chapter 10

Chapter 10

SWBAT solve problems using the Pythagorean Theorem.

SWBAT perform operations with radical expressions.SWBAT graph square root functions.

10 – 1 The Pythagorean TheoremVocabulary:

HypotenuseLegPythagorean TheoremConditional HypotenuseConclusionConverse

10 – 1 The Pythagorean TheoremVocabulary:

Hypotenuse: The side opposite the right angle. Always the longest side

Leg: Each side forming the right anglePythagorean Theorem: relates the lengths of legs

and length of hypotenuseConditional: If-then statementsHypothesis: The part following the if Conclusion: The part following then Converse: Switches the hypothesis and conclusion

10-1 The Pythagorean Theorem

Find the length of a HypotenuseA triangle as leg lengths 6-inches. What is the

length of the hypotenuse of the right triangle?

a2 + b2 = c2 Pythagorean Theorem 62 + 62 = c2 Substitute 6 for a and b72 = c2 Simplify√72 = c Find the PRINCIPAL square root8.5 = c Use a CalculatorThe length of the hypotenuse is about 8.5 inches.

You DO!

What is the length of the hypotenuse of a right triangle with legs of lengths 9 cm and 12 cm?

15 cm

Finding the Length of a LegWhat is the side length b in the triangle below?

a2 + b2 = c2 Pythagorean Theorem 52 + b2 = 132 Substitute 5 and 13 25 + b2 = 169 Simplify b2 = 144 Subtract 25 b = 12 Find the PRINCIPAL square

root

You Do!What is the side length a in the triangle below?

9

The Converse of the Pythagorean Theorem

Identifying a Right TriangleWhich set of lengths could be the side lengths of

a right triangle?a) 6 in., 24 in., 25in.b) 4m, 8m, 10mc) 10in., 24in., 26in.d) 8 ft, 15ft, 16ft

Identifying a Right TriangleWhich set of lengths could be the side lengths of

a right triangle?Determine whether the lengths satisfy a2 + b2 = c2. The greatest length = c.a) 6 in., 24 in., 25in. 62 + 242 =? 252

36 + 576 =? 625 612 ≠ 625

Identifying a Right TriangleWhich set of lengths could be the side lengths of

a right triangle?Determine whether the lengths satisfy a2 + b2 = c2. The greatest length = c.b) 4m, 8m, 10m 42 + 82 =? 102

16 + 64 =? 100 80 ≠ 100

Identifying a Right TriangleWhich set of lengths could be the side lengths of

a right triangle?Determine whether the lengths satisfy a2 + b2 = c2. The greatest length = c.c) 10in., 24in., 26in. 102 + 242 =? 262

100 + 576 =? 676 676 = 676

Identifying a Right TriangleWhich set of lengths could be the side lengths of

a right triangle?Determine whether the lengths satisfy a2 + b2 = c2. The greatest length = c.By the Converse of the Pythagorean Theorem,

the lengths 10 in., 24 in., and 26 in. could be the side lengths of a right triangle. The correct answer is C.

You Do!

Could the lengths 20 mm, 47 mm, and 52 mm be the side lengths of a right triangle? Explain.

No; 202 + 472 ≠ 522

Homework

• Workbook Pages:pg. 288 1 – 31 odd

10-5 Graphing Square Root Functions

Vocabulary:Square Root Function

Square Root FunctionsA square root function is a function containing a square root

with the independent variable in the radicand. The parent square root function is:y = √x .

The table and graph below show the parent square root function.

Essential Understanding

• You can graph a square root function by plotting points or using a translation of the parent square root function.

• For real numbers, the value of the radicand cannot be negative. So the domain of a square root function is limited to values of x for which the radicand is greater than or equal to 0.

Finding the Domain of a Square Root Function

• What is the domain of the function y = 2√(3x-9) ?3x – 9 ≥ 0 The radicand cannot be

negative3x ≥ 9 Solve for x x ≥ 3The domain of the function is the set of real

numbers greater than or equal to 3.

