Form 3-Chapter 12 Graphs of Functions

7/29/2019 Form 3-Chapter 12 Graphs of Functions

1/16

Math 233 - Spring 2009

Chapter 3 - Graphs and Functions

3.1 Graphs

3.1.1 The Cartesian Coordinate System

Definition 1. Cartesian Coordinate System - (or Rectangular coordinate system) consists oftwo number lines in a plane drawn perpendicular to each other.

x-axis - the horizontal axis is called the x-axis.

y-axis - the vertical axis is called the y-axis.

Origin - the point of intersection of the two axes is called the origin.

On the coordinate system we will be plotting points. To describe the points in this coordinatesystem we use and ordered pair of numbers (x, y). The two numbers, x and y are called thex-coordinate and y-coordinate respectively.

EX 1. Plot the following points on the same set of axes:

A(2, 3) B(0,1) C(-5, 0) D(-2, -1) E(3,-2)

-

6

3.1.2 Graphs

Definition 2. The graph of an equation is an illustration of the set of points whose coordinatessatisfy the equation.

EX 2. -

1. Determine whether the following ordered pairs are solutions of the equation y = 2x + 5.(a) (1, 3) (b) (2, 3)

2. Graph y = 2x.

-

6

1


2/16

3. Graph y = 12x + 3.

-

6

REMARK 1. For the graphs above:

The above graphs are called linear because they are straight lines.

Any equation whose graph is a straight line is called a linear equation.

They are also called first degree equations because the highest exponent on any of thevariables is 1.

3.1.3 Nonlinear Graphs

Equations whose graphs are not straight lines are called nonlinear equations. The key to graphingnonlinear equations is to be sure to plot enough points so we can be sure of what it will look like.

EX 3. -

1. Graph y = x2 1.

-

6

2. Graph y = 2x

.

-

6

2


3/16

3. Graph y = |x| + 1.

-

6

3.2 Functions

The concept of a function is one of the most important in all of mathematics. We will discuss severalways of thinking about and defining functions. But first:

EX 4. Suppose you are driving your car at a constant 40 mph. Can we find a correspondencebetween the number of hours driven with the distance travelled?

Definition 3. We have the following terminology:

The set of all possible times driven is called the domain.

The set of all possible distances travelled is called the range

EX 5. We have the following schematic:

Definition 4. A function is a correspondence between the first set of elements, the domain, anda second set of elements, the range, such that each element of the domain corresponds to exactly

one element in the range.EX 6. Consider the following.

1. (Blackboard)

3


4/16

2. (Blackboard)

3. Consider children and biological mothers. The correspondence of children to biological mothersis a function since for each child there is only one biological mother. However the correspon-dence between biological mothers to children is not a function, since one biological mothercould have multiple children.

An Alternate Definition

Definition 5. A function is a set of ordered pairs in which no first coordinate is repeated.

EX 7. Determine whether the following are functions:

1. {(1, 6), (2, 3), (4, 3), (5, 7)}

2. {(1, 6), (2, 3), (1, 3), (5, 7)}

4


5/16

3.2.1 The Vertical Line Test

Most of our functions we will have a domain and range that is either the real numbers or a subsetof the real numbers. For such functions we can graph them on the cartesian coordinate system.

The graph of a function is the graph of its set of ordered pairs.The vertical line test: If a vertical line can be drawn through any part of the graph and the

line intersects another part of the graph, the graph does not represent a function. If a vertical linecannot be drawn to intersect the graph at more than one point, the graph represents a function.

Stated more simply: if, on the graph of a function, we can draw a line that intersects at morethan one point, it is not a function. If we cant, it is a function.

EX 8. Determine whether the following are functions:

1. Consider the following graphs:

(a)

-

6

(b)

-

6

2. Use the vertical line test to determine whether the following graphs represent functions. Alsodetermine the domain and range of each function or relation.

(a)

-

6

(b)

-

6

3. Critical thinking: Consider the graph (drawn on blackboard) which represents the speed versustime of a student driving to school in the morning. Describe what might be occuring for this

function.

5


6/16

3.2.2 Function Notation

Many of the equations we graphed in sections 3.1 were functions. See examples 2 and 3 from section3.1. We notice that each of them passes the vertical line test. What are the domain and range ofeach of them?

In this class, most of the equations we will encounter will be functions. When an equation iswritten in terms of x and y we will frequently wrtie the equation in function notation

f(x) read as f of x

Warning: This is NOT multiplication.

