Quadratic Functions- Word Problems

Lesson 37

In the warm up for lesson 37 your students will be finding the parts of a parabola described by the following quadratic equation. They will find the axis of symmetry, the coordinates of a vertex, the X intercept, and the Y intercept. This is a very important skill because today they are going to do applications with quadratic equations; and they must know how to find all of these specific parts of the of the parabola.

A baseball was hit at a height of 3 feet off the ground with an initial velocity of 64 feet per second. If “T” represents the time in seconds, and “Y” represents the height in feet, the height of the ball can be determined over a limited period of time using the equation; Y equals negative 16T squared plus 64T plus 3. What is the maximum height that the ball reached?

In “S” study the problem, we will first underline our question. What is the maximum height that the ball reached? We will also answer the question what is this problem asking me to find? This problem is asking you to find the maximum height of the ball.

We will begin our lesson by looking at a quadratic equation which has been graphed. We see that we have a parabola and our graph below represents the relationship between the time in seconds after a projectile is propelled vertically into the air and the height in feet that it reaches. So in other words we are looking at our time in seconds, so this is 1 second, 2 second, 3 second, we are going to go across and we can see that if our projectile started here, it went up up up, got to it’s maximum height, and then came back down and hit the ground. Because just like anything, what goes up must come down, and so this is like your flight pattern of your projectile. We ask ourselves several questions, number 1, what is the initial height of the projectile? And if you notice our graph starts here at zero, zero. So zero is our initial height, it starts at zero. We can ask out selves what the height of the projectile is after 4 seconds, we see that our seconds is represented on the X-axis, so we move to 4 and if you run your finger up at 4, it’s in between 400 and 500, so 450 is our height; our height is represented on the Y-axis. Anytime you want to find after how many seconds, you can just find that number of seconds, and you can find the height by finding where it intersects the parabola. What is the maximum height that the projectile reaches? Our projectile goes all the way up to this point right here, so if you notice the Y-coordinate, which represents the height, is 800 in feet, so 800 feet is the maximum height. What is the amount of time it takes the projectile to reach its maximum height? You have to look at time so you are looking at seconds on your X-axis, and our maximum height which was 800 feet actually happened at 10 seconds. And from the time that the projectile has been fired, how long does it take the projectile to hit the ground? So we see that it goes up to the maximum, comes back down and it hits the ground at 18. This means that the projectile was in the air 18 seconds.

One of the hardest parts of solving word problems with quadratic equations is your students have to actually figure out what the problem is asking them to find and so they have to identify are we looking for an X-intercept, are we looking for a coordinate, are we looking for the maximum, are we looking for our axis of symmetry; all of those part could be asked depending on what the question in the problem.

The height “H” in feet of a rock falling, from a height of 80 feet for “T” seconds, can be found using the equation “H” equals negative 16T squared plus 80. What is the height after 2 seconds? In “S” we will study the problem, we will first underline our questions. What is the height after 2 seconds? This problem is asking me to find the height after 2 seconds. In “O” organize the facts, we will read out problem and we will decide which facts are necessary or unnecessary. The height “H” in feet, is a fact; of a rock falling from a height of 80 feet, that is a fact; for “T” seconds; can be found using the equation H equals negative 16T squared plus 80. So now we have to decide if our facts are necessary or unnecessary. We know that the height equals “H” and this is pretty important so we are going to write down, that “H” represents height; are rock is falling from a height of 80 feet, this also can be necessary even though we have it in our equation, if you want to write it you can but some student may say it is in the equation, some students may want to cross it out. I’m not going to write it down because it is in our equation. But that is one of those facts that it depends just on how the students justify. For “T” seconds, I am gong to write down that “T” represents the time; we next look at the fact can be used in this equations, and our equations defiantly important, so we are going to write that fact down. In “L” line up a plan, we have to think about what our plan is before we decide what our operations will be. If you have your student think back to their original parabola they looked at, we are trying to find the height after 2 seconds, so we know the 2 seconds, or we know the time or T, so what we can do in order to find the height is plug that time in. so we are going to be plugging into our quadratic equations and in order to simplify that to find the height; we will use multiplication and addition. So if we write in words what our plan of action will be, we are going to substitute in the time into the equation for “T” and simplify for the height. In “V” verify your plan with action, we will first estimate, we see that when we substitute in our time it is squared and this is multiplied by a negative, so anytime you add a negative to 80 it’s going to get smaller, unless it’s zero and it will stay the same, so because we have a time other than zero, which is greater than zero, then our answer is going to have to be less than 80. To carry out our plan we said we are going to have to substitute into our equation, so we are going to write our original equation H equals negative 16T squared plus 80, and we are trying to find the height after 2 seconds, which means T equals 2, so we are going to plug in a 2 for T; please remind your students to use parentheses when they plug in. You’re going to simplify your exponent first, then your going to multiply, and add to find a height of 16. In “E” examine your results, we will first ask does our answer does our answer make sense? If we go back to our questions, or what we were looking for, the height after 2 second, then we do have our height. We have a value for “H” which represents height, so our answer makes sense. Is our answer reasonable, we said in our estimate that our answer must be less than 80 and 16 is less than 80, so our answer is reasonable. And is your answer accurate, your students could check this by graphing the parabola, graphing the quadratic equation, or they could rework the problem in another place. So now we will rewrite our answer as a complete sentence. The height of the rock after 2 seconds is 16 feet.