You Do!

What is the domain of: y =

x ≤ 2.5

Graphing a Square Root FunctionGraph the function: I = ⅕√P, which gives the

current I in amperes for a certain circuit with P watts of power. When will the current exceed 2 amperes?

Step 1: Make a Table

Graphing a Square Root FunctionGraph the function: I = ⅕√P, which gives the

current I in amperes for a certain circuit with P watts of power. When will the current exceed 2 amperes?

Step 1: Plot the points on a graph.

The current will exceed 2 amperes when the power is more than 100

watts

You Do!When will the current in the previous example

exceed 1.5 amperes?

56.25 watts

1.5 = ⅕√P Substitute 1.5 for I7.5 = √P Multiply by 5(7.5)2 = (√P)2 Square both sides56.25 = P Simplify

Graphing a Vertical Translation

For any number k, graphing y = (√x )+ k translates the graph of y = √x up k units. Graphing y = (√x)– k translates the graph of y = √x down k units.

Graphing a Vertical Translation

What is the graph of y = (√x) + 2 ?

You Do!What is the graph of y = (√x) – 3 ?

Graphing a Horizontal TranslationFor any positive number h, graphing y = translates the graph of y = √x to the left h units.

Graphing y = translates the graph of y = √x to the right h units.

Graphing a Horizontal TranslationWhat is the graph of y = ?

You Do!

What is the graph of y =

Homework

Workbook pages 303-3041 – 25 odd; 29*

10-2 Simplifying Radicals

1. Simplify Find a perfect square that goes into 147.

147

147 349

147 349

147 7 3

2. SimplifyFind a perfect square that goes into

605.

605

121 5

121 5

11 5

Simplify

1. .

2. .

3. .

4. .

2 18

72

3 86 236 2

Look at these examples and try to find the pattern…

How do you simplify variables in the radical?

x7

1x x2x x3x x x4 2x x5 2x x x6 3x x

What is the answer to ? x7

7 3x x xAs a general rule, divide the

exponent by two. The remainder stays in the

radical.

Find a perfect square that goes into 49.4. Simplify 49x2

249 x7x

5. Simplify 258x254 2x

122 2x x

Simplify 369x

1. 3x6

2. 3x18

3. 9x6

4. 9x18

Multiply the radicals.6. Simplify 6 10

604 15

4 152 15

7. Simplify 2 14 3 21Multiply the coefficients and radicals.

6 2946 49 6

6 649

42 66 67

Simplify

1. .

2. .

3. .

4. .

24 3x44 3x

2 48x448x

36 8x x

How do you know when a radical problem is done?

1. No radicals can be simplified.Example:

2. There are no fractions in the radical.Example:

3. There are no radicals in the denominator.Example:

8

14

15

8. Simplify.

Divide the radicals.

108

3

1083

366

Uh oh…There is a radical in the denominator!

Whew! It simplified!

9. Simplify

8 2

2 8

4 14

422

Uh oh…Another radical in the

denominator!

Whew! It simplified again! I hope they all are like this!

10. Simplify

57

57

757 7

3549

357

Since the fraction doesn’t reduce, split the radical up.

Uh oh…There is a fraction

in the radical!

How do I get rid of the radical in the

denominator?

Multiply by the denominator to make the denominator a perfect square!

Homework

Workbook Pages pg. 291 – 2921 – 35 odd

Chapter 11

OBJECTIVES: SWBAT to solve rational equations and

proportions.SWBAT write and graph equations for inverse

variations.SWBAT compare direct and inverse variations.SWBAT graph rational functions.

11-5 Solving Rational Equations

Vocabulary: Rational Equation

11-5 Solving Rational Equations

Vocabulary: A rational equation is an equation that contains

one or more rational expression.

