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Chapter Twelve: Radicals, Functions, and Coordinate Geometry Section One: Operations with Radicals The square root of a number is the number that we can multiply by itself to give us that number. For example, the square root of 16 is 4 since 4 times itself gives us 16 16 4 . We discussed in chapter 10 some of the principles of radicals or roots. We also discussed the meaning of the symbol . EX1: Try to find the square root of the following. Estimate if necessary. a. 121 b. 225 c. 22 d. 45 We can add or subtract radicals if they have the same radicand (the number under the radical). EX2: Simplify the radical expressions. a. 7 3 9 3 b. 6 4 2 9 3 2 c. 4 5 7 2 7 9 5 d. m n n We can also simplify roots by splitting them. All roots can be multiplied together and therefore can also be factored. This is the multiplication property of square roots. a b ab which means ab a b . An example of this is as follows: 2 3 6 or we can break a root apart like this 12 4 3 . We use this method to write square roots in simplest form. EX3: Write each in simplest form. (Keep in mind that we are finding principle roots so use the absolute value bars where necessary). a. 18 b. 72 c. 40 d. 500 e. y6 f. r 6 s3 g. 45x 6 y 3 z 4 EX4: Simplify. a. 2 3 2 b. 8 14 c. 3 6 5 d. 2 7 2 7 We can split division the same way we do multiplication (ex. EX5: Simplify. 9 a. 121 5 b. 16 4 c. 7 d. x3 y2 z4 12 12 ). 13 13 Section Two: Square-Root Functions and Radical Equations A square root function is a function that contains at least one radical. The simplest of which is y x (This function is the inverse of y x 2 ). Think about what we can take the square root of. The domain of this function is zero and positive numbers. Since we are talking about the principle root, the range is positive real numbers. The graph is seen to the right. One example of a radical function deals with a pendulum. The motion of a pendulum is l described by the equation t 2 where t is the time it takes to make one full swing, g l is the length in centimeters, and g is the acceleration due to gravity (980 centimeters per second squared). EX1: Determine the time in seconds that is takes a 120-centimeter pendulum to make one complete swing. Determine the length in centimeters of a pendulum that takes 1.5 seconds to make one complete swing. We can solve radical equations using the fact that squaring is the inverse of square rooting. We use the steps below to solve radical equations: 1. Isolate the square root if there is only one 2. Separate the square roots if there is more than one 3. Square both sides (FOIL if necessary) 4. Clean up and solve now if possible 5. Start over with step one if necessary 6. Check the solution(s)! EX2: Solve the equations. a. x 3 3 b. 2 x 3 4 c. 3x 4 x d. 4 x 5 x Remember that we solve some quadratic equations using radicals. EX3: Solve each equation with radicals. a. x 2 72 b. x 2 200 c. x 2 82 152 d. x 2 y 2 z 2 EX4: The area of a circular flower bed is 120 square feet. What is the diameter of the bed? (The area of a circle: A r 2 ) EX5: Solve the equations by using radicals. a. x 2 10 x 25 49 b. x 2 12 x 36 1 Section Three: The Pythagorean Theorem The Pythagorean Theorem is a method used to find a missing side length of a right triangle when we know the lengths of the other two sides. The longest side of a right triangle is called the hypotenuse. It is always across from the right angle. The other two sides are called the legs of the triangle. The Pythagorean Theorem says that the sum of the squares of the legs is equal to the square of the hypotenuse, or a 2 b 2 c 2 where a and b are the legs and c is the hypotenuse. EX1: Find the unknown lengths. EX2: An airplane leaves an airport and flies 48 km due north and 20 km due east. At that point, how far is the airplane from the airport? EX3: Find the missing part of the right triangles. a. a 3, b ?, and c 5 b. a ?, b 6, and c 25 c. a 0.75, b ?, and c 1.25 We can also use the theorem to determine if we have a right triangle by looking at the side lengths. If a 2 b 2 c 2 then the triangle is a right triangle. (c is the longest side) EX4: Do the following lengths define a right triangle? a. 5, 12, and 13 b. 6, 7, and 8 c. 6, 12, and 6 3 Section Four: The Distance Formula We can find the distance between two points on a coordinate plane by using the distance formula. The distance formula is simply a variation of the Pythagorean Theorem. d x2 x1 y2 y1 2 2 EX1: Find the distance between each pair of points. Round answers to the nearest hundredth. a. 11,7 and 5, 1 b. 3,6 and 2,8 We can prove that three points form a right triangle by using the inverse of the Pythagorean Theorem along with the distance formula. EX2: Given vertices 3, 4 , 2, 2 , and 0,3 , determine whether the points form a right triangle. The midpoint formula tells us the point that is directly between two other points in a coordinate plane. x x y y2 midpoint 1 2 , 1 2 2 We are simply averaging the x’s and averaging the y’s EX3: Find the midpoint of the following pair of points. a. 11,7 and 5, 1 b. 3,6 and 2,8 EX4: The streets of a city are laid out like a coordinate plane, with City Hall at the origin. Alan lives 5 blocks west and 4 blocks south of City hall, and his friend Mona lives 7 blocks east and 2 blocks north of City Hall. They want to meet midway between their homes. At what location should they meet? EX5: The center of a circle is 3, 5 . One endpoint of a diameter is at 2, 3 . What are the coordinates of the other endpoint of the diameter? Section Five: Geometric Properties In this lesson we will discuss several properties of a circle. A circle is the set of all points that are an equal distance from a given point called the center of the circle. The distance from the center to any of these points is called the radius. We can use the distance formula to derive the equation of a circle. x h y k 2 2 r2 where (h,k) is the center and r