Graphing Equations and Inequalities

Chapter 10

Chapter Sections

10.1 Reading Graphs and the Rectangular

Coordinate System 10.2 Graphing Linear

Equations 10.3 Intercepts 10.4 Slope and

Rate of Change 10.5 Equations of Lines 10.6

Introduction to Functions 10.7 Graphing Linear

Inequalities in Two Variables 10.8 Direct and

Inverse Variation

10.1

- Reading Graphs and the Rectangular Coordinate

System

Vocabulary

- Ordered pair a sequence of 2 numbers where the

order of the numbers is important - Axis horizontal or vertical number line
- Origin point of intersection of two axes
- Quadrants regions created by intersection of 2

axes - Location of a point residing in the rectangular

coordinate system created by a horizontal (x-)

axis and vertical (y-) axis can be described by

an ordered pair. Each number in the ordered pair

is referred to as a coordinate

Graphing an Ordered Pair

Graphing an Ordered Pair

Note that the order of the coordinates is very

important, since (-4, 2) and (2, -4) are located

in different positions.

Vocabulary

- Paired data are data that can be represented as

an ordered pair - A scatter diagram is the graph of paired data as

points in the rectangular coordinate system - An order pair is a solution of an equation in two

variables if replacing the variables by the

appropriate values of the ordered pair results in

a true statement.

Solutions of an Equation

Example

Determine whether (3, 2) is a solution of 2x

5y 4. Let x 3 and y 2 in the equation.

2x 5y 4 2(3) 5(2) 4

(replace x with 3 and y with 2) 6

(10) 4 (compute the products)

4 4 (True)

So (3, 2) is a solution of 2x 5y 4

Solutions of an Equation

Example

Determine whether ( 1, 6) is a solution of 3x

y 5. Let x 1 and y 6 in the equation.

3x y 5 3( 1) 6 5 (replace

x with 1 and y with 6) 3 6 5

(compute the product) 9 5

(False)

So ( 1, 6) is not a solution of 3x y 5

Solving an Equation

Solving an Equation

Example

Complete the ordered pair (4, ) so that it is a

solution of 2x 4y 4.

Let x 4 in the equation and solve for y.

2x 4y 4 2(4) 4y 4

(replace x with 4) 8 4y 4

(compute the product)

So the completed ordered pair is (4, 3).

4y 12 (simplify both sides)

y 3 (divide both sides by 4)

Solving an Equation

Example

Complete the ordered pair (__, 2) so that it is

a solution of 4x y 4.

Let y 2 in the equation and solve for x.

4x y 4 4x ( 2) 4

(replace y with 2) 4x 2 4

(simplify left side)

So the completed ordered pair is (½, 2).

4x 2 (simplify both sides)

x ½ (divide both sides by 4)

10.2

- Graphing Linear Equations

Linear Equations

- Linear Equation in Two Variables
- Ax By C
- A, B, C are real numbers, A and B not both 0
- This is called standard form
- Graphing Linear Equations
- Find at least 2 points on the line
- y mx b crosses the y-axis at b (called

slope-intercept form)

Graphing Linear Equations

Example

Graph the linear equation 2x y 4.

Continued.

Graphing Linear Equations

Example continued

Graph the linear equation 2x y 4. Let x

1. Then 2x y 4 becomes

2(1) y 4 (replace x with 1) 2

y 4 (simplify the left side) y

4 2 6 (subtract 2 from both sides)

y 6 (multiply both sides by

1) So one solution is (1, 6)

Continued.

Graphing Linear Equations

Example continued

Graph the linear equation 2x y 4. For the

second solution, let y 4. Then 2x y 4

becomes

2x 4 4 (replace y with 4)

2x 4 4 (add 4 to both sides)

2x 0 (simplify the right side)

x 0 (divide both sides by

2) So the second solution is (0, 4)

Continued.

