Precalculus Functions DEV Project In learning you will

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Precalculus Functions DEV Project “In learning you will teach, and in teaching you will

Precalculus Functions DEV Project “In learning you will teach, and in teaching you will learn. ” -Phil Collins By: Emma Goebel, Leaha Sinaveve, and Hannah-Rose Rizer

Question 1: Tony Stark is testing out a new Iron Man suit. With his

Question 1: Tony Stark is testing out a new Iron Man suit. With his new suit he can project himself vertically upward from the top of the Stark tower with the initial velocity of 500 ft/sec. It’s distance d(s) in feet above the ground after s seconds is given by the equation below. What is Tony Stark’s max height his new Iron Man suit can reach? In how many seconds? How tall is the Stark tower?

Step 1: Replace your old c value with a new perfect c value Think

Step 1: Replace your old c value with a new perfect c value Think of the equation in a² + bx + c form. Get rid of your c value (we will be creating a new perfect c value). The c value is 150, so I subtracted 15 o on both sides to make the equation equal. Divide the right part of the equation by -25 to get s² 20 s + ___. You can only make a perfect square if your squared variable is 1. Next step is to find your perfect c value.

To find your new perfect c value, you take your b value and divide

To find your new perfect c value, you take your b value and divide it by two. Then multiply the whole thing by itself by squaring it. So I took my b value (20), divided it by two, squared it and got 100. Then rewrite your equation with your new c value. Because you added 100 to one side, you must not forget to add the 100 to the other side of the equation as well. When you add something to one side, you always add to the other to keep the equation balanced. I multiplied the 100 by -25 before adding the 100 to the other side then subtracted the number I got from the other side. In this case you would subtract 2, 500 from the -150 to get -2650.

Step 2: Factor your new equation Now that you have a perfect square equation,

Step 2: Factor your new equation Now that you have a perfect square equation, you can simply factor it by dividing your b value by 2. Being perfect, your c value can factor out into two equal factor pairs (s-10) and (s-10). Then you add your new c value from the other side of the equation to find your new equation.

Step 3: Find your vertex, then using that, your max height Next, find your

Step 3: Find your vertex, then using that, your max height Next, find your vertex by using your vertex form of your equation you just found (y= (a-h)² + c). Vertex = (x, y), so you find x by looking at your h value. Whatever your h value is, your x is the opposite. So if your h value is -2, your x value is 2. In this case your x will be 10 because your h value is -10. Then to find y, just look at your c value. Your c value is y. Your max height is your y value of your vertex and the amount of time needed to get to that max height. In this case, it takes the new Iron Man suit 10 seconds to reach it’s max height of 2, 650 feet.

Step 4: Finding out the height of the Stark Tower All you need to

Step 4: Finding out the height of the Stark Tower All you need to do is put 0 into the original equation to find out how tall the Stark tower is. This is because you have not gone anywhere and no seconds have passed. When you put 0 into the equation -25 s² + 500 s + 150, you get 150. Therefore, the Stark tower is 150 feet tall.

Question 2: Black Widow is being held by super villain, Mad Thinker. Black Widow

Question 2: Black Widow is being held by super villain, Mad Thinker. Black Widow persuades Mad Thinker to make a math problem she would never be able to solve. She proposes that if she gets the answer right to the problem, she should be set free. This is the question Black Widow was asked: Complete the square of the function below. Using that, find the y intercept, line of symmetry and the vertex of this equation. Solve this problem so that Black Widow can be set free.

Step 1: Replace your old c value Don’t be intimidated by the big numbers!

Step 1: Replace your old c value Don’t be intimidated by the big numbers! First subtract your c value from both sides to get rid of it. In this equation, your c value is 18, 505. Then divide your entire equation by 1, 234, 567 because you can only find a perfect square if you’re a value is one. Now find your new c value by using the equation (b/2)². When you use that equation you should get 20. 25. Put that in your new equation like the picture shows.

Step 2: Replace your old c value Subtract your new c value to the

Step 2: Replace your old c value Subtract your new c value to the other side of the equation to make the equation equal. Now that you have your perfect square, you can simply factor your quadratic into two factor pairs (x-4. 5). Then add the 43, 518, 486. 75 to both sides to complete the vertex form of this equation.