Tyrone built a model rocket for his science project. The equation h equals negative 16t squared plus 260T models the flight of the rocket launched from ground level at a velocity of 260 feet per second. Where “h” is the height of the rocket in feet after “t” seconds. How long does it take the rocket to hit the ground? In “S” study the problem we will first underline the question, how long does it take the rocket to hit the ground? We will also ask ourselves, what is this problem asking me to find? This problem is asking me to find the time it takes for the rocket to hit the ground. In “O” identify our facts, we are going to identify our facts and then decide which ones are necessary and unnecessary. Tyrone built a model rocket for his science project, fact; the equations h equals negative 16T squared plus 260T models the flight of the rocket; launched at a velocity of 260 feet per second; where “h” is the height of the rocket in feet; after “t” seconds. We now have to decided which of these facts are necessary and unnecessary. Tyrone built a model rocket for his science project, this is one of those facts that many students will say is necessary and some will say is unnecessary, most of them will not want to write it down so I’m going to draw a line through it. The equation h equals negative 16T squared plus 260T models the flight of the rocket, is necessary so it needs to be written down; launched at a velocity of 260 feet per second this fact actually deals with how the equation was created, it tells you about the 260 feet that is used as your “V”, however it is not a necessary fact so I’m going to cross it out. Where “h” is the height of the rocket in feet, this is necessary, we need to know what the variable stands for; after “t” seconds, this is also necessary so we know what “t” stands for. In “L” line up a plan, we have to identify our operations and write in words what our plan of action will be. If we think back to what the question is asking us to find, how long does it take the rocket to hit the ground, and we think back to that original discussion your students had about the parabola it started at zero, at ground level, and then the parabola went up into the air and came back down at another X intercept. So we are actually doing is we have to find the X intercept of the parabola. So in order to do this we are going to have to factor and then set those factors equal to zero, and we will use subtraction and division in order to solve those equation. So our operations are going to be factoring, subtraction, division. Your students may not be able to know that the subtraction and division are going to be used after we factor because they don’t know what the factors are going to be so factoring rule would be sufficient. If we write in words what our plan of action will be, we are going to set both factors equal to zero and solve for the variable. In “V” verify your plan with action, our first step will be to estimate. We can look at our equation and we know that the parabola is going to go up and come back down. This one is kind of hard we are going to be factoring things out, but some students may be able to estimate. I’, going to say that my estimate will be around 15; I actually got this estimate because I know that after I factor our a T from our equation, if we were to set that equal to zero and have 260 divided by 16, it’s kind of like 225 divided by 15, which would be 15 so I’m just estimating around 15. Let’s see what the real answer is. If we factor, in this case we don’t have a trinomial we have a binomial; we do have a greatest common factor, so we can take that out, our greatest common factor is going to be 4T, so that leaves us with negative 4T plus 65; and we are going to set each of those equal to zero; so we have 4T equal to zero and negative 4T plus 65 equal to zero. We divide by 4 on both sides and we have T equal to zero, and that is where we actually initially started, we initially started at a height of zero, that is a time of zero and a height of zero, and then it went up from there and then it came back down, and if we solve our second equation by subtracting 65 from, and now we have negative 4T equal to negative 65; and we divide both sides by negative 4; the we have T equal to 16 point 25 or time of 16 and 25 hundredth seconds, 16 and a fourth some students may get; so there is our answer. Zero is not going to make sense for it to be in the air zero, that is actually our first X intercept where the parabola begins, and it goes up and it comes back down with a time of 16 point 25. In “E” examine our results, we have to ask our selves does our answer make sense, so we go back to our questions and we were looking at how long is take for the rocket to hit the ground. We have 16 point 25 as a T or a time, so that does make sense. Is your answer reasonable, we estimated around 15, and so yes our answer is reasonable; and is your answer accurate, your students could check their answer by reworking the problem in a separate place, some students may want to graph parabola, to see where the X intercepts fall, these are all options when checking for accuracy. And we will then write our answer as a complete sentence. It takes the rocket 16 point 25 seconds to hit the ground.