Graphing Linear Equations

Example continued

Graph the linear equation 2x y 4. For the

third solution, let x 3. Then 2x y 4

becomes

2( 3) y 4 (replace x with 3)

6 y 4 (simplify the left side)

y 4 6 2 (add 6 to both sides)

y 2 (multiply both sides by

1) So the third solution is ( 3, 2)

Continued.

Graphing Linear Equations

Example continued

Now we plot all three of the solutions (1, 6),

(0, 4) and ( 3, 2).

And then we draw the line that contains the three

points.

Graphing Linear Equations

Example

Continued.

Graphing Linear Equations

Example continued

Let x 4.

y 3 3 6 (simplify the right

side) So one solution is (4, 6)

Continued.

Graphing Linear Equations

Example continued

For the second solution, let x 0.

y 0 3 3 (simplify the right

side) So a second solution is (0, 3)

Continued.

Graphing Linear Equations

Example continued

For the third solution, let x 4.

y 3 3 0 (simplify the

right side) So the third solution is ( 4, 0)

Continued.

Graphing Linear Equations

Example continued

Now we plot all three of the solutions (4, 6),

(0, 3) and ( 4, 0).

And then we draw the line that contains the three

points.

10.3

- Intercepts

Intercepts

- Intercepts of axes (where graph crosses the axes)
- Since all points on the x-axis have a

y-coordinate of 0, to find x-intercept, let y 0

and solve for x - Since all points on the y-axis have an

x-coordinate of 0, to find y-intercept, let x 0

and solve for y

Intercepts

Example

- Find the y-intercept of 4 x 3y
- Let x 0.
- Then 4 x 3y becomes
- 4 0 3y (replace x with 0)
- 4 3y (simplify the right

side)

Intercepts

Example

- Find the x-intercept of 4 x 3y
- Let y 0.
- Then 4 x 3y becomes
- 4 x 3(0) (replace y with 0)
- 4 x (simplify the right side)
- So the x-intercept is (4,0)

Graph by Plotting Intercepts

Example

- Graph the linear equation 4 x 3y by plotting

intercepts.

Plot both of these points and then draw the line

through the 2 points. Note You should still

find a 3rd solution to check your computations.

Continued.

Graph by Plotting Intercepts

Example continued

Graph the linear equation 4 x 3y. Along with

the intercepts, for the third solution, let y

1. Then 4 x 3y becomes

4 x 3(1) (replace y with 1)

4 x 3 (simplify the right side) 4

3 x (add 3 to both sides) 7 x

(simplify the left side) So the third

solution is (7, 1)

Continued.

Graph by Plotting Intercepts

Example continued

And then we draw the line that contains the three

points.

Graph by Plotting Intercepts

Example

- Graph 2x y by plotting intercepts
- To find the y-intercept, let x 0
- 2(0) y
- 0 y, so the y-intercept is

(0,0) - To find the x-intercept, let y 0
- 2x 0
- x 0, so the x-intercept is

(0,0) - Oops! Its the same point. What do we do?

Continued.

Graph by Plotting Intercepts

Example continued

- Since we need at least 2 points to graph a line,

we will have to find at least one more point - Let x 3
- 2(3) y
- 6 y, so another point is (3, 6)
- Let y 4
- 2x 4
- x 2, so another point is (2, 4)

Continued.

Graph by Plotting Intercepts

Example continued

Now we plot all three of the solutions (0, 0),

(3, 6) and (2, 4).

And then we draw the line that contains the three

points.

Graph by Plotting Intercepts

Example

- Graph y 3
- Note that this line can be written as y 0x

3 - The y-intercept is (0, 3), but there is no

x-intercept! - (Since an x-intercept would be found by letting

y 0, and 0 ? 0x 3, there is no x-intercept) - Every value we substitute for x gives a

y-coordinate of 3 - The graph will be a horizontal line through the

point (0,3) on the y-axis

Continued.