EX 9. Lets consider the equation y = 2x + 1

Notice that the value of y depends on x.

If we plug in a value for x we get a value for y, different values of x give different values of x.

We say that y is a function of x.

In this case, we can substitute f(x) for y. This tells us what the independent variable is. Itexplicitly states that the value depends on x.

Our function becomes f(x) = 2x + 1

We will use both notations interchangeably.

REMARK 2. We dont always us the letter f. Sometimes we use different letters for both ourfunction and our independent variable. For example g(x), h(x), P(t), etc. . .

EX 10. We also will evaluate functions with this notation.

1. Iff(x) = 3x2 5x + 1 find(a) f(4) (b) f(a)

2. Determine each function value: (a) g(3) for g(t) = 1t+4 (b) h(6) for h(x) = 2|x 10|

3. An application: The Celsius temperature, C, is a function of the Fahrenheit temperature, F.

C(F) =5

9(F 32)

Determine the Celsius temperature that corresponds to 131F

6


7/16

3.3 Linear Functions

3.3.1 Graph Linear Functions

A linear function is a function of the form

f(x) = ax + b

The graph of a linear function is a straight line. Also, for linear functions, the domain is the set ofall real numbers R. Recall: When graphing y = f(x).

EX 11. Graph f(x) = 12x + 1

-

6

3.3.2 Intercepts and Standard Form

The standard form of a linear equation is

ax + by = c

where a, b, and c are real numbers and a and b are not both 0.In this form it is frequently easier to graph the equation using the intercepts. The x-intercept

is the point where the graph crosses the x-axis. The y-intercept is the point where the graphcrosses the y-axis.

To find the y-intercept, set x = 0 and solve for y.

To find the x-intercept, set y = 0 and solve for x.

EX 12. 1. Graph 3x = 6y + 12 using the x- and y-intercepts.

-

6

2. Graph f(x) = 12x + 2 using the x- and y-intercepts.

7


8/16

-

6

3. Graph 2x + y = 0

-

6

3.3.3 Vertical and Horizontal Lines

Horizontal Lines

Any equation of the form y = b will always be a horizontal line.

EX 13. Graph the equation y = 4 (or written f(x) = 4)

-

6

Vertical Lines

Any equation of the form x = a will always be a vertical line.

EX 14. Graph the equation x = 4

-

6

8


9/16

3.3.4 An Application

EX 15. Suppose a store owner sells widgets for $30 each. If her monthly expenses are $3,000,answer the following:

1. Construct a function that relates the number of widgets sold to the profits.

2. How many widgets must she sell to break even?

3. Graph the profit function.

-

6

9


10/16

3.4 Slope-Intercept Form of a Linear Equation

Our goal in this section will be to completely describe a line using two numbers which reveal certaincharacteristics of the line. The characteristics we will use are the y-intercept and the slope.

3.4.1 Understand Translations

Consider the graph of the function y = 12x. What happens if we add 2 to the right hand side? Howabout if I subract 2? Lets graph the following functions on the same coordinate system.

y =1

2x + 2

y =1

2x

y =1

2x 2

-

6

What are the y-intercepts? Each line is parallel to the other, but the new lines are shifted, ortranslated, up or down by two.

3.4.2 Slope

As was mentioned we wish to describe lines using two numerical characteristics. One of those is the

slope.Definition 6. The slope of a line is the ration of the vertical change (or rise) to the horizontalchange (or run).

slope =vertical change

horizontal change=

rise

run

EX 16. We examine how to find slope:

1. Look at the graphs from the previous example, find the slope of the lines.

y =1

2x + 2

y =1

2

x

y =1

2x 2

10


11/16

2. Graph the equations y = 2x and and y = 23x and find their slopes.

-

6

The slope of the line through the distinct points (x1, y1) and (x2, y2) is

slope =change in y

change in x=

y2 y1x2 x1

provided that x1 = x2. We usually use the lowercase letter m to denote the slope.

EX 17. Calculate the slope for the following lines:

(a)

-

6

(b)

-

6

(c)

-

6

REMARK 3. From the example we notice the following:

Lines with positive slope increase as we go from left to right.

Lines with negative slope decrease as we go from left to right.

Any horizontal line has zero slope.

What would the slope of a vertical line be?

3.4.3 Slope-Intercept Form

The slope-intercept form of a linear equation is

y = mx + b

where m is the slope of the line and (0, b) is the y-intercept of the line.To write an equation in slope-intercept form, solve the equation for y.