Solving Equations With Rational Expressions

What is the solution of (5/12) – (1/2x) = (1/3x)?(5/12) – (1/2x) = (1/3x) The denominators

are 12, 2x, and 3x, so the LCD is 12x

12x[(5/12)-(1/2x)] = 12x(1/3x) Multiply by LCD12x(5/12) – 12x(1/2x) = 12x( 1/3x) Dist. Prop.5x – 6 = 4 Simplify5x = 10 Solve for x x = 2

6 4

Solving Equations With Rational Expressions

What is the solution of (5/12) – (1/2x) = (1/3x)?

Check:(5/12) – (1/2x) = (1/3x) See if x = 2 is true(5/12) –(1/2×2) = (1/3×2) Substitute 2(5/12) – (1/4) = ? (1/6) Simplify(5/12) – (3/12) = (1/6) Same denominator(2/12) = (1/6) Simplify(1/6) = (1/6)

You Do!What is the solution of each equation? Check

your solution?

a) (1/3) + (3/x) = (2/x)

x = -3

Solving by FactoringWhat are the solutions of: 1 – (1/x) = (12/x2) ?1 – (1/x) = (12/x2) The denominators

are x and x2 the LCD is x2. x2[1 - (1/x)] = (12/x2)x2 Multiply by x2

x2(1) – x2(1/x) = (12/x2)x2 Distributive Property x2 – x = 12 Simplify x2 – x -12 = 0Subtract 12(x – 4) (x + 3) = 0 Factor x – 4 = 0 or x + 3 = 0 Zero – Product Prop. x = 4 or x = -3 Solve for x

x

Solving by FactoringWhat are the solutions of: 1 – (1/x) = (12/x2) ?Check: Determine whether 4 and -3 both make 1 –

(1/x) = (12/x2) a true statement. When x = 4; 1 – (1/4) = (12/42) 1 – (1/4) = (12/16) ¾ = ¾When x = -31 – (1/-3) = (12/(-3)2)1 + (1/3) = (12/9)4/3 = 4/3

The solutions are 4 and -3!

You DO!

What are the solutions the equation: d + 6 = (d + 11)/(d+3)

-7, -1

Solving a Work ProblemAmy can paint a loft apartment in 7 h. Jeremy can

paint a loft apartment of the same size in 9 h. If they work together, how long will it take them to paint a third loft apartment of the same size?

Know: Amy painting time is 7h. Jeremy painting time is 9h.

Need: Amy and Jeremy's combined painting time.Plan: Find what fraction of a loft each person can

pain in 1 h. Then write and solve an equation.Define: Let t = the painting time, in hours. If Amy and

Jeremy work together.

Solving a Work ProblemAmy can paint a loft apartment in 7 h. Jeremy can

paint a loft apartment of the same size in 9 h. If they work together, how long will it take them to paint a third loft apartment of the same size?

Define: Let t = the painting time, in hours. If Amy and Jeremy work together.Write: (1/7) + (1/9) = (1/t) (1/7) + (1/9) = (1/t) The denominators are 7, 9, and t so the LCD is 63t.63t(1/7) + 63t(1/9) = (1/t)63t Multiply by LCD9t + 7t = 63 Distributive Property16t = 63 Simplify t = 63/16 Divide each side by 16

It will take Amy and Jeremy about 4h to paint the loft apartment together.

You DO!

One hose can fill a pool in 12 h. Another hose can fill the same pool in 8 h. How long will it take for both hoses to fill the pool together?

4.8 h

Solving a Rational ProportionWhat is the solution of 4/(x + 2) = 3/(x + 1) ? 4__ = 3__(x + 2) (x + 1)4(x+1) = 3(x+2) Cross Product

Prop.4x+4 = 3x+6 Distributive Property x = 2 Solve for xCheck:4/2+2 = 3/2+14/4 = 3/3

You DO!

Find the solution(s) of the equation:

A) c__ = 7_ 3 c – 4

-3, 7

Do Now

Solve each equation.

