Ch. 7.6 Squares, Squaring & Parabolas Learning Intentions: Learn about the squaring & square root function. Graph parabolas. Compare the squaring function with other functions. Relate the squaring function to finding the area of a square.
Learning Goal: Students will work with radicals & solve quadratic equations using the absolute value function. 4 3 2 1 0 In addition to Level 3, I go above & beyond what was taught in class, Examples: - Make connection with other concepts in math (i.e. System of Equations, Absolute Value Function, Other parent functions, etc. - Make connection with other content areas. - 7.6 Practice: GRAPH all given problems Students will work with radicals and integer exponents. - Use square root symbols to solve equations in the form x 2 = y. - Evaluate roots of small perfect square. - Evaluate roots of small cubes. - Apply square roots & cube roots as it relates to volume and area of cubes and squares. Students will be able to: - Understand that taking the square root & squaring are inverse operations. - Understand that taking the cube root & cubing are inverse operations. With help from the teacher, I have partial success with level 2 and 3. Even with help, students have no success with the unit content.
Perfect Squares 25, 16 and 81 are called perfect squares. This means that if each of these numbers were the area of a square, the length of one side would be a whole number. Area = 25 units 2 5 Area = 4 16 units 2 Area = 81 units 2 9 5 4 9
s 2 = Area 1 2 = 1 2 2 = 4 3 2 = 9 4 2 = 16 Perfect Squares - iff s is an integer, then A = a perfect square. Recursive Sequence: common b or d? A: {1, 4, 9, 16, 25, 36, 49, 64, 81, 100, } D 1 : 3 5 7 9 11 13 D 2 : 2 2 2 2 2 Common Differences occur in the 2 nd round: Thus, y = degree 2 polynomial y = ax 2 + bx + c Since there is no transformation, this sequence equals the parent function: y = x 2 5 2 = 25 6 2 = 36 7 2 = 49 8 2 = 64 9 2 = 81 10 2 = 100 x length y Area 1 1 2 4 3 9 4 16 x x 2
Non-Perfect Squares - numbers whose square root equals an irrational value. Why isn t 20 a perfect square? 20 does NOT equal an Area with equivalent whole number dimensions. l = 1 unit Area = 20 units 2 w = 20 units l = 2 units Area = 20 units 2 w = 10 units The square root of 20 must be an irrational number between 4 and 5. l = 4 units A = 20 units 2 w = 5 units l = 20 units A = 20 units 2 w = 20 units
How to find the approximate square root of 20 1. What two perfect squares does 20 lie between? a) 16 and 25 b) The square root of 16 is 4, so the square root of 20 must be a little more than 4. 2. How to find the little more a) Is the non-perfect square 20 closer to 16 or 25? b) It seems to be right in the middle. So pick a number in between 4 and 5. c) Multiply 4.4 times 4.4. What do you get? i. 19.36 ii. 20 19.36 = 0.64 d) Lets see if we can get closer to 20. Multiply 4.5 times 4.5. What do you get? i. 20.25 ii. 20 20.25 = -0.25 e) 4.5 is the best estimate for the square root of 20.
A x Area = x 2 A x Recall: A A = A Because any same number x square rooted equals that given number.
Square Root (Radical) Function: Only gives positive solutions unless otherwise noted. Simplifying 9 vs. ± 9 9 = 3 vs. ± 9 = ±3 Radical vs. Quadratic x = y vs. x 2 = y Let x = 16 Let y = 16 16 = y vs. x 2 = 16 4 = y x 2 = 16 x = 16 x = 4 or x = -4
Vocabulary: Squaring: the process of multiplying a number by itself. x 2 x to the power of 2 or x squared Parabola: the graph of y = x 2 NOTE: the graph of the squaring function is a parabola. Although every positive number has TWO square roots, the square root function ( x ) in your calculator gives only the POSITIVE square root.
