Mth 95 Module 4 Chapter 8 Spring 04 Review - Solving quadratic equations using the quadratic formula Write the quadratic formula. The NUMBER of REAL and COMPLEX SOLUTIONS to a quadratic equation ( a b c 0) can be determined b evaluating the discriminant, b 4ac, the radicand in the quadratic formula. If b 4ac > 0, then there are TWO REAL solutions. If b 4ac = 0, then there is ONE REAL solution. If b 4ac < 0, then there are NO REAL solutions. Instead there are TWO imaginar solutions. Use the discriminant to determine the number of real or imaginar solutions for each equation. 40 4 40 0 8. - Solving Quadratic Equations b Square Root and/or Completing the Square A quadratic equation is a degree equation that can be written in the standard form as where a, b, and c are real numbers and Which of the following are quadratic equations? 44 8 e) 57 ( ) 6 4 0 f ) 7 Remember the Square Root Propert: If b is a real number and if a b, then a b. Use square root to solve each equation over the set of Comple Numbers. First isolate or the binomial squared, and then take the square root on both sides of the equation. 9 6 4 60
Mth 95 Module 4 Chapter 8 Spring 04 ( 4) 5 ( ) 0 Applications: Height of a Free-falling Object Neglecting air resistance, the distance s(t) in feet traveled b an object freel falling is given b s( t) 6t, where t is the number of seconds elapsed How long would it take a pebble to hit the water if dropped off a bridge 00 feet above? A flower pot is knocked off the window sill of a nd stor apartment. The sill is 0 feet above the ground. How long will it take to hit the ground? Solving Quadratic Equations b Completing the Square What does a perfect square trinomial look like? Remember the square of a binomial creates a perfect square trinomial. ( ) ( 5) When a is one, the value of c in a perfect square trinomial can be found b squaring half of b, the coefficient of. Complete each perfect square trinomial. 0 7 Fill in the correct c to make each of the following a perfect square trinomial and then write it b as a binomial squared. Remember c 4 8 6 5
Mth 95 Module 4 Chapter 8 Spring 04 Steps in Completing the Square for ) Isolate the quadratic and linear terms 6 4 0 6 4 ) If a, it is necessar to divide the equation b the coefficient of the quadratic term, a. (step not needed). ) Divide the coefficient of b and square it. Add this result to BOTH SIDES of the equation. 4) Factor the perfect square trinomial. 5 5) Take the square root of both sides. 5 69 4 9 6) Isolate. 5 (eact answers) = 5 5.4 or = 5 0.76 (approimate answers) Solve the follow quadratic equations where a =, b completel the square. 6 0 5 Eample of Completing the Square where a. Solve for : 5 0 ) Isolate the quadratic and linear terms 5 ) If necessar, when a, divide the equation b the coefficient of the quadratic term. 5 5 5 ) Divide the coefficient of b and square it. 5 5 4 6 5 5 5 Add this result to BOTH SIDES of the equation 6 6 4) Factor the perfect square trinomial 5 7 5) Take the square root of both sides 6) Isolate. 4 6 5 7 7 4 6 4 5 7 5 7 4 4 4
Mth 95 Module 4 Chapter 8 Spring 04 Solve the following quadratic equations b completing the square. + 7 = 4 4 + 5 = 0 Another Application Simple interest is used when loans are made. It uses the formula I = prt. Most investments are computed with compound interest. In this case, the interest is accrued on both the principal and the interest. It produces an eponential growth function. The formula used for the total amount of mone when the interest is compounded annuall is: AP( r ) t where P is the principal, r is the annual rate, t is the amount of time in ears and A is the amount of mone at the end of t ears. Find the amount of mone in an account that has $500 invested at 6%, compounded annuall for ears. Find the interest rate, r, if $600 compounded annuall grows to $764 in ears. Review: Evaluate the discriminant for each quadratic equation and tell how man real solutions or comple solutions there are. 5 6 0 4 4 0 4
Mth 95 Module 4 Chapter 8 Spring 04 Use the completing the square and/or the square root propert to solve each equation. Simplif completel. Give solutions in eact form, no decimals. Use a + bi form where appropriate. 0 0 40 0 8.- The Quadratic Formula The quadratic formula can be derived from the standard form of a quadratic equation b completing the square. a b c 0 Standard form, a 0 a b c Isolate quadratic and linear terms. b c a a Divide each term b a. b b b c a a a a b b c 4a a 4a a 4a b b 4ac a 4a 4a b b 4ac Divide the coefficient of the linear term b, square it, and add the result to both sides of the equation. Factor the perfect square trinomial on the left side. Take the square root of both sides. Simplif. Simplif. Isolate and simplif again. a a a a b b 4ac b b 4ac a Use the quadratic formula to find the real solutions (-intercepts) of each equation. You ma check our solution(s) b graphing each equation on our calculator and finding the -intercepts. Give the eact solution and a decimal approimation rounded to two decimal places where appropriate. In the case of comple solutions, write them in standard form. 97 0 0 5 5
Mth 95 Module 4 Chapter 8 Spring 04 Use Completing the Square AND the Quadratic Formula to solve the following. Make sure our answers match!: 4 50 8 4 6 45 Applications Use the Pthagorean Theorem to find the distance saved b cutting across the lawn. The sidewalks meet at a right angle. One leg of the sidewalk is feet longer than the other. The path across the lawn measures 0 feet. How man feet are saved b the cut off instead of going around on the sidewalk? 6
Mth 95 Module 4 Chapter 8 Spring 04 Bob has a picture that is 8 centimeters longer than it is wide. The area of the picture is 900 square centimeters. What are the length and width of his picture? Give our answer in a complete sentence with our dimensions rounded to the nearest tenth of a centimeter. Suppose that a stone is thrown upward with an initial velocit of 44 feet per second (0 miles per hour) and is released 4 feet above the ground. Its height (h) in feet after t seconds is given b the model. h( t) 6t 44t 4 a) When does the stone reach a height of feet? Approimate that time to the nearest tenth of a second if it is not eact. b) After how man seconds does the stone hit the ground? Approimate that time to the nearest tenth of a second if it is not eact. 7
Mth 95 Module 4 Chapter 8 Spring 04 8.5 - Quadratic Functions and Their Graphs A QUADRATIC function can be written in the form f a b c, where a, b, and c are real numbers and a 0. The graph of a quadratic function is called a PARABOLA. The VERTEX is the lowest point on a parabola that opens UPWARD or the highest point on a parabola that opens DOWNWARD. Graph f. Label it. Name the verte. To determine when the graph is increasing and decreasing, walk along the -ais, left to right. The graph is increasing when.the graph is decreasing when. The ais of smmetr (AOS) is the vertical line What is f()? f( )? Use our calculator to help sketch graphs of the following on the above set of aes. 4 5 Basic Transformations of Graphs in the form = a When a > 0, the graph of When a < 0, the graph of a is a parabola that OPENS. a is a parabola that OPENS. As the value a increases, the parabola becomes. For a, is the graph is than =. As the value a decreases, the parabola becomes. For 0 a, the graph is than =. For all graphs of the form a, the verte is, and the ais of smmetr is. 8
Mth 95 Module 4 Chapter 8 Spring 04 Quadratics of the form: f ( ) a k Use our calculator to help sketch graphs of the following on the same set of aes. 4 4 4 Basic Transformations of Graphs in the form = a + k For all graphs a k, the verte is. If k is positive, the graph shifted k units. If k is negative, the graph shifted k units. The ais of smmetr is. Quadratics of the form: f ( ) ( h) Use our calculator to help sketch graphs of the following on the same set of aes. 4 ( ) ( 4) ( ) Basic Transformations of Graphs in the form f ( ) ( h). The graph has the same shape as and the verte is. If h is positive, the graph shifted to the h units. If h is negative, the graph shifted to the h units. The ais of smmetr is. 9
Mth 95 Module 4 Chapter 8 Spring 04 Quadratics of the form: f ( ) ( h) k Use our calculator to help sketch graphs of the following on the same set of aes. ( ) ( ) Label each verte. Sketch and label each ais of smmetr (AOS). Basic Transformations of Graphs in the form f ( ) ( h) k, The graph has the same shape as and the verte is. The ais of smmetr is the vertical line.. Summarize the VERTICAL and HORIZONTAL translations of the parabola in terms of h and k. Write each equation in the formf ( ) ( h) k. Identif each verte. 4 Verte Form of a Parabola: f ( ) a( h) k The verte form of a parabola with verte hk, is a h k, where a is a constant and a 0. When a > 0, the parabola opens. When a < 0, the parabola opens. For a, is the graph is than =. For 0 a, the graph is than =. The ais of smmetr is the vertical line. 0
Mth 95 Module 4 Chapter 8 Spring 04 Graph the following: f ( ) 5 Label the verte. Sketch and label the ais of smmetr (AOS). Graph the following: f ( ) ( ) 4 Label the verte. Sketch and label the ais of smmetr (AOS). Sec. 8.6 - Further Graphing of Quadratic Functions The verte is not obvious when the quadratic equation is written in standard form, but we can find the verte using the following formula. b The -coordinate of the graph a b c, a 0, is given b. a b To find the -coordinate, substitute the -value into the equation, i.e. find f a. Using the above formulas, find the verte for f ( ) -intercept -coordinate -coordinate Verte Ais of Smmetr Parabola opens
Mth 95 Module 4 Chapter 8 Spring 04 g 5 6 Complete the following steps to help ou graph Using the above formulas, find the verte -coordinate -coordinate Verte -intercept Parabola opens Use factoring, completing the square, or the quadratic formula to algebraicall find an -intercepts that are real numbers, -intercept(s) g 5 6. Now graph Indicate the verte and the ais of smmetr. Find the verte and an or intercepts to help ou graph the following equations. Indicate the verte and the ais of smmetr on our graphs. f Verte Ais of smmetr -intercept Parabola opens -intercept(s)