We have already “S” the problem so we know that the problem is asking me to find the maximum height of the ball. If we want to organize our facts, or “O” the problem. We will being by identifying each fact and we then decide if they are necessary or unnecessary. The baseball was hit at a height of 3 feet off the ground, this is a fact; with initial velocity of 64 feet per second, fact; if T represents the time in seconds, fact; and Y represents the height in feet; the height of the ball can be determined over a limited period of time using the equation Y equals negative 16T squared plus 64T plus 3. Now we have to decided whether our facts are necessary or unnecessary. This is going to be tricky for some students because of the first two facts. The base ball was hit at a height of 3 feet off the ground, a lot of students may want to say that his fact is necessary but we are not actually going to use it in our problem because of the 3 we don’t really want to have that 3 there, many student think that they need to add, or subtract, or divide by 3, or multiply by 3; so I going to say that this fact is unnecessary. With an initial velocity of 64 feet per second, once again we are going to use the formula or the equation that we’re given; and then these first two facts are used in the equation; so we don’t really need this fact to be written down; if T represents the time in seconds, this is necessary because we want to know what the variables represent, and Y represents the height in feet, this is also necessary, and then our last fact gives us our equations, which is necessary, so we are going to write that down. In “L” line up a plan, we will first decided what our plan will be, and this will help us decided what operations we are going to use. We are trying to find the maximum height, and if we are trying to find the height we will be looking for Y, ask you students to go back and think about the first parabola that they looked at and how they found that height, and they actually found the Y coordinate of the vertex. In order for us to find our vertex we are going to find our axis of symmetry because we have a formula to find our X coordinate. Then we will plug our X coordinate in to find our Y coordinate or the height. So we will be using the formula for the axis of symmetry, which involves multiplication and division, then we will also be plugging that back in to the formula which will multiplication and addition in order to simplify. So we will use multiplication, division, and addition. Our plan will be to find the axis of symmetry and use the values of X for T to find the maximum height. In “V” verify your plan with action, we will first estimate, your students are going to be finding your axis of symmetry, which will give them their X value and they should know that this is negative B over 2A. You’re just guessing at this point, or they are rounding in their head, if I have a negative B over 2A it is going to give them a positive answer and some students may say that it’s close to 1, some students may say that it is close to 2. If they do say 1 then they are looking at plugging a 1 back in that’s going to give you a negative 16 plus 64 plus 3. I think it would be fair to say that if it is between 1 and 2 that it could be around 100, so our estimate could be around 100 or less than 100. The harder the problems the harder it is for your students to estimate, but if they have some ball park it will help. We have X equals negative B over 2A, carry out our plan. Our equation is Y equals negative 16T squared plus 64T plus 3; so A equals negative 16 and B equals 64; and C equals 2. Negative 64 divided by 2 times negative 16, equals negative 64 divided by negative 32, which equals 2; from here we are going to plug that 2 back into our original equation, use parentheses please, we will evaluate our exponents, then multiply, and then add so that we see is height or Y is equal to 67. From here we are going to examine our results, does our answer make sense? We will compare our answer to the question, which is what is the maximum height the ball reached, so it is possible for a ball to read 67 feet. Is our answer reasonable, we will look at our estimate, we said our answer should be around 100 or less than 100, so yes our answer is reasonable because 67 is close to that, and is your answer accurate, your students can check for accuracy by reworking the problem on a separate sheet of paper, some students may want to graph the parabola and what the maximum is when they graph it. We will not write our answer as a complete sentence. The maximum height is 67 feet.