Graph by Plotting Intercepts

Example continued

Graph by Plotting Intercepts

Example

- Graph x 3
- This equation can be written x 0y 3
- When y 0, x 3, so the x-intercept is (

3,0), but there is no y-intercept - Any value we substitute for y gives an

x-coordinate of 3 - So the graph will be a vertical line through the

point ( 3,0) on the x-axis

Continued.

Graph by Plotting Intercepts

Example continued

Vertical and Horizontal Lines

- Vertical lines
- Appear in the form of x c, where c is a real

number - x-intercept is at (c, 0), no y-intercept unless

c 0 (y-axis) - Horizontal lines
- Appear in the form of y c, where c is a real

number - y-intercept is at (0, c), no x-intercept unless

c 0 (x-axis)

10.4

- Slope and Rate of Change

Slope

- Slope of a Line

Slope

Example

- Find the slope of the line through (4, -3) and

(2, 2) - If we let (x1, y1) be (4, -3) and (x2, y2) be (2,

2), then

Note If we let (x1, y1) be (2, 2) and (x2, y2)

be (4, -3), then we get the same result.

Slope of a Horizontal Line

- For any 2 points, the y values will be equal to

the same real number. - The numerator in the slope formula 0 (the

difference of the y-coordinates), but the

denominator ? 0 (two different points would have

two different x-coordinates). - So the slope 0.

Slope of a Vertical Line

- For any 2 points, the x values will be equal to

the same real number. - The denominator in the slope formula 0 (the

difference of the x-coordinates), but the

numerator ? 0 (two different points would have

two different y-coordinates), - So the slope is undefined (since you cant divide

by 0).

Summary of Slope of Lines

- If a line moves up as it moves from left to

right, the slope is positive. - If a line moves down as it moves from left to

right, the slope is negative. - Horizontal lines have a slope of 0.
- Vertical lines have undefined slope (or no slope).

Parallel Lines

- Two lines that never intersect are called

parallel lines. - Parallel lines have the same slope
- unless they are vertical lines, which have no

slope. - Vertical lines are also parallel.

Parallel Lines

Example

- Find the slope of a line parallel to the line

passing through (0,3) and (6,0)

So the slope of any parallel line is also ½

Perpendicular Lines

- Two lines that intersect at right angles are

called perpendicular lines - Two nonvertical perpendicular lines have slopes

that are negative reciprocals of each other - The product of their slopes will be 1
- Horizontal and vertical lines are perpendicular

to each other

Perpendicular Lines

Example

- Find the slope of a line perpendicular to the

line passing through (?1,3) and (2,-8)

Parallel and Perpendicular Lines

Example

- Determine whether the following lines are

parallel, perpendicular, or neither. - ?5x y ?6 and x 5y 5
- First, we need to solve both equations for y.
- In the first equation,
- y 5x 6 (add 5x to both sides)
- In the second equation,
- 5y ?x 5 (subtract x from both sides)

10.5

- Equations of Lines

Slope-Intercept Form

- Slope-Intercept Form of a line
- y mx b has a slope of m and has a

y-intercept of (0, b). - This form is useful for graphing, since you have

a point and the slope readily visible.

Slope-Intercept Form

Example

- Find the slope and y-intercept of the line 3x

y 5. - First, we need to solve the linear equation for

y. - By adding 3x to both sides, y 3x 5.
- Once we have the equation in the form of y mx

b, we can read the slope and y-intercept. - slope is 3
- y-intercept is (0, 5)

Slope-Intercept Form

Example

- Find the slope and y-intercept of the line 2x

6y 12. - First, we need to solve the linear equation for

y. - 6y 2x 12 (subtract 2x from both

sides)

Point-Slope Form

- The slope-intercept form uses, specifically, the

y-intercept in the equation. - The point-slope form allows you to use ANY point,

together with the slope, to form the equation of

the line.

m is the slope (x1, y1) is a point on the line

Point-Slope Form

Example

- Find an equation of a line with slope 2, through

the point (11, 12). Write the equation in

standard form. - First we substitute the slope and point into the

point-slope form of an equation. - y ( 12) 2(x ( 11))
- y 12 2x 22 (use distributive

property) - 2x y 12 22 (add 2x to both sides)
- 2x y 34 (subtract 12 from both

sides)

Point-Slope Form

Example

- Find the equation of the line through (4,0) and

(6, 1). Write the equation in standard form. - First find the slope.

