Math 2 1 Lesson 4-5: Completing the Square

Targets: I can identify and complete perfect square trinomials. I can solve quadratic equations by Completing the Square.

When a=1 in a perfect square trinomial, then cb

2

21

.

On your own:

a. x2 – 18x + ____________________ = _______________________

b. x2 + 2x + _____________________ = _______________________

c. x2 + 21x + ____________________ = _______________________

Math 2 2 Rewriting a Quadratic Equation from standard form to vertex form when a = 1 Here is an example: Rewrite the following quadratic in vertex form: y = x2 - 16x + 5 Step 1 Subtract c from both sides -5 -5

Step 2 Take 2

21

b and add to both sides y – 5 + 64 = x2 - 16x + 64

Step 3 Factor the perfect square trinomial y + 59 = (x - 8)2 The vertex is (8,-59) y = (x – 8)2 - 59

4. y = x2 + 4x + 10 5. y = x2 + 5x + 34 **Could you do #3 using formula instead of completing the square?

Math 2 3 Homework 4-5: Completing the Square (a = 1) Write each quadratic function in vertex form by completing the square. 1) 푦 = 푥 − 14푥 + 54 2) 푦 = 푥 + 8푥 + 7 3) 푦 = 푥 + 10푥 + 28 4) 푦 = 푥 − 4푥 − 2 5) 푦 = 푥 − 12푥 − 41 6) 푦 = 푥 − 8푥 + 15 7) 푦 = 푥 − 4푥 8) 푦 = 푥 + 8푥 + 18

Math 2 4 Lesson 4-6: Solving Equations by Completing the Square

Targets: I can solve quadratic equations by Completing the Square

You can also solve equations by COMPLETING THE SQUARE. The reason you can do this is because you can square root a binomial squared. In other words: baba 2)(

Be careful when you square root binomials. Find the mistake in this statement. Error Analysis: 242 xx

Solve these equations by completing the square. 1. x2 + 9x + 15 = 0 2. m2 + 16m = -59 3. t2 - 4t -165 = 0 4. m2 + 7m - 294=0

Math 2 5 Homework 4-6: Solving Quadratics by Completing the Square Solve each quadratic equation by completing the square and taking the square root. 1) x2 + 6x – 7 = 0 2) x2 + 6x + 10 = 0 3) x2 – 4x – 8 = 0 4) x2 – 48 = 0 5) x2 – 2x – 1 = 0 6) x2 – 8x + 24 = 0 7) 5x2 – 6x = 8 8) x2 – 4x – 165 = 0

Math 2 6 Lesson 4-7: Introduction to the Quadratic Formula

Targets: I can solve quadratics by using the quadratic formula.

In order to use the quadratic formula to solve a quadratic: you must first set the quadratic equal to 0.

Math 2 7

8.) A batter hits a baseball. The equation y= -.005x2 + 0.7x + 3.5 models the path of the ball, where x is the horizontal distance, in feet, the ball travels and y is the height, in feet, of the ball. How far does the ball land from the batter. Round to nearest tenth of a foot. What was the maximum height of the ball? ________________ How far from the batter was the ball when it hit the maximum height? ________________ How far from the batter was the ball when the ball was 15 feet high? ________________ 9.) Pete hit a fly ball whose path can be described by the function (풙) = −ퟎ.ퟎퟎퟓퟒ풙ퟐ + ퟐ.ퟏ풙 + ퟑ.ퟓ . In this function, x is the distance on the ground (in feet) of the ball from home plate and h(x) is the height (in feet) of the ball. The ball is traveling toward the outfield fence, which is 7 feet high and 389 feet from home plate. Could the ball go over the fence? Explain your reasoning. ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 10.) Consider the equation 풚 = ퟐ풙ퟐ + ퟒ풙+ ퟏ. Write a brief description of the graph of the equation. (You do not have to draw the graph.) Your description should include facts such as the shape of the graph, the number of x-intercepts, the y-intercept, the vertex, and the equation of the axis of symmetry.

Math 2 8 Homework 4-7 Quadratic Formula Use the quadratic formula to find the zeros of each function. Approximate the zeros to the nearest tenth 1) 푦 = 푥 − 2푥 − 3 5) 5 = 3푥 − 4푥 Zeros: ______, ______ Zeros: ______, ______ 2) 푦 = −푥 − 4푥 − 3 6) 푦 = 2푥 − 16푥 + 28 Zeros: ______, ______ Zeros: ______, ______ 3) 푦 = 푥 − 8푥 + 15 7) 6 = 푥 Zeros: ______, ______ Zeros: ______, ______ 4) 푦 = −2푥 − 16푥 − 23 8) −12 = −2푥 + 6푥 Zeros: ______, ______ Zeros: ______, ______

Math 2 9 Lesson 4-9: Solving Quadratics

Targets: I can solve quadratics by factoring, using the quadratic formula, using my graphing calculator, and completing the square. Some quadratic equations can be solved by using inverse operations. This can be done when b=0. Remember that when you square root both sides that xx 2 , so you will need to give both the positive and negative values for x. 1. x2 – 16 = 0 2. 3x2 + 6 = 0 3. (x - 7)2 = 0 To solve quadratics where 0b it is slightly more complicated, here are some important things to remember. _____________ to quadratics are also called ________, ________, and _____________. A _________ is defined as the x value for which f(x) = 0. We have several different methods that we can use to solve quadratics but setting the equation equal to zero is always our first step. This list is not exhaustive but in this class this is how we will solve quadratics.

