6.3 Creating and Comparing Quadratics

Transcription

1 6.3 Creating and Comparing Quadratics Just like with exponentials and linear functions, to be able to compare quadratics, we ll need to be able to create equation forms of the quadratic functions. Let s examine each modality individually. Creating Equations from Tables We ll start by examining a table and looking for any patterns that we see What things do you notice about this table? Here are some things you might notice: There is no constant rate of change or constant difference. There is no constant growth factor that comes out of the differences. The vertex must be at 2, 7 since the values are symmetrical on either side of that point. What you might not have noticed is that there is a constant SECOND difference in the table. Did you notice that the differences in outputs are all the odd numbers? That means the rate of change is changing by a constant value of two. Check this out: If there is a constant second difference, that s our hint that it is a quadratic function. Notice that this is not a constant growth rate. The growth rate for exponential functions was multiplication of the rate of change. This is a constant difference between the different rates of change which we call a constant second difference. This gives us a way to determine whether a function is quadratic or not. Since we know the vertex is at 2, 7 in this table, we know that the equation must be something of the form 2 7. What we don t know is the value, but we can find it by using an input/output pair from the table. We shouldn t use the vertex because it will make the parentheses zero meaning we can t solve for the value. Instead, let s use a different point such as 1, 2. Plug in what you know and solve for what you don t From this we can see that the value must be one. In other words, this quadratic has not been stretched (widened or narrowed) and has the equation 2 7.

2 Let s look at an example where the value is something other than one We have a constant second difference, so it is a quadratic function. The vertex is at 2, 4 so we can solve for using the point 3, 7 as follows: So we know that the equation of this function is Astute observers may have noticed that the second difference is actually equal to twice the value. In the first example, 1 and the constant second difference was 2. In the second example, 3 and the constant second difference was 6. Explore on your own or with a teacher whether this works all the time or not. Creating Equations from Graphs With graphs, we follow the same process as with tables except that we ll usually be able to spot the vertex immediately thereby making the process shorter. Consider the following graph: Notice that the vertex is at 6, 8 and that we have a nice point at 3, 1 which lets us immediately set up an equation to solve for the value as follows: This gives us the equation 6 8. Creating Equations from Descriptions We can describe quadratics without a context by giving the zeros and a point or the vertex and a point. Given the vertex and a point is exactly the same as the table or graph method for finding the value of the function in vertex form. Given the zeros and a point, it may be prudent to at least find the coordinate of the vertex and use that to solve for the coordinate of the vertex and value. 267

3 Let s say we know that the zeros of a quadratic are at1 8 and 2 and there is a point on the quadratic at 4, 6. Thinking of the graph, it should look something like this: We can t quite see where the vertex is, but we should be able to solve for it because we know that the coordinate of the vertex must be 3. This leads us to following equation as a partial set up. 3 Now we can use two of the points, one zero and the other given point, to set up a system of equations to solve for the missing values. Why can t you use both zeros? Because you would end up with infinite parabolas that could have those zeros. We need to use the other given point to uniquely define a quadratic function. So we end up with the following equations: 6 43 Simplify these equations and solve for the missing and as follows: Now we know that our equation is 3. It s OK that the vertex doesn t come out as an integer! That happens in real life! What to Compare? Now that we have equations, what exactly do we compare? We can compare where vertices are using the point,. We can compare which quadratic has the highest average rate of change within a given interval by substituting into those equations. We can see which quadratic will eventually always have a higher average rate of change using the value. We can also see when each quadratic will have a certain output by solving the quadratics using factoring, completing the square, or the quadratic formula. 268

4 Real Life Modeling with Quadratics In real life, we use quadratics to model the distance something has fallen because the acceleration of gravity is in meters per second squared. Squared. That means quadratic. In general we use one of the following two formulas depending on whether we are using feet or meters: Height in terms of time using 16 Height in terms of time using 4.9 In both of these equations, the represents the initial velocity the object is projected at. If the object is thrown upwards, then is positive. If it is thrown down, then is negative. The represents the initial height of the object. Finally, represents the time since the object was thrown. So, let s say you were standing on a platform 300 in the air and throw a ball straight up at 20 /. We can model this situation with the equation: If you friend is standing on a different platform 200 in the air and throws a ball straight up at 40 /. We can model this situation with the equation: Now can ask some interesting comparison questions like which ball will hit the ground first? You threw the ball from a higher point, but your friend threw his up in the air twice as fast as you. Since we consider the ground to have a height of zero, just plug in 0 and use your favorite solution method (probably the quadratic formula in this case). Here s the work for your ball Since we can t have a negative time, your ball hit the ground at 5. Here s the work for your friend s ball Again, we can t have a negative time, so your friend s ball hit the ground at 5 as well. You tied! 269

5 Lesson 6.3 Create equations for the following situations. 1. Vertex: 3, Zeros: Point: 5, Zeros: Vertex: 7, Point: 3,

6 10. Vertex: 1, Point: 2, Answer the following questions about comparing functions. 13. Which type of function will eventually always be growing at the fastest rate: linear or quadratic? Why do you think so? 14. Which type of function will eventually always be growing at the fastest rate: quadratic or exponential? Why do you think so? 15. Is it possible for a linear function to be higher than a quadratic function in some spots but lower than the quadratic in others? Sketch an example to the left. Answer the following questions using the given information. To see who was stronger, The Vision and Thor had a throwing contest. They took turns throwing Thor s hammer straight up into the air, and the winner was the person who kept the hammer in the air longer. Since The Vision has a definite advantage with an infinity gem, he allowed Thor to stand at the top of a cliff 320 in the air before throwing his hammer up at 128 /. The Vision then threw the hammer up at 160 / from the bottom of a hole that was 144 deep. 16. Write an equation to represent the height of the hammer in terms of time when Thor threw the hammer. 17. Write an equation to represent the height of the hammer in terms of time when The Vision threw the hammer. 18. When did Thor s throw reach a maximum height and what was that height? 271

7 19. When did The Vision s throw reach a maximum height and what was that height? 20