Algebra 1 Key Vocabulary Words Chapter 11

Transcription

1 Section : radical expression radical function square root function parent square root function Section 11.2: radical expression (see Ch ) simplest form of a radical expression rationalizing the denominator Section 11.3: radical equation extraneous solution Section : right triangle hypotenuse legs of a right triangle Pythagorean theorem converse of Pythagorean theorem Algebra 1 Key Vocabulary Words Chapter 11

2 Radical Expression A radical expression is an expression containing a radical such as square root, cube root, or other root. The following are examples of Radical Expressions: is a radical expression. is a radical expression. is a radical expression. is a radical expression. Radical Function A Radical Function contains a radical expression with the independent variable in the radicand. The following are examples of Radical Functions: is a radical function. and are radical functions. and are radical functions. and are radical functions.

3 Square Root Function If the radical is a square root, then the function is a square root function. The following are examples of Square Root Functions: is a Square Root Function. is a Square Root Function. is a Square Root Function. is a Square Root Function. Parent Square Root Function The parent square root function is the most basic square root function. The Parent Square Root Function is The Parent Square Root Function is In function notation the Parent Square Root Function however, is not. The Parent Square Root Function is however, is not. The Parent Square Root Function is, however is not. The Parent Square Root Function is, however is not.

4 Simplest Form of a Radical Expression The Simplest Form of a Radical Expression is characterized by: No perfect square factors other than 1 are in the radicand No fractions are in the radicand No radicals appear in the denominator of a fraction. The following are examples of Radical Expressions in Simplest Form. The Radical expression is said to be in simplest form because no perfect square factors other than 1 are in the radicand The Radical expression is said to be in simplest form because The Radical expression is said to be in simplest form because no perfect square factors other than 1 are in the radicand. The Radical expression is said to be in simplest form because. Rationalizing the denominator: Rationalizing the To rationalize, the first step is to multiply Denominator, Given 11.2 Rationalizing the Denominator is the process of eliminating a radical from the denominator of a radical expression by multiplying the expression by an appropriate from of 1., multiply by, Product Property of Radicals by. To rationalize, the first step is to multiply by., Simplify To rationalize, the first step is to multiply by. To rationalize, the first step is to multiply by.

5 Radical Equation 11.3 An equation that contains a radical expression with a variable in the radicand is a radical equation. Here are examples of radical equations: The expression equation. is a radical The expression is a radical equation. The expression is a radical equation. The expression is a radical equation. Extraneous Solution 11.3 A solution that does not satisfy the original equation is an extraneous solution. This happens when squaring both sides of a = b. In solving, the solution x = -3 is an extraneous solution because when substituted, it does not yield a true statement (3-3) In solving, the solution x = -2 is an extraneous solution because when substituted, it does not yield a true statement. In solving, the solution x = is an extraneous solution because when substituted, it does not yield a true statement. In solving, the solution x = is an extraneous solution because when substituted, it does not yield a true statement

6 Right Triangle A right triangle is a triangle that contains a right angle. (Or with a 90 0 angle) Examples: A C Z B ΔABC is a right triangle because it has a right angle. 90 o X Y ΔXYZ is a right triangle because A right triangle is a triangle that contains a. Hypotenuse The hypotenuse of a right triangle is the side opposite the right angle. (It is the longest side of the right triangle) A Hypotenuse A C B In ΔABC the hypotenuse is segment AB C B Z In ΔXYZ the hypotenuse is segment X Y The hypotenuse of the right triangle is opposite the.

7 Legs of a Right Triangle The Legs of a Right Triangle are the other two sides that form the right triangle other than the hypotenuse. Leg Leg Hypotenuse A C Z X B Y In ΔABC the legs are segment AC and segment BC In ΔXYZ the legs are segment and segment The legs of the right triangle are and. (Teacher supplies the figure) Pythagorean Theorem The Pythagorean Theorem states that if a triangle is a right triangle, then the sum of the squares of the length of the legs equals the square of the length of the hypotenuse. a Given a right triangle with length of sides a, b, and c. Since the triangle above is a right triangle then it follows that b c If a and b are lengths of the legs of a right triangle and c is the length of the hypotenuse, the Pythagorean Theorem states that a 2 + b 2 = c 2. If x and y are lengths of the legs of a right triangle and z is the length of the hypotenuse, the Pythagorean Theorem states that. a 2 + b 2 = c 2 If h and i are lengths of the legs of a right triangle and j is the length of the hypotenuse, the Pythagorean Theorem states that h 2 + i 2 = j 2. If and are lengths of the legs of a right triangle and is the length of the hypotenuse, the Pythagorean Theorem states that.

8 Converse of a Pythagorean Theorem The Converse of a Pythagorean Theorem states that if a triangle has side lengths as a, b, and c such that a 2 + b 2 = c 2, then the triangle is a right triangle Since the lengths of the sides satisfy the equation a 2 + b 2 = c 2, we can conclude that the triangle above is a right triangle. ( = 5 2 ) The Converse of a Pythagorean Theorem states that if a triangle has side lengths as x, y, and z such that x 2 + y 2 = z 2, then the triangle is a right triangle. The Converse of a Pythagorean Theorem states that if a triangle has side lengths as,, and such that, then the triangle is a right triangle. The Converse of a Pythagorean Theorem states that if a triangle has side lengths as c, d, and e such that c 2 + d 2 = e 2, then the triangle is a right triangle. The Converse of a Pythagorean Theorem states that if a triangle has side lengths as,, and such that, then the triangle is a right triangle.