Factoring Quadratic Formula Technology Completing the Square

Zero Product Property: If ab = 0, then either a or b must equal zero Solve by factoring. You will need to use the zero product property to find solutions. Let's practice the zero product property first: 1.) (x - 2)(x+9)=0 2.) (2x+3)(x-4) =0 3.) (7n-2)(5n-4)=0 4.) x(x-3) = 0

Math 2 10 Solve by factoring. 5.) 24100 2 xx 6.) yy 220 7.) 406 2 nn

8.) 5x2-10x = 0 9.) c2 = 5c

Can you graph by factoring. Since you can find the solution to a quadratic equation by factoring, the x intercepts are the same as the solutions. Using symmetry, you can find the axis of symmetry. Graph: Factor to find the zeros and use symmetry to find the vertex. 10. 1892 xxy Explain how to find the axis of symmetry using the roots. ___________________________________ __________________________________________________________________________________

Math 2 11 Solve using quadratic formula. Round answers to nearest hundredth if necessary. 11.) 031310 2 zz 12.) nn 11212 2 13.) 722 mm

14.) 04113 2 xx 15.) 827 2 xx Go back and do problems 11-13 using your calculator to find your solutions. Make sure you use your calc function and don't just use trace. Trace does not give as good of an approximation. Graph these quadratics using their zeros and vertices. 16.) y = 6x2 + 13x - 5 17.) y = 8x2 + 14x - 15

Math 2 12 Homework 4-9: Solving Quadratics 1. 푥 = −18− 9푥 2. 7푥 − 42 = −35푥 3. 2푥 + 2푥 − 12 = 0 4. 푥 = 16 5. 푥 − 5푥 − 14 = 0 6. 푥 + 8 = 44 7. 2푥 − 3푥 − 5 = 0 8. −4푥 + 6푥 − 16 = −5푥

Math 2 13 9. 4푥 + 8푥 + 7 = 4 10. −5푥 = −500 11. (푥 − 6) = 121 12. 3푥 + 72 = 33푘 13. 푥 + 2푥 = 80 14. 9푥 + 10 = 91 *15. 0.7푥 − 8푥 − 7.2 = 0 *16. 4.3푥 = 2.6푥 + 5.8 Solve with graphing calculator. Round your answer to the nearest hundredth.

Math 2 14 Lesson 4-10: Applications with Quadratics

Gravitational constant (g) = 2/8.9 sm or 2/32 sft

The height of a projectile given as a function of time is given by the formula: 002

21 htvgth where

0v is the initial velocity and 0h is the initial height. If an object is dropped, the height can be given by the

formula 02

21 hgth

1.) An acorn drops form a tree branch 70 ft above the ground. The function h= -16t2+70 gives the height h of the acorn (in feet) after t seconds. What is the graph of this function. At what time does the acorn hit the ground?

________________

a.) What is a reasonable domain for this function? _________________

b.) When is the vertex of a parabola a minimum? _________________

c.) When is the vertex of a parabola a maximum? _________________ 2.) Suppose a person is riding in a hot- air balloon, 154 ft above the ground. He drops an apple. The height, h, in feet, of the apple above the ground is given by the formula h = -16t2+154, where t is time in seconds.

a.) What time does the apple hit the ground? ______________

b.) How far has to apple fallen from t=0 to t=1? ______________

c.) Does the ball fall the same distance from t=1 to t=2 as it does from t=0 to t=1? Explain ______________________________________________________________________________________

Math 2 15 3.) A blueprint for a 15ft-by-9ft rectangular wall has a square window in the center. If each side of the window is x ft, the function y=135 - x2 gives the area (in sq. ft)of the wall minus the area of the window.

a.) Graph the function.

b.) What is a reasonable domain of the function? Explain. ____________________________________________________________________________________________________________

c.) What is the range of the function? Explain.

____________________________________________________________________________________________________________

d.) Estimate the side length of the window if the area of the wall is 117 ft2. ______________

4.) A baseball is thrown into the air with an upward velocity of 30 ft/s. Write a function that gives the height

h, in feet, after t seconds of the baseball. Remember ( 002

21 htvgth )

_____________

b.) When will the baseball reach its maximum height? _____________

c.) What is the maximum height of the ball? _____________

d.) When will the ball hit the ground? _____________

e.) When will the ball hit a height of 12 feet? _____________

Math 2 16 5.) Suppose you have 100 ft of string to rope of a rectangle section for a bake sale at a school fair. The function A= -x2+50x give the area of the section in sq. ft, where x is the width in feet. What width gives you the maximum area you can rope off? _____________

b.) What is the maximum area? _____________

c.) What is the range of the function? _____________ 6.) What translation maps y= x2 onto f(x)= x2 + 2x - 3? 6. _____________ 7.) 8.) You have enough shrubs to cover an area of 100 ft2. What is the radius of the largest circular region you can plant with these shrubs?

Math 2 17 9.) Suppose a ball is thrown upward from a height of 5 feet with an initial velocity of 35 ft/sec. a.) Write an equation relating the time t and the height h of the ball. b.) Find the height of the ball after 2 seconds. c.) Is the ball still in the air after 3 seconds? Explain. d.) When did the ball reach its max height? e.) How high was the ball when it reached its maximum height? 10.) For what values of n will the equation x2=n have two solutions? 10. ___________

b.) What values of n will have one solution? b. ____________

c.) What values of n will have no solution? c.____________ 11.) The trapezoid has an area of 1960 cm2. Use

the formula )(21

21 bbhA , where A represents

the area of the trapezoid, he represents its height, and 21andbb represent its bases, to find the value of y. 11. ___________

Math 2 18 12.) You have a rectangular koi pond that measures 6ft by 8ft. You have enough concrete to cover 72 ft2 for a walkway, as shown in the diagram. Write a function for find the area of the concrete walkway.

b.) What is the maximum width of the walkway? 13.) Find the equation of a quadratic congruent to y = 2x2 that has roots of 2 and -5. Give the equation in standard form. 13. _________________