Step 3: Answer Mad Thinker’s problem! (y= (a-b)²+c) Looking at your vertex form of

Step 3: Answer Mad Thinker’s problem! (y= (a-b)²+c) Looking at your vertex form of your equation you just found, take your b and put it into the x of the vertex. Since it was subtracted in the equation, that means the 4. 5 is positive. Then take your c and put that into the y of the vertex. Your y intercept is what your equation equals when you put 0 in for x. In this case it’s 18, 505. Your line of symmetry is whatever your x is in your vertex because that is where your parabola is able to be cut in half (so that each side is equal). In this case its 4. 5. Congrats you saved Black Widow!

Question 3 Thor is trying a new method of charging his hammer. This method

Question 3 Thor is trying a new method of charging his hammer. This method involves a machine that generates electricity that powers the hammer. The machine is powered by math problems. Solve this math problem to help Thor charge his hammer. Solve for f(x)/g(x), then find the domain and range.

Step 1: Set up the problem First set up the problem and set the

Step 1: Set up the problem First set up the problem and set the top ≥ 0, and the bottom > 0. ≥ 0 means that the numerator can equal 0. However, because it’s under a radical it cannot be a negative number because a negative number cannot be under a radical. The bottom is >0 because it can equal 0, but cannot be a negative number as previously mentioned.

Step 2: Split up the numerator and denominator and factor both First, take your

Step 2: Split up the numerator and denominator and factor both First, take your numerator and divide by four to make your a value equal to one. Now you are able to factor. Since 6 and -7 multiply to equal 42 and add up to equal -13, these are your factor pairs. Next, take your denominator and divide that by 100 so that your a value is equal to one (like we did for the numerator). Now we can factor! Since -5 and -8 multiply to equal 40 and add up to equal -13, these are your factor pairs.

Step 3: Find your intercepts and put them on a number line Find the

Step 3: Find your intercepts and put them on a number line Find the intercepts for the numerator and denominator and put them on a number line to find the domain. We find the intercepts by setting the factor pairs equal to 0 and solving for x. x=5, x=6, x=7, x=8 If your factor pairs are ≥ 0, then you use ∙ which means the numbers are included. If your factor pairs are > 0, then you use ○ which means the numbers are not included. The line includes -∞ to 6 and 7 to ∞. Numbers between 6 and 7 (but not including 6 and 7) are not included in the domain because if you put those numbers into the equation, you will get a negative number under a radical (which creates an imaginary number that does not graph).

Step 4: Find the domain using your number line Using the number line below,

Step 4: Find the domain using your number line Using the number line below, find your domain. Use [ when the numbers are included, and ) when they are not.

Step 5: Find the range To find the range, graph the numerator of the

Step 5: Find the range To find the range, graph the numerator of the problem. Since it cannot be a negative and goes on forever the range is [0 -∞). Congrats ! You helped Thor solve his math problem, so now he can charge his hammer!

Question 4 Bruce Banner (the Hulk) is trying to figure out how to create

Question 4 Bruce Banner (the Hulk) is trying to figure out how to create a new nuclear bomb. In order to put it together he has to solve the domain of problems. Help him out and solve one of the many domain problems he has to solve. Here is one of the domain problems he has to solve in order to create the new nuclear bomb. Graph this equation below and find the domain.

Step 1: Find the intercepts and degree of the equation First, find the intercepts

Step 1: Find the intercepts and degree of the equation First, find the intercepts of the equation. We do this by setting the factor pairs equal to 0. Then write if they go straight through, curve, or bounce. You determine this by counting the number of degrees, or x’s. If the degree is 1 it goes straight through, if it is even, it bounces, and if it is odd, it curves. Then, find the degree of the entire equation. All you need to do is add all of your number of degrees, or your x’s.

Step 2: Sketch your graph using your intercepts Sketch the ideal graph shape by

Step 2: Sketch your graph using your intercepts Sketch the ideal graph shape by looking at the total degree. Since the total degree is even and the equation is positive, the graph will have a shape similar to this… Sketch your graph using your intercepts. Remember bounces and curves. If you have a straight through, the line goes straight through the x intercept. If you have a curve, the line curves through the x intercept. If you have a bounce, the line “bounces” on the x intercept.

