SWBAT graph square root functions. Chapter 10 SWBAT solve problems using the Pythagorean Theorem. SWBAT perform operations with radical expressions. SWBAT graph square root functions.

10 – 1 The Pythagorean Theorem Vocabulary: Hypotenuse Leg Pythagorean Theorem Conditional Conclusion Converse

10 – 1 The Pythagorean Theorem Vocabulary: Hypotenuse: The side opposite the right angle. Always the longest side Leg: Each side forming the right angle Pythagorean Theorem: relates the lengths of legs and length of hypotenuse Conditional: If-then statements Hypothesis: 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 Hypotenuse A 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 b 72 = c2 Simplify √72 = c Find the PRINCIPAL square root 8.5 = c Use a Calculator The 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 Leg What 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 Triangle Which set of lengths could be the side lengths of a right triangle? 6 in., 24 in., 25in. 4m, 8m, 10m 10in., 24in., 26in. 8 ft, 15ft, 16ft

Identifying a Right Triangle Which set of lengths could be the side lengths of a right triangle? Determine whether the lengths satisfy a2 + b2 = c2. The greatest length = c. 6 in., 24 in., 25in. 62 + 242 =? 252 36 + 576 =? 625 612 ≠ 625

Identifying a Right Triangle Which 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 Triangle Which 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 Triangle Which 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 Functions A 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 negative 3x ≥ 9 Solve for x x ≥ 3 The 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 Function Graph 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 Function Graph 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 I 7.5 = √P Multiply by 5 (7.5)2 = (√P)2 Square both sides 56.25 = 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 Translation For 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 Translation What is the graph of y = ?

You Do! What is the graph of y =

Homework Workbook pages 303-304 1 – 25 odd; 29*

10-2 Simplifying Radicals

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

Find a perfect square that goes into 605. 2. Simplify Find a perfect square that goes into 605.

Simplify .

How do you simplify variables in the radical? Look at these examples and try to find the pattern… What is the answer to ? As a general rule, divide the exponent by two. The remainder stays in the radical.

Find a perfect square that goes into 49. 4. Simplify Find a perfect square that goes into 49. 5. Simplify

Simplify 3x6 3x18 9x6 9x18

6. Simplify Multiply the radicals.

Multiply the coefficients and radicals. 7. Simplify Multiply the coefficients and radicals.

Simplify .

How do you know when a radical problem is done? No radicals can be simplified. Example: There are no fractions in the radical. Example: There are no radicals in the denominator. Example:

There is a radical in the denominator! 8. Simplify. Divide the radicals. Uh oh… There is a radical in the denominator! Whew! It simplified!

9. Simplify Uh oh… Another radical in the denominator! Whew! It simplified again! I hope they all are like this!

10. Simplify Since the fraction doesn’t reduce, split the radical up. Uh oh… There is a fraction in the radical! 10. Simplify Since the fraction doesn’t reduce, split the radical up. 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 – 292 1 – 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 LCD 12x(5/12) – 12x(1/2x) = 12x( 1/3x) Dist. Prop. 5x – 6 = 4 Simplify 5x = 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? (1/3) + (3/x) = (2/x) x = -3

Solving by Factoring What 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 = 0 Subtract 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 Factoring What 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 = ¾ When x = -3 1 – (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 Problem Amy 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 Problem Amy 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 LCD 9t + 7t = 63 Distributive Property 16t = 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 Proportion What 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 x Check: 4/2+2 = 3/2+1 4/4 = 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 Solution What is the solution of: 6 = x + 3 x + 5 x + 5 6(x+5) = (x +3)(x+5) Cross Product Prop. 6x + 30 = x2 + 8x + 15 Simplify 0 = x2 + 2x – 15 0 = (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 + 5 x = 3 or x = -5 Solve for x. Check: 6 = 3 + 3 = 6/8 = 6/8 3+ 5 3 + 5 6 = -5 + 3 = 6/0 ≠ -2/0 X Undefined -5+ 5 -5 + 5 The equation has one solution, 3.

You DO! What is the solution of: – 4 = -2__ x2 – 4 x – 2 0

Homework Workbook Page 327 1 – 17 odd

11-6 Inverse Variation Vocabulary: Inverse Variation Constant 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 Point Suppose 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 variation 3(8) = k Substitute for x and y 24 = k Simplify xy = 24 Write an equation. Substitute 24 for k in xy = k An 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 Variation The 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 Variation The 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 ∙ distance1 = weight2 ∙ distance2

Using Inverse Variation The 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 ∙ distance1 = weight2 ∙ distance2 1000 ∙ 7 = 160 ∙ x 7000 = 160x Simplify x = 43.75 Divide by 160 The person should sit 43.75 ft from the fulcrum to balance the lever.

Using Inverse Variation The 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 1 2 4 8 y Undefined

Graphing an Inverse Variation What 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 = -5 The ratio y/x is the same for all data pairs. So this is a 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) = 18 4(4.5) = 18 6(3) = 18 The 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 Y 2 12 6 4 8 3

Identifying Direct or Inverse Variation Does each situation represent a direct variation or an inverse variation? Explain your reasoning. 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 variations To compare direct and inverse variations

Homework Workbook Pages. Pg. 331-332 1-27 odd Pg. 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 Function A 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? f(x) = 5_ x – 2 x – 2 = 0 Set the number equal to 0 x = 2 Solve for x The 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 x The 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 – 3 The 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 x The 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 3 Y -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 function x = 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 2 3 4 Y -2.5 -3 -3.5 1 -0.5

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 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. 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 Function 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 contest Write: y = 350 + 30  equation models the situation x

Using a Rational Function b) 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. x Since both y and x must be nonnegative numbers, use only the part of the graph in the 1st quadrant.

Using a Rational Function b) 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 Function c) 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