is the radius EX1: Find the equation of the circle with the given center and radius. a. C 0,0 and r 1 b. C 2,5 and r 5 c. C 2,3 and r 4 We can use the midpoint formula to derive another theorem of geometry called the triangle midsegment theorem. Triangle Midsegment Theorem: The segment joining the midpoints of two sides of a triangle is parallel to the third side and is half the length of the third side. EX2: Test the Triangle Midsegment Theorem for the triangle with the given coordinates: 0,0 , 2,7 , and 6,0 . We will now discuss the geometry of a parabola. Another definition of a parabola is as follows: A parabola is the set of all points that are an equal distance from a given point (the focus) and a given line (the directrix). EX3: Use the distance formula to write an equation that describes the set of all points that are an equal distance from the point 4,2 and the x-axis. Section Six and Seven: The Sine, Cosine, and Tangent Function Trigonometry is a branch of mathematics that deals mainly with triangles and special ratios between the sides of the triangles. The most basic trigonometry deals only with right triangles. Before discussing these ratios we will look at the names for the three sides of a right triangle. The longest side of a right triangle is the side across from the right angle. It is called the hypotenuse. The next two sides we define depending on which of the two acute angles we are referencing. The side that makes up the angle along with the hypotenuse is called the adjacent side. The side that does not touch the angle is called the opposite side. In the figure above, angle C is the right angle. If referencing angle A: Side 1 is the opposite side Side 2 is the adjacent side Side 3 is the hypotenuse If referencing and B: Side 1 is the adjacent side Side 2 is the opposite side Side 3 is the hypotenuse We will discuss three trigonometric functions that describe the relationship between the 3 sides. (We will use the Greek letter theta, , to name our angle in these formulas.) Sine of an angle Cosine of an angle Tangent of an angle opp adj opp sin cos tan hyp hyp adj EX1: If opposite is 3, adjacent is 4, and hypotenuse is 5, find the three trig function values. EX2: Find the values of the six trigonometric functions of X and Y in the triangle at the right. Give the exact and approximate answers rounded to the nearest thousandth. EX3: Use the calculator to evaluate the following expressions. a. sin30 b. cos60 c. tan75 d. sin 1 0.235 e. cos 1 0.168 f. tan 1 1.5 EX4: Use your trig functions to find the missing side length. a. Use sine if 30 and the hypotenuse is 12 to find the opposite. b. Use cosine if 48 and the hypotenuse is 23 to find the adjacent. c. Use tangent if 65 and the adjacent is 35 to find the opposite. We can use the six trig functions to find missing pieces of our triangle. EX4: For the triangles below, find the missing side lengths. a. b. We can use these procedures to find an angle of depression or angle of elevation (inclination). EX5: The height of an observation tower in a state park is 30 feet. A ranger at the top of the tower sees a fire along a line of sight that is at a 1° angle of depression. How far is the fire from the base of the tower? Round your answer to the nearest foot. When given the sides of a triangle and we are looking for the sides we use the inverse functions: sin 1 , cos 1 , and tan 1 . We sometimes call these functions arcsine, arccosine, and arctangent. EX6: Solve the following triangles (Find all sides and angles). Keep in mind that the angles of all triangles add up to 180 degrees. a. fj b. Section Eight: Introduction to Matrices A matrix (plural: matrices) is simply a way of displaying data in a table form. We draw a matrix in brackets. Matrices are usually named with a capital letter. Woodville High School Enrollment Girls Boys Freshmen 34 29 Sophomores 40 26 Junior 23 30 Senior 17 22 34 40 23 17 E 29 26 30 22 Matrices are divides into rows (horizontal) and columns (vertical). The dimensions tell how big the matrix is. We measure dimension as row column . Matrix E above is a 2x4 matrix. Therefore it has eight entries or pieces. We name each entry in the following way: e12 40 (first row, second column) and e24 22 (second row, fourth column). A matrix with the same number of rows and columns are called square matrices. Two matrices are equal only if every entry in each are equal. EX1: Are the following matrices equal? 3 0.06 64 ? 23 50 1 11 48 6 110 12 We can add or subtract matrices simply by adding or subtracting the corresponding entries. Therefore, we cannot add or subtract unless the dimensions of the matrices are the same. 3 5 1 4 0 1 3 1 . EX2: Let X and Y 4 6 2 1 8 1 2 5 a. Find X Y b. Find X Y There are two types of matrix multiplication. The first type of multiplication is scalar multiplication. It is simply the same thing as the distributive property. EX3: Perform the scalar multiplication. 3 0 2 a. 4 5 2 1 3 0 5 2 b. 1 6 4 1 1 The next type of multiplication is when we try to multiply two matrices together. Dimensions do not have to be the same in order to multiply two matrices. However, not all matrices can be multiplied. Two matrices can only be multiplied if the first matrix has the same number or columns as the second matrix has rows. Can Multiply 2 4 4 3 Can’t Multiply 2 43 4 3 55 1 1 7 7 9 1 25 1 6 2 4 5 The number one rule to remember when multiplying matrices is to multiply rows by columns and add. EX4: Multiply the following matrices. 2 1 1 3 a. 4 2 2 4 3 4 1 2 3 b. 2 5 2 1 3 6 3 c. 1 0 1 2 0 4 5 3 5 d. 4 1 2 1 0 There is an identity element for matrix multiplication. The identity element is a square matrix with ones on its main diagonal and zeros everywhere else. Below are the 2 2 , 3 3 , and 4 4 identity matrices. 1 0 0 0 0 1 0 0 1 0 I I 22 I 33 44 0 1 0 0 1 0 0 0 0 1 EX5: State the identity matrix for the matrix and verify by using multiplication. 3 2 1 2 3 1 4 4 5 1 0 0 0 1 0 0 0 1