_1_ + _2_ = _1_ 2 x x

_5_ = x + 2 x + 1 x + 1

Checking to Find an Extraneous Solution

The process of solving a rational equation may give a solution that is extraneous because it makes a denominator in the original equation equal to 0. An extraneous solution is a solution of an equation that is derived from the original equation, but is not a solution of the original equation itself. So you must check your solution.

Checking to Find an Extraneous SolutionWhat is the solution of:

6 = x + 3 x + 5 x + 56(x+5) = (x +3)(x+5) Cross Product Prop.6x + 30 = x2 + 8x + 15 Simplify0 = x2 + 2x – 150 = (x – 3)(x + 5) Factor. x – 3 = 0 or x + 5 = 0 Zero-Product Prop. x = 3 or x = -5 Solve for x.

Checking to Find an Extraneous Solution

What is the solution of: 6 = x + 3

x + 5 x + 5x = 3 or x = -5 Solve for x.Check:

6 = 3 + 3 = 6/8 = 6/83+ 5 3 + 5

6 = -5 + 3 = 6/0 ≠ -2/0 X Undefined-5+ 5 -5 + 5The equation has one solution, 3.

You DO!

What is the solution of: – 4 = -2__ x2 – 4x – 2

0

Homework

Workbook Page 327 1 – 17 odd

11-6 Inverse Variation

Vocabulary:Inverse VariationConstant of Variation for an Inverse Variation

11-6 Inverse Variation

Vocabulary:Inverse Variation: an equation of the form xy = k or y = k/x, where

k≠0.Constant of Variation for an Inverse Variation is k, the product of x y for an ordered pair ∙

(x,y) that satisfies the inverse variation

Writing an Equation Given a PointSuppose y varies inversely with x, and y = 8 when x =

3. What is an equation for the inverse variation? xy = k Use general formula for inverse variation3(8) = k Substitute for x and y24 = k Simplify xy = 24Write an equation. Substitute 24 for k in

xy = kAn equation for the inverse variation is xy =24 or y = 24/x

You Do!

Suppose y varies inversely with x, and y = 9 when x = 6. What is an equation for the inverse variation?

xy = 54

Using Inverse VariationThe weight needed to balance a lever varies

inversely with the distance from the fulcrum to the weight. How far away from the fulcrum should the person sit to balance the lever?

Using Inverse VariationThe weight needed to balance a lever varies

inversely with the distance from the fulcrum to the weight. How far away from the fulcrum should the person sit to balance the lever?

Relate: The 1000-lb elephants is 7 ft. from the fulcrum. The 160-lb person is x ft from the fulcrum. Weight and Distance Varies inversely.

Define: Let weight1 = 1000 lb, Let distance1 = 7 ft. Let weight2 = 160 and let distance2 = x ft.

Write: weight1 distance∙ 1 = weight2 distance∙ 2

Using Inverse VariationThe weight needed to balance a lever varies

inversely with the distance from the fulcrum to the weight. How far away from the fulcrum should the person sit to balance the lever?

Write: weight1 distance∙ 1 = weight2 distance∙ 2

1000 7 = 160 x∙ ∙7000 = 160x Simplifyx = 43.75 Divide by 160

The person should sit 43.75 ft from the fulcrum to balance the lever.

Using Inverse VariationThe weight needed to balance a lever varies

inversely with the distance from the fulcrum to the weight. How far away from the fulcrum should the person sit to balance the lever?

The person should sit 43.75 ft from the fulcrum to balance the lever.

You Do!

A 120-lb weight is placed on a lever, 5 ft from the fulcrum. How far from the fulcrum should an 80-lb weight be placed to balance the lever?

7.5ft

Graph of Inverse Function

Each graph has two unconnected parts. When k > 0, the graph lies in the 1st and 3rd quadrants.

When k < 0, the graph lies in the 2nd and 4th quadrants. Since k is a nonzero constant, xy≠0. So neither x nor y can equal 0.

