7.1 Review of Graphs and Slopes of Lines

• Standard form of a linear equation:

• The graph of any linear equation in two variables is a straight line. Note: Two points determine a line.

• Graphing a linear equation:1. Plot 3 or more points (the third point is used

as a check of your calculation)2. Connect the points with a straight line.

CByAx

7.1 Review of Graphs and Slopes of Lines

• Finding the x-intercept (where the line crosses the x-axis): let y=0 and solve for x

• Finding the y-intercept (where the line crosses the y-axis): let x=0 and solve for y

Note: the intercepts may be used to graph the line.

7.1 Review of Graphs and Slopes of Lines

• If y = k, then the graph is a horizontal line (slope = 0):

• If x = k, then the graph is a vertical line (slope = undefined):

7.1 Review of Graphs and Slopes of Lines

• Slope of a line through points (x1, y1) and (x2, y2) is:

• Positive slope – rises from left to right.Negative slope – falls from left to right

run

rise

xx

yym

)(

)(

in x change

yin change

12

12

7.1 Review of Graphs and Slopes of Lines

• Using the slope and a point to graph lines:Graph the line with slope passing through the point (0, 0)

Go over 5 (run) and up 3 (rise) to get point (5, 3) and draw a line through both points.

run

risem

5

3

5

3

7.1 Review of Graphs and Slopes of Lines

• Finding the slope of a line from its equation:

1. Solve the equation for y2. The slope is given by the coefficient of x

• Parallel and perpendicular lines:1. Parallel lines have the same slope2. Perpendicular lines have slopes that are

negative reciprocals of each other

7.1 Review of Graphs and Slopes of Lines

• Example: Decide whether the lines are parallel, perpendicular, or neither:

1. solving for yin first equation:

2. solving for yin second equation:

3. The slopes are negative reciprocals of each other so the lines are perpendicular

32

72

yx

yx

2

3232

m

xyyx

21

27

21

7272

mxy

xyyx

7.2 Review of Equations of Lines

• Standard form:

• Slope-intercept form:(where m = slope and b = y-intercept)

• Point-slope form: The line with slope m going through point (x1, y1) has the equation:

CByAx bmxy

)( 11 xxmyy

7.2 Review of Equations of Lines

• Example: Find the equation in slope-intercept form of a line passing through the point (-4,5) and perpendicular to the line 2x + 3y = 6

1. solve for y to get slope of line

2. take the negative reciprocal to get the slope32

32 2

623632

mxy

xyyx

23m

7.2 Review of Equations of Lines

• Example (continued):

3. Use the point-slope form with this slope and the point (-4,5)

4. Add 5 to both sides to get in slope intercept form:

11

645

)4(5

23

23

23

23

xy

xxy

xy

23m

7.3 Functions Relations

• Relation: Set of ordered pairs:

Example: R = {(1, 2), (3, 4), (5, 1)}

• Domain: Set of all possible x-values

• Range: Set of all possible y-values

• What is the domain of the relation R?

7.3 FunctionsRelations

Domain:x-values(input)

Range:y-values(output)

Example: Demand for a product depends on its price.Question: If a price could produce more than one demand would the relation be useful?

7.3 Functions - Determining Whether a Relation or Graph is a Function

• A relation is a function if: for each x-value there is exactly one y-value– Function: {(1, 1), (3, 9), (5, 25)}– Not a function: {(1, 1), (1, 2), (1, 3)}

• Vertical Line Test – if any vertical line intersects the graph in more than one point, then the graph does not represent a function

7.3 Functions

• Function notation: y = f(x) – read “y equals f of x”note: this is not “f times x”

• Linear function: f(x) = mx + b

Example: f(x) = 5x + 3

• What is f(2)?

7.3 Functions - Graph of a Function

• Graph of

• Does this pass the vertical line test?What is the domain and the range?

xxf )(

7.3 Functions - Graph of a Parabola

Vertex

2)( xxf

7.4 Variation

• Types of variation:1. y varies directly as x:2. y varies directly as the

nth power of x:

3. y varies inversely as x:

4. y varies inversely as the nth power of x: nx

ky

nkxy

x

ky

kxy

7.4 Variation

• Solving a variation problem:1. Write the variation equation.

2. Substitute the initial values and solve for k.

3. Rewrite the variation equation with the value of k from step 2.

4. Solve the problem using this equation.

7.4 Variation• Example: If t varies inversely as s and

t = 3 when s = 5, find s when t = 5

1. Give the equation:

2. Solve for k:

3. Plug in k = 15:

4. When t = 5: 315515

5 sss

s

kt

155

3 kk

st

15

9.2 Review – Things to Remember

• Multiplying/dividing by a negative number reverses the sign of the inequality

• The inequality y > x is the same as x < y• Interval Notation:

– Use a square bracket “[“ when the endpoint is included

– Use a round parenthesis “(“ when the endpoint is not included

– Use round parenthesis for infinity ()

9.2 Review - Compound Inequalities and Interval Notation

Solve eachinequality for x:

Take the intersection:(why does the order change?)Express in interval notation:

422 and 1013 xx22 and 93 xx

1 and 3 xx

3,1

1 3

31 x

9.2 Review - Compound Inequalities and Interval Notation

Solve eachinequalityfor x:

Take the union:

Express ininterval notation

33-or 32 xxx1or 03 xx1or 3 xx

-1

1x

),1[

9.2 Review - Absolute Value Equations

• Solving equations of the form: kbax

35or 1

53or 33

143or 143

143

xx

xx

xx

x

9.2 Absolute Value Inequalities

• To solve where k > 0, solve the compound inequality (intersection):

• To solve where k > 0, solve the compound inequality (union):

Why can’t you say ?

kbax

kbaxk

kbax

kbaxkbax or

kbaxk

9.2 A Picture of What is Happening

• Graphs of

and f(x) = k

The part below the line f(x) = k is where

The part above the line f(x) = k is where

baxxf )(

)0,( ab

kbax

x

y

f(x) = k

kbax

9.2 Absolute Value Inequalities - Form 1

• Solving equations of the form:

1. Setup the compoundinequality

2. Subtract 4 all the wayacross

3. Divide by 3

4. Put into intervalnotation

kbax

135 x

1,35

143 x1431 x

335 x

9.2 Absolute Value Inequalities - Form 2

• Solving equations of the form:

1. Setup the compoundinequality

2. Subtract 4 all the wayacross

3. Divide by 34. Put into interval notation

What part of the real line is missing?

kbax

35or x 1 x

,1(), 35

143 x143or 143 xx

53or 33 xx

9.2 Absolute Value Inequalitythat involves rewriting

• Example:

Add 3 to both sides (why?):

Set up compound equation:

Add 2 all the way across:

Put into interval notation

132 x

22 x

222 x

40 x

4,0

9.2 Absolute Value Inequalities

• Special case 1 when k < 0:

Since absolute value expressions can never be negative, there is no solution to this inequality. In set notation:

435 x

9.2 Absolute Value Inequalities

• Special case 2 when k = 0:Since absolute value expressions can never be negative, there is one solution for this:

In set notation: What if the inequality were “<“?

035 x

53

35

035

035

x

x

x

x

53

9.2 Absolute Value Inequalities

• Special case 3:

Since absolute value expressions are always greater than or equal to zero, the solution set is all real numbers. In interval notation:

135 x

,

9.2 A Picture of What HappensWhen k is Negative

• Graphs of

and f(x) = k

never gets below the line f(x) = k so there is no solution toand the solution to is all real numbers

baxxf )(

)0,( ab

baxxf )(

x

y

f(x) = k

kbax kbax

9.2 Relative Error

• Absolute value is used to find the relative error of a measurement. If xt represents the expected value of a measurement and x represents the actual measurement, then

relative error in t

t

x

xxx

9.2 Example of Relative Error

• A machine filling quart milk cartons is set for a relative error no greater than .05. In this example, xt = 32 oz. so:

Solving this inequality for x gives a range of values for carton size within the relative error specification.

05.32

32

x

9.2 Solution to the Example

1. Simplify:

2. Change into acompound inequality

3. Subtract 1

4. Multiply by –32

5. Reverse the inequality

6. Put into interval notation

05.32

132

32 xx

05.32

105. x

95.32

05.1 x

4.306.33 x6.334.30 x

6.33,4.30