Continued.

Point-Slope Form

Example continued

Now substitute the slope and one of the points

into the point-slope form of an equation.

Point-Slope Form

Example

Find the equation of the line passing through

points (2, 5) and (4, 3). Write the equation

using function notation.

Continued.

Point-Slope Form

Example continued

10.6

- Introduction to Functions

Vocabulary

- An equation in 2 variables defines a relation

between the two variables. - A set of ordered pairs is also called a relation.
- The domain is the set of x-coordinates of the

ordered pairs - The range is the set of y-coordinates of the

ordered pairs

Domain and Range

Example

- Find the domain and range of the relation (4,9),

(4,9), (2,3), (10, 5) - Domain is the set of all x-values, 4, 4, 2, 10
- Range is the set of all y-values, 9, 3, 5

Domain and Range

Example

Find the domain and range of the function graphed

to the right. Use interval notation.

Domain and Range

Example

Find the domain and range of the function graphed

to the right. Use interval notation.

Functions

- Some relations are also functions.
- A function is a set of order pairs that assigns

to each x-value exactly one y-value.

Functions

Example

- Given the relation (4,9), (4,9), (2,3), (10,

5), is it a function? - Since each element of the domain is paired with

only one element of the range, it is a function. - Note Its okay for a y-value to be assigned to

more than one x-value, but an x-value cannot be

assigned to more than one y-value (has to be

assigned to ONLY one y-value).

Vertical Line Test

- Relations and functions can also be described by

graphing their ordered pairs. - Graphs can be used to determine if a relation is

a function. - If an x-coordinate is paired with more than one

y-coordinate, a vertical line can be drawn that

will intersect the graph at more than one point. - If no vertical line can be drawn so that it

intersects a graph more than once, the graph is

the graph of a function. This is the vertical

line test.

Vertical Line Test

Example

Use the vertical line test to determine whether

the graph to the right is the graph of a function.

Since no vertical line will intersect this graph

more than once, it is the graph of a function.

Vertical Line Test

Example

Use the vertical line test to determine whether

the graph to the right is the graph of a function.

Since no vertical line will intersect this graph

more than once, it is the graph of a function.

Vertical Line Test

Example

Use the vertical line test to determine whether

the graph to the right is the graph of a function.

Since vertical lines can be drawn that intersect

the graph in two points, it is NOT the graph of a

function.

Vertical Line Test

- Since the graph of a linear equation is a line,

all linear equations are functions, except those

whose graph is a vertical line

Function Notation

- Specialized notation is often used when we know a

relation is a function and it has been solved for

y. - For example, the graph of the linear equation

y 3x 2 passes the vertical line

test, so it represents a function. - We often use letters such as f, g, and h to name

functions. - We can use the function notation f(x) (read f of

x) and write the equation as f(x) 3x 2. - Note The symbol f(x) is a specialized notation

that does NOT mean f x (f times x).

Function Notation

- When we want to evaluate a function at a

particular value of x, we substitute the x-value

into the notation. - For example, f(2) means to evaluate the function

f when x 2. So we replace x with 2 in the

equation. - For our previous example when f(x) 3x 2,

f(2) 3(2) 2 6 2 4. - When x 2, then f(x) 4, giving us the order

pair (2, 4).

Function Notation

Example

- Given that g(x) x2 2x, find g( 3). Then

write down the corresponding ordered pair. - g( 3) ( 3)2 2( 3) 9 ( 6) 15.
- The ordered pair is ( 3, 15).

