1. Section 11 – 1 Simplifying Radicals Multiplication Property of Square Roots: For every number a > 0 and b > 0, You can multiply numbers that are both under the radical and you can separate a number under a radical into two radicals being multiplied by each other First 10 perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 (know these)

2. Simplifying a radical is when you rewrite a radical so that all of the perfect squares have been factored out The end result is said to be in simplified radical form Simplify each expression Ex1. Ex2. Ex3. Ex4.

3. Division Property of Square Roots: For every number a > 0 and b > 0, You use this property just as you would with multiplication Fractions are not allowed to have radicals in the denominator You must rationalize the denominator 1) Simplify the radical(s) 2) Multiply the numerator and denominator by the radical remaining in the denominator

4. Simplify Ex5. Ex6. Ex7.

5. Section 11-2 The Pythagorean Theorem The Pythagorean Theorem is applied only to RIGHT triangles You can use this theorem to find the length of missing sides The two shortest sides of a right triangle are called the legs (they must meet at a 90° angle) The longest side is called the hypotenuse (it is directly across from the 90° angle)

6. The Pythagorean Theorem: a² + b² = c² where a and b are the legs and c is the hypotenuse Find the length of the missing side (to the nearest tenth) Ex1. a = 8, b = 12, c = ? Ex2. a = 20, b = ?, c = 41 Ex3. 25 ft 67 ft x

7. Section 11 – 3 The Distance and Midpoint Formulas The Pythagorean Distance Formula: The distance d between any two points (x 1, y 1 ) and (x 2, y 2 ) is Ex1. Find the distance between A(-3, 7) and B(5, -4). Show work. Ex2. Find the perimeter of ∆XYZ with X(-3, 4) Y(-1, -5) and Z(2, 2). Show work. The Pythagorean Distance formula can be derived from the Pythagorean Theorem

8. If you are asked to give an answer in exact form, you are to give it in simplified radical form The midpoint of a segment is the point that divides the segment into two equal segments The Midpoint Formula: The midpoint M of a line segment with endpoints A(x 1, y 1 ) and B(x 2, y 2 ) is Ex3. If segment CD has endpoints C(-4, 7) and D(3, -2), find the midpoint of CD. Show work.

9. Section 11 – 4 Operations with Radical Expressions Radicals are like radicals if they have the same radicand (the same number under the radical symbol) Unlike radicals have different numbers under the radical You can add and subtract like radicals, just as you could with like terms Ex1. Simplify A) B)

10. You can distribute with radicals as well (remembering that you can multiply radicals together and then simplify them if possible) Ex2. Simplify Ex3. Use FOIL & then simplify Conjugates are the sum and the difference of the same two terms (i.e. and are conjugates) The product of two conjugates results in the difference of two squares (an integer)

11. To rationalize the denominator of an expression that has an addition or subtraction radical expression in the denominator, you must multiply the numerator and denominator by the conjugate of the denominator Ex4. Rationalize the denominator You should never leave a radical in the denominator of the a fraction!

12. Section 11 – 5 Solving Radical Equations A radical equation is an equation that has a variable as a radicand Remember that the expression under a radical must be nonnegative Ex1. Solve each equation. a) b) If an equation has radical expressions on both sides, square each side and then solve

13. Ex2. Solve When you solve an equation by squaring each side, you create a new equation. This new equation may have solutions that do not solve the original equation. See page 609 These solutions that do not solve the original equation are called extraneous solutions Ex3. Solve a) b) You should make a table of values to create an accurate graph

14. Section 11 – 6 Graphing Square Root Functions A square root function is a function that contains the independent variable in the radicand The parent function for square root functions is The graph of the parent function is the positive half (because radicands can’t be negative) of a sideways parabola (see page 614)

15. The domain of a function contains all possible values of the independent variable The domain of the parent function is {x: x > 0} You can find the domain by graphing and looking at the graph or you can determine algebraically what values can meaningfully be substituted for x Ex1. Find the domain of The equation is a translation of the parent function by k units up

16. The equation is a translation of the parent function by k units down The equation is a translation of the parent function by h units to the left The equation is a translation of the parent function by h units to the right Ex2. Graph each equation a)b)

18. Section 11 – 7 Trigonometric Ratios There are three trigonometric ratios: sine (sin), cosine (cos), and tangent (tan) These ratios describe a specific relationship between an angle in a RIGHT triangle and two of the sides of that triangle SOHCAHTOA should help you remember these ratios (if you spell it correctly)

19. Use a capital letter to represent an angle Open to page 621 to see how to identify adjacent leg vs. opposite leg vs. hypotenuse Ex1. Use the triangle below to find a) sin Xb) cos Xc) tan X 3 ft 4 ft 5 ft X Y Z

20. You can use your calculator to find the value of trigonometric functions Make sure your calculator mode is in degrees! Ex2. Find the value of each expression. Round to the nearest thousandth. a) sin 130° b) cos 130° c) tan 130° You can use SOHCAHTOA to find the lengths of missing sides of a right triangle Ex3. Find the length of x. 37° x 29

21. An angle of elevation is an angle from the horizontal up to a line of sight (see page 623) An angle of depression is an angle measured below the horizontal line of sight (see page 624) You can use angle of elevation and angle of depression with trigonometric functions to solve for missing lengths (see example 4 on page 623 and example 5 on page 624)