EX 18. -

11


12/16

1. Consider the equation y = 23x + 2 and determine the slope and y-intercept.

2. Write the equation 3x + 4y = 8 in slope-intercept form and determine the slope and y-intercept.

3.4.4 Graphing Linear Equations Using Slope and y-Intercept

EX 19. Graph the following equations using the slope and y-intercept:

1. 2x + 4y = 8

-

6

2. f(x) = 23x + 1

-

6

3.5 Point-Slope Form of a Linear Equation

We investigate one more method of expressing a linear equation. For this, we take the perspectivethat we know a point on the line and we know the slope, how can we write an equation for the line?

3.5.1 Point-Slope Form

The point-slope form of a linear equation is

y y1 = m(x x + 1)

where m is the slope of the line and (x1, y1) is a point on the line.

EX 20. -

12


13/16

1. Write the equation of the line with slope 4 and passing through the point (2, 5).

2. Write, in slope-intercept form, the equation of the line that passes through the points (1, 5)and (3, 9).

3.5.2 Parallel and Perpendicular Lines

Definition 7. -

Two lines are parallel when they have the same slope.

Two lines are perpendicular when their slopes are negative reciprocals.

REMARK 4. For any number a, its negative reciprocal is 1a

or 1a

.

EX 21. Some problems involving parallel and perpindicular lines.

1. Suppose (0, 3) and (3, 0) are two points on line 1 also (7, 4) and (9, 2) are points one line 2.Determine whether line 1 and line 2 are perpendicular or parallel.

2. Consider the equation 3x + y = 7. Find an equation of a line that has y-intercept of 4 and is(a) parallel to the given line and(b) perpendicular to the given line.

3. Consider the equation 3y = 2x + 8(a) Determine an equation of a line that passes through (6, 1) that is perpendicular to thegraph of the given equation. Write the equation in standard form.(b) Write the equation from (a) using function notation.

13


14/16

3.6 The Algebra of Functions

There are several ways in which we can combine functions to get new functions. We examine a few.Warning: We introduce some new notation, the actual mathematics is pretty straightforward butyou must keep the notation clear.

Operations on Functions

If f(x) represents one function, g(x) represents a second function, and x is in the domain of bothfunctions, then we have the following operations:

Sum of Functions: (f + g)(x) = f(x) + g(x)

Difference of Functions: (f g)(x) = f(x) g(x)

Product of Functions: (f g)(x) = f(x) g(x)

Quotient of Functions: (f /g)(x) =f(x)g(x) , provided g(x) = 0

EX 22. Evaluate the following:

1. Iff(x) = x2 x + 3 and g(x) = x + 2(a) (f + g)(x) (b) (f g)(x) (c) (g f)(x)

2. Iff(x) = x2 16 and g(x) = x 4(a) (f + g)(3) (b) (f g)(5) (c) (g/f)(10)

3.7 Graphing Linear Inequalities

First a brief review: Recall that when we graph an equation in two variables, we are marking onthe Cartesian Coordinate system all points which satisfy the equation:

EX 23. Graph the equation, 2x + 4y = 8.

-

6

We notice that any pair of numbers, (x, y) that satisfies the equation falls on the line. Theimportant idea is we are graphically indicating which points satisfy the equation.

14


15/16

3.7.1 Graph Linear Inequalities in Two Variables

A linear inequality is what results when the equal sign in a linear equation is replaced by aninequality sign.

EX 24. The following are example of linear inequalities:

1. (a) 2x + 4y < 8 (b) 2x + 4y > 82. (a) y x 1 (b) y x 1

Our goal will be two graph all points which satisfy the inequality. How can we do this?For this notice that when we graph a linear equation it splits the plane into three regions: the

two sides of the line, and the line itself.Steps to graphing linear inequalities

1. Replace the inequality with an equal sign.

2. Draw the graph of the equation from step 1.

(a) If the original inequality contains or , draw a solid line.

(b) If the original inequality contains < or >, draw adashed line

.3. Select any point not on the line drawn in step 2 and check whether the chosen point solves

the original inequality:

(a) If it does, shade the side of the line containing the point

(b) If it does not, shade the side of the line not containing the point.

EX 25. Graph the following linear inequalities:

1. 2x + 4y < 8

-

6

2. y x 1

-

6

15


16/16

3. y 14x

-

6

16

Documents

Form 3-Chapter 12 Graphs of Functions