10.7

- Graphing Linear Inequalities in Two Variables

Linear Equations in Two Variables

- Linear inequality in two variables
- Written in the form Ax By lt C
- A, B, C are real numbers, A and B are not both 0
- Could use (gt, ?, ?) in place of lt
- An ordered pair is a solution of the linear

inequality if it makes the inequality a true

statement.

Linear Equations in Two Variables

- To Graph a Linear Inequality
- Graph the related linear equality (forms the

boundary line). - ? and ? are graphed as solid lines
- lt and gt are graphed as dashed lines
- Choose a point not on the boundary line

substitute into original inequality. - If a true statement results, shade the half-plane

containing the point. - If a false statements results, shade the

half-plane that does NOT contain the point.

Linear Equations in Two Variables

Example

- Graph 7x y gt 14

- Graph 7x y 14 as a dashed line.

- Pick a point not on the graph

(0,0)

- Test it in the original inequality.
- 7(0) 0 gt 14, 0 gt 14
- True, so shade the side containing (0,0).

Linear Equations in Two Variables

Example

- Graph 3x 5y ? 2

- Graph 3x 5y 2 as a solid line.

- Pick a point not on the graph

(0,0), but just barely

- Test it in the original inequality.
- 3(0) 5(0) gt 2, 0 gt 2
- False, so shade the side that does not contain

(0,0).

Linear Equations in Two Variables

Example

- Graph 3x lt 15

- Graph 3x 15 as a dashed line.

- Pick a point not on the graph

(0,0)

- Test it in the original inequality.
- 3(0) lt 15, 0 lt 15
- True, so shade the side containing (0,0).

Linear Equations in Two Variables

Warning!

- Note that although all of our examples allowed us

to select (0, 0) as our test point, that will not

always be true. - If the boundary line contains (0,0), you must

select another point that is not contained on the

line as your test point.

10.8

- Direct and Inverse Variation

Direct Variation

- y varies directly as x, or y is directly

proportional to x, if there is a nonzero constant

k such that y kx. - The family of equations of the form y kx are

referred to as direct variation equations. - The number k is called the constant of variation

or the constant of proportionality.

Direct Variation

- If y varies directly as x, find the constant of

variation k and the direct variation equation,

given that y 5 when x 30. - y kx
- 5 k30
- k 1/6

Direct Variation

Example

- If y varies directly as x, and y 48 when x 6,

then find y when x 15. - y kx
- 48 k6
- 8 k
- So the equation is y 8x.
- y 815
- y 120

Direct Variation

Example

- At sea, the distance to the horizon is directly

proportional to the square root of the elevation

of the observer. If a person who is 36 feet

above water can see 7.4 miles, find how far a

person 64 feet above the water can see. Round

your answer to two decimal places.

Continued.

Direct Variation

Example continued

We substitute our given value for the elevation

into the equation.

So our equation is

Inverse Variation

- y varies inversely as x, or y is inversely

proportional to x, if there is a nonzero constant

k such that y k/x. - The family of equations of the form y k/x are

referred to as inverse variation equations. - The number k is still called the constant of

variation or the constant of proportionality.

Inverse Variation

Example

- If y varies inversely as x, find the constant of

variation k and the inverse variation equation,

given that y 63 when x 3. - y k/x
- 63 k/3
- k 633
- k 189

Powers of x

- y can vary directly or inversely as powers of x,

as well. - y varies directly as a power of x if there is a

nonzero constant k and a natural number n such

that y kxn

Powers of x

Example

- The maximum weight that a circular column can

hold is inversely proportional to the square of

its height. - If an 8-foot column can hold 2 tons, find how

much weight a 10-foot column can hold.

Continued.

Powers of x

Example continued

We substitute our given value for the height of

the column into the equation.

So our equation is