Graphing an Inverse Variation

What is the graph of y = 8/x ?Step 1: Make a Table

x -8 -4 -2 -1 0 1 2 4 8

y -1 -2 -4 -8 Undefined 8 4 2 1

Graphing an Inverse VariationWhat is the graph of y = 8/x ?Step 2: Plot the points from the table. Connect

the points in Quadrant I with a smooth curve. Do the same for the points in Quadrant III.

You DO!

What is the graph of y = -8/x ?

Direct and Inverse Variations

Determining Direct or Indirect Variation

Do the data in each table represent a direct variation or an inverse variation? For each table, write an equation to model the data?

The values of y seem to vary directly with the values of x. Check each ratio.

X Y

3 -15

4 -20

5 -25

Determining Direct or Indirect Variation

Do the data in each table represent a direct variation or an inverse variation? For each table, write an equation to model the data?-15/3 = -5-20/4 = -5-25/5= -5The ratio y/x is the same for all data pairs. So this isa direct variation and k = -5. An equation is y = -5x

X Y

3 -15

4 -20

5 -25

Determining Direct or Indirect Variation

Do the data in each table represent a direct variation or an inverse variation? For each table, write an equation to model the data?

The values of y seem to vary inversely with the values of x. Check each product xy.

X Y

2 9

4 4.5

6 3

Determining Direct or Indirect Variation

Do the data in each table represent a direct variation or an inverse variation? For each table, write an equation to model the data?2(9) = 184(4.5) = 186(3) = 18The product xy is the same for all data points. So this is an inverse variation,and k = 18.An equation is xy = 18, or y = 18/x

X Y

2 9

4 4.5

6 3

You Do!Do the data in the table represent a direct

variation or an inverse variation? For the table, write an equation to model the data.

direct; y = -3x

X Y

4 -12

6 -18

8 -24

Do Now

Do the data in each table represent a direct variation or an inverse variation? For each table, write an equation to model the data?

Inverse, y = 24/x

X Y2 126 48 3

Identifying Direct or Inverse VariationDoes each situation represent a direct variation

or an inverse variation? Explain your reasoning.

a) The cost of a $120 boat rental is split amount several friends.

The cost per person times the number of friends equals the total cost of the boat rental. Since the total cost is a constant product of $120, the cost per person varies inversely with the number of friends. This is an inverse variation.

Identifying Direct or Inverse Variation

Does each situation represent a direct variation or an inverse variation? Explain your reasoning.

b) You download several movies for $14.99 each.The cost per download times the number of

movies downloaded equals the total cost of the downloads. Since the ratio (total cost)/(number of movies downloaded) is constant at $14.99, the total cost varies directly with the number of movies downloaded. This is a direct variation.

Summary

Our objectives in 11-6 were to:Write and graph equations for inverse variationsTo compare direct and inverse variations

Homework

Workbook Pages.Pg. 331-332 1-27 oddPg. 333 1-5 all

11-7 Graphing Rational Functions.Objective: SWBAT illustrate rational functions graphically

11-7 Graphing Rational Functions.Vocabulary: Rational Function

Asymptote:

11-7 Graphing Rational Functions.Vocabulary: Rational FunctionA rational function can be written in the form of

f(x) = polynomial , where the denominator polynomial

cannot be 0.Asymptote: A line is an asymptote of a graph if

the graph gets closer to the line as x or y gets larger in absolute value.

Identifying Excluded Values

Since division by zero is undefined, any value of x that makes the denominator equal to 0 is excluded.

Identifying Excluded Values

What is the excluded value of each function?

a) f(x) = 5_ x – 2

x – 2 = 0 Set the number equal to 0

x = 2 Solve for xThe excluded value is x = 2.

Identifying Excluded Values

What is the excluded value of each function?

b) f(x) = -3_ x + 8

x + 8 = 0 Set the number equal to 0

x = -8 Solve for xThe excluded value is x = -8.