Absolute Value vs. Quadratic Functions y = x 2 y = x Let y = 4 y = x vs. y = x 2 4 = x 4 = x 2 4 = x 2 x = -4 or x = 4 2 = x x = -2 or x = 2
Ex.) 0 0 Integers Square Roots
SOLUTION: Ex.) 1 1.41 1.73 2 3 2 1 4 85 9.219544457 9 10 81 85 100 Integers Square Roots Since: 81 < 85 < 100 9 < 85 < 10
Ex.) Find the side of the square whose area is 6.25 cm 2. Use a graph to check your answer. x 6.25cm 2 x
SOLUTION: Ex.) Find the side of the square whose area is 6.25 cm 2. Use a graph to check your answer. Let x = side length of the square (cm.) Solve the equation: x 2 = 6.25 x 2 = 6.25 x 2 = 6.25 x = 2.5 x = -2.5 or x = 2.5 The equation has two solutions, but because the side of the square must be POSITIVE, the only realistic solution is 2.5cm. y = 6. 25 Graph: y = x 2 x 6.25cm 2 x
Absolute Value Functions vs. Quadratic Equations 1.) Graph the function f(x) = x 2 What other equation produces the same graph? 2.) Solve each equation for x. a.) x = 6 b.) x 2 = 36 c.) x = 3.8 d.) x 2 = 14.44 3.) Solve each equation, if possible. a.) 4.7 = x - 2.8 b.) -41 = x 2 28 c.) 11 = x 2 14 4.) Solve each equation for x. Sketch a graph of each. a.) x 2 = 4 b.) (x 2) 2 = 16 c.) x + 3 = 7 d.) (x + 3) 2 = 49
SOLUTIONS: Absolute Value Functions vs. Quadratic Equations 1.) Graph the function f(x) = x 2 What other equation produces the same graph? The absolute value function (y = x ) 2.) Solve each equation for x. a.) x = 6 b.) x 2 = 36 c.) x = 3.8 d.) x 2 = 14.44 x 2 = 36 x 2 = 14.44 x = -6 or x = 6 x = 6 x = -3.8 or x = 3.8 x = 3.8 x = -6 or x = 6 x = -3.8 or x = 3.8 3.) Solve each equation, if possible. a.) 4.7 = x - 2.8 b.) -41 = x 2 28 c.) 11 = x 2 14 7.5 = x -13 = x 2 25 = x 2 x = -7.5 or x = 7.5 x = no solution x = -5 or x = 5 y = x 2 AND y = x
SOLUTIONS: Absolute Value Functions vs. Quadratic Equations y = x 2 AND y = x 4.) Solve each equation for x. Sketch a graph of each. a.) x 2 = 4 b.) (x 2) 2 = 16 c.) x + 3 = 7 d.) (x + 3) 2 = 49 x 2 = -4 or x 2 = 4 (x 2) 2 = 16 x + 3 = -7 or x + 3 = 7 (x + 3) 2 = 49 x = -2 or x = 6 x 2 = 4 x = -10 or x = 4 x + 3 = 7 x 2 =-4 or x 2 = 4 x + 3 =-7 or x + 3 = 7 x = -2 or x = 6 x = -10 or x = 4 y = x 2 y = 4
Absolute Value & Quadratic Function Notation Ex.) Consider the function f(x) = x. Each point: (x, y) = (x, f(x)) a.) What is f(-3)? b.) What is f(-2)? c.) Solve f(x) = 10
Solutions: Absolute Value & Quadratic Function Notation Ex.) Consider the function f(x) = x. Each point: (x, y) = (x, f(x)) a.) What is f(-3)? b.) What is f(-2)? c.) Solve f(x) = 10 f(x) = x f(-3) = 3 f(-3) = 3 (x, f(x)) = (-3, 3) f(x) = 10 f(x) = x f(-2) = 2 f(-2) = 2 (x, f(x)) = (-2, 2) -3 f(10) f(-3) f(x) = x f(x) = 10 10 = x x = -10 or x = 10 (x, f(x)) = (-10, 10) or (10, 10)
SOLUTIONS: Remember Square Root Function Square Roots of a Number (e.) (d.)
Ex.) Equations of Parabolas
SOLUTION: Equations of Parabolas y = a(x h) 2 + k (h, k) = vertex h = horizontal shift k = vertical shift a = size change