Step 3: Find the domain To find the domain we look at the graph

Step 3: Find the domain To find the domain we look at the graph we made and see where it is positive. Where it is positive is what the domain is. We only use brackets with these graphs because we found the x intercepts (where x is 0) and under a radical, it can be equal to zero. For this example the domain would be -∞ to -6, then the next time the line hits the x axis is at -4, so the -4 would be included in it’s own bracket. Then it would be -2 to 3, and 5 to +∞. Great job! You helped out Bruce Banner (the Hulk) very much! One less domain to find!

Question 5: Hawkeye is aiming his bow and arrow at a TTBS (thing to

Question 5: Hawkeye is aiming his bow and arrow at a TTBS (thing to be shot). With the two given equations, determine which equation to use to find how much time it takes for Hawkeye’s arrow to reach its maximum height. Also, figure out the maximum height, the xintercepts , graph the equation and give the domain and range.

Step 1: Figure out which equation to use We chose the quadratic equation representing

Step 1: Figure out which equation to use We chose the quadratic equation representing the vertical distance because when Hawkeye shoots an arrow it makes an arc which looks like a quadratic.

Step 2: Find the vertex The question told us to find the maximum height

Step 2: Find the vertex The question told us to find the maximum height and the amount of time it takes to get there. Thinking of the trajectory of the arrow in graph form, the maximum height and its time will be the highest point on the graph (also known as the vertex). To find the x-coordinate of the vertex we use the equation –b/2 a. In this case b= -99. 995 and a= -4. 9. After plugging in those numbers we discover that the x-coordinate is 10. 204 which is also our time for the arrow. Lastly, we need to find the y-coordinate. We find that by plugging the x-coordinate into our original quadratic equation and solving. The y-coordinate is 510. 153 which is our maximum height. These two numbers are our vertex at (10. 204, 510. 153)

Step 3: Find the x-intercepts To find the x-intercepts we are going to use

Step 3: Find the x-intercepts To find the x-intercepts we are going to use the quadratic formula. In this case our a= -4. 9, b= 100, and our c= -79. 9. After plugging in these numbers we are left with. To get our x-intercepts we do -100+ -9. 37 which equals -109. 37. Then we subtract those numbers and are left with 90. 63. These two numbers are our xintercepts.

Step 4: Graph and find the domain and range To graph this we are

Step 4: Graph and find the domain and range To graph this we are going to plot our xintercepts on the x-axis. Then we are going to graph our vertex at (10. 204, 510. 153) and then connect the points making an arch (hopefully your graph looks better than mine). After we have graphed, we can use the graph to find our domain and range. Our domain is all real numbers because our graph is a quadratic. The quadratic isn’t in a fraction or under a radical so you don’t have to worry about having zeroes or negatives. To find the range we use our maximum “y” at 510. 153 because it is the highest our graph goes and negative infinity because our graph opens down.

Question 6: Captain America is in a fire fight shooting his pistol at different

Question 6: Captain America is in a fire fight shooting his pistol at different rates. He’s letting out only so many bullets per second. Using the given formula, find the graph and then use the graph to find the following: maximum number of bullets, how long was Captain America firing before changing his clip, and the domain.

Step 1: Find the graph Due to the complexity of the problem we are

Step 1: Find the graph Due to the complexity of the problem we are going to plug in our equation into our “Y=“ on our graphing calculators. The following graph should appear

Step 2: Find the max bullets and how long Captain America was firing Our

Step 2: Find the max bullets and how long Captain America was firing Our max is always going to be the highest point on our graph as shown by the turqoise dot which is at about 8 bullets per second. How long the Captain America was firing before he changed his clip was indicated by the point at the far right, the yellow dot, which is at about 3. 8 seconds.

Step 3: Find the domain Our domain is found in our calculator by going

Step 3: Find the domain Our domain is found in our calculator by going to our graph and doing 2 nd to calc to zero at the following points on the graph. By doing this we are left with a domain of