You Do!

What is the excluded value for y = _3_ x + 7

x = -7

Asymptote

A line is an asymptote of a graph if the graph gets closer to the line as x or y gets larger in absolute value.

Asymptote

y = _1_ x – 3The x-axis and x = 3 are the asymptotes.

Using a Vertical Asymptote What is the vertical asymptote of the graph of y = _5_ ? Graph the function.

x + 2 x + 2 = 0 Since the numerator and

denominator have no common factors. To find the vertical asymptote, find the excluded value.

x = -2 Solve for xThe vertical asymptote is the line x = -2.

Using a Vertical Asymptote What is the vertical asymptote of the graph of y = _5_ ? Graph the function.

x + 2 To graph the function, first make a table of

values. Use values of x near -2, where the asymptote occurs.

X -7 -4 -3 -1 0 3

Y -1 -2.5 -5 5 2.5 1

Using a Vertical Asymptote What is the vertical asymptote of the graph of y = _5_ ? Graph the function.

x + 2 Use the points from the table to make the

graph. Draw a dashed line for the vertical asymptote.

You Do!!What is the vertical asymptote of the graph of

h(x) = -3_ x – 6

Graph the functionx = 6

Identifying Asymptotes

Using Vertical and Horizontal Asymptotes

What are the asymptotes of the graph of f(x) = 3_ - 2 ? Graph the function.

x - 1 Step 1: From the form of the function, you can

see that there is a vertical asymptote at x = 1 and a horizontal asymptote at y = -2.

Using Vertical and Horizontal Asymptotes

What are the asymptotes of the graph of f(x) = 3_ - 2 ? Graph the function.

x - 1 Step 2: Make a table of values using values of x

near 1.

X -5 -2 -1 0 2 3 4

Y -2.5 -3 -3.5 -5 1 -0.5 -1

Using Vertical and Horizontal Asymptotes

What are the asymptotes of the graph of f(x) = 3_ - 2 ? Graph the function.

x - 1 Step 3: Sketch the asymptotes. Graph the

function.

You Do!

What are the asymptotes of the graph of y = -1_ - 4? Graph the function.

x + 3 x = -3, y = -4

Using a Rational Function

Your dance club sponsors a contest at a local reception hall. Reserving a private room costs $350, and the cost will be divided equally among the people who enter the contest. Each person also pays a $30 entry fee.

Using a Rational FunctionYour dance club sponsors a contest at a local

reception hall. Reserving a private room costs $350, and the cost will be divided equally among the people who enter the contest. Each person also pays a $30 entry fee.

a) What equation gives the total cost per person y of entering the contest as a function of the number of people x who enter the contest?

Using a Rational Functiona) What equation gives the total cost per person y

of entering the contest as a function of the number of people x who enter the contest?

Relate: total cost per person = cost renting private room + entry fee per person# of people entering contestWrite: y = 350 + 30 equation models the situation

x

Using a Rational Functionb) What is the graph of he function in part (A)?

Use the graph to describe the change in the cost per person as the number of people who enter the contest increases.

Use a graphing calculator to graph y = 350 + 30. xSince both y and x must be nonnegative

numbers, use only the part of the graph in the 1st quadrant.

Using a Rational Functionb)

You can see from the graph that as the number of people who enter the contest increases, the cost per person decreases. Because the graph has horizontal asymptote at y = 30, the cost per person will eventually approach $30.

Using a Rational Functionc) Approximately how many people must enter

the contest in order for the total cost per person to be about $50?

Use the trace key or the TABLE feature. When y = 50, x = 18. So if 18 people enter the contest, the cost per person will be about $50.

You DO!

Suppose the cost to rent a private room increases to $400. Approximately how many people must then enter the contest in order for the total cost per person to be about $50?

About 20 people

Summary

Our objective was to:Illustrate rational functions graphically

Homework

Workbook Pages: 335-336 1 – 23 odd