacts only the “y”

values of the function– No changes are made

to the “x” values

• Horizontal shifts– Moves the graph left

or right– Impacts only the “x”

values of the function– No changes are made

to the “y” values

Recognizing the shift from the equation, examples of shifting the

function f(x) =

• Vertical shift of 3 units up

• Horizontal shift of 3 units left (HINT: x’s go the opposite direction that you might believe.)

3)(,)( 22 xxhxxf

22 )3()(,)( xxgxxf

2x

Combining a vertical & horizontal shift

• Example of function that is shifted down 4 units and right 6 units from the original function.

46)(,)( xxgxxf

Reflecting

• Across x-axis (y becomes negative, -f(x))

• Across y-axis (x becomes negative, f(-x))

Horizontal stretch & shrink• We’re MULTIPLYING

by an integer (not 1 or 0).

• x’s do the opposite of what we think they should. (If you see 3x in the equation where it used to be an x, you DIVIDE all x’s by 3, thus it’s compressed horizontally.)

4)3()( 2 xxg

VERTICAL STRETCH (SHRINK)

• y’s do what we think they should: If you see 3(f(x)), all y’s are MULTIPLIED by 3 (it’s now 3 times as high or low!)

43)( 2 xxf

Sequence of transformations• Follow order of operations.• Select two points (or more) from the original function and

move that point one step at a time.

f(x) contains (-1,-1), (0,0), (1,1)1st transformation would be (x+2), which moves the function left

2 units (subtract 2 from each x), pts. are now (-3,-1), (-2,0), (-1,1)

2nd transformation would be 3 times all the y’s, pts. are now (-3,-1), (-2,0), (-1,3)

3rd transformation would be subtract 1 from all y’s, pts. are now (-3,-2), (-2,-1), (-1,2)

1)2(31)2(3

)(3

3

xxf

xxf

Graph of Example

1)2(31)2(3)(

)(

3

3

xxfxg

xxf

1.7 Combinations of Functions;Composite Functions

• Objectives– Find the domain of a function– Combine functions using algebra.– Form composite functions.– Determine domains for composite functions.– Write functions as compositions.

Using basic algebraic functions, what limitations are there when

working with real numbers?

• A) You canNOT divide by zero.– Any values that would result in a zero

denominator are NOT allowed, therefore the domain of the function (possible x values) would be limited.

B) You canNOT take the square root (or any even root) of a negative number.Any values that would result in negatives under an even radical (such as square roots) result in a domain restriction.

Example

• Find the domain

• There are x’s under an even radical AND x’s in the denominator, so we must consider both of these as possible limitations to our domain.

65

22

xx

x

}3,2:{:

3,2,0)2)(3(

065

2,022

xxxDomain

xxx

xx

xx

Algebra of functions

• (f + g)(x) = f(x) + g(x)• (f - g)(x) = f(x) – g(x)• (fg)(x) = f(x)g(x)

0)(,)(

)()( xg

xg

xfx

g

f

Composition of functions

• Composition of functions means the output from the inner function becomes the input of the outer function.

• f(g(3)) means you evaluate function g at x=3, then plug that value into function f in place of the x.

• Notation for composition:

))(())(( xgfxgf

1.8 Inverse Functions

• Objectives– Verify inverse functions– Find the inverse of a function.– Use the horizontal line test to deterimine one-

to-one.– Given a graph, graph the inverse.– Find the inverse of a function & graph both

functions simultaneously.

What is an inverse function?

• A function that “undoes” the original function.• A function “wraps an x” and the inverse would

“unwrap the x” resulting in x when the 2 functions are composed on each other.

xxffxff ))(())(( 11

How do their graphs compare?• The graph of a function

and its inverse always mirror each other through the line y=x.

• Example:y = (1/3)x + 2 and its inverse = 3(x-2)

• Every point on the graph (x,y) exists on the inverse as (y,x) (i.e. if (-6,0) is on the graph, (0,-6) is on its inverse.

Do all functions have inverses?

• Yes, and no. Yes, they all will have inverses, BUT we are only interested in the inverses if they ARE A FUNCTION.

• DO ALL FUNCTIONS HAVE INVERSES THAT ARE FUNCTIONS? NO.

• Recall, functions must pass the vertical line test when graphed. If the inverse is to pass the vertical line test, the original function must pass the HORIZONTAL line test (be one-to-one)!

How do you find an inverse?

• “Undo” the function.

• Replace the x with y and solve for y.

1.9 Distance & Midpoint Formulas;Circles

• Objectives– Find distance between 2 points.– Find midpoint of a line segment.– Write standard form of a circle’s equation.– Give center & radius of a circle whose

equation is in standard form.– Convert general form of a circle’s equation to

standard form.

Distance formula is an application of the Pythagorean theorem.

• Recall for any right triangle, the sum of the squares of the legs equals the square of the hypotenuse.

• For any 2 points, you can imagine a rt. Triangle that would have those 2 pts. The length of the vertical leg is the difference in the y values. The length of the horizontal leg is the difference between the x’s.

• Distance between the points= 212

212 )()( yyxx

222 bac

Midpoint is the point in the middle, therefore the x-value is the average of the 2 given x’s and the y-value is

the average of the 2 given y’s.

• Midpoint of

2

,2

),(),,( 21212211

yyxxyxyx

Standard form or a circle with radius=r and center at (h,k)

222 )()( rkyhx

• A circle centered at (-2,5) with a radius=7 has what equation?

• A circle whose equation is

has what as its center and radius?

center = (2,-6) and radius=2

49)5()2( 22 yx

4)6()2( 22 yx

What if the equation is not in standard form?

You may have to complete the square.

222

22

22

22

63691611)3()4(

119)3(16)4(

13)6()8(

01168

yx

yx

yyxx

yxyx