This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide.

Transcription

1 COLLEGE ALGEBRA UNIT 2 WRITING ASSIGNMENT This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide. 1) What is the difference between quadratics and lines? Explain the differences between a linear expression and a quadratic one. Also describe something in the real world that is described a quadratic. Lines have the x only to the 1 st power, while quadratics have an x to the 2 nd power. The height of a projectile like a baseball is described by a quadratic. 2) How do I graph a quadratic? Enumerate the important features of the graph of quadratic using bullets. For each bullet explain the meaning and how to find it algebraically. The Vertex: We find the x-coordinate of the vertex by computing b/(2a), and then to find the y-coordinate of the vertex we plug that number into the quadratic. The vertex represents the minimum or maximum of the quadratic. The x-intercepts: We find the x-intercepts by making y = 0 and then solving for x. X- intercepts are the value of x which makes the y-coordinate zero. There could be 0, 1, or 2 x-intercepts. The y-intercept: We find the y-intercept by making x = 0 and then solving for y. y- intercept are the value of y which have an x-coordinate of zero. There is exactly one y- intercept. 3) What is factoring? Give a short description of what it means to factor an expression. Contrast the meaning of factoring with multiplication. Factoring means to write something as the product of factors. Factoring is the opposite of multiplication. For example foiling is a type of factoring which turns a product into a polynomial ( )( ) While factoring does the opposite, taking a polynomial and writing it as a product of factors. ( )( )

2 4) What are the different ways we factor? In class we have discussed several ways of factoring. Using bullets describe each type of factoring including when it s used and describe the mechanics of how to actually do the factoring. Use examples to explain where needed. Factoring a trinomial with the easy method. This is used to factor something of the form. You come up with two numbers that multiply to c and add to b. Then you put those numbers in the form (x + #)(x + #) Factoring a trinomial with the AC method. This is used to factor something of the form. First you multiply a*c, then you determine two numbers that multiply to a*c and add to b. Then you break up the middle term (the bx) into two terms such as. Then you perform factor by grouping Factor by grouping is done when you have 4 terms with no GFC. You look at the first two terms and take a GCF out of those. Then you do the same for the second set of two terms. Often these two results will have a GCF between them. You factor our that GCF and you re done. Factoring out a GCF can be done for a polynomial with any number of terms. You simply look for what is common (what is each term divisible by). Write down the GCF followed by a set of parenthesis containing what is left after dividing out the GCF.

3 5) Why are we learning to factor here and now? While factoring has many uses, this unit has focused on factoring for one purpose. What is that purpose? Also explain why we were completely unable to accomplish this task before learning the skills we have in this unit. Factoring allows us to solve quadratic equations. We learned that if the product of two factors equals zero, then at least one of the factors must equal zero. We were unable to solve trinomials before this unit because we couldn t isolate the x. 6) What is the quadratic formula and what is it for? Explain what the quadratic formula does and when we can use it. We can use the quadratic formula to solve a quadratic equation in the form. 7) What can we expect from the quadratic formula? How many solutions can we expect from the quadratic formula and what type are they? It is possible to get 2 real solutions, or 2 complex solutions, or 1 real solution. 8) Why were imaginary numbers created? Explain the situation where mathematicians were stuck and needed to come up with what are now called imaginary numbers. Imaginary numbers were created so we can take the square-root of negative numbers.

4 9) What do imaginary numbers mean to us now? Explain how complex numbers relate to the parabolas that you have learned about in this unit. If we use the quadratic formula to find the x-intercepts of parabolas, and get a complex/imaginary result that means that the parabola does not intersect the x-axis. 10) How do we perform mathematical operations on complex numbers? For each of the following operations explain in words how perform them. a. Addition/Subtraction Combine the real parts, and combine the imaginary parts. b. Multiplication Foil or multiply them together like you would for any variable. The difference will be that one terms may have an. We then replace the with a negative one. c. Division Multiply the fraction by the complex conjugate of the denominator over itself. When we do this the denominator will come out purely real. At the end we make sure to separate the fraction into its real and imaginary parts. 11) What is completing the square? Explain the purpose of completing the square, and how it helps us accomplish it. Completing the square is a process that allows us to solve quadratic equations. It helps us because it changes the form from into something where we can isolate the x. Ex. I cannot isolate the x because there is an x^2 and an x. ( ) I can isolate the x by subtracting 1, square rooting, then subtracting 2.

5 12) Time to demonstrate everything I know! Use the following situation to demonstrate as many of the applicable skills you have learned in this unit. Be sure to explain your results in the context of the situation described. A projectile is launched from the top of a tower when a timer hits t = 0 seconds. The height of the projectile is described by the following: +36t+200 Height is measured in meters, and time is measured in seconds. 1) It opens down 2) The y-intercept is found by plugging in x = 0. The y-intercept is (0,200). This means that at t = 0 the height is 200 feet. 3) I will find the vertex ( ) ( ) The vertex is (3.67, ). This means that the projectile reaches a maximum height of about 266 feet after 3.67 seconds have passed. 4) Find the x-intercepts by making y = 0 ( )( ) ( ) The x-intercepts are (-3.96,0) and (11.04,0) We throw out the first negative value because it makes no sense. The projectile hits the ground after seconds have passed. My discriminant ( ( )( )) was positive, so I expected 2 real answers. (3.67,266) (0,200) (-3.96,0) (11.04,0)

6 I can answer a questions such as What time does the projectile reach a height of 50 feet. ( )( ) ( ) The projectile reaches a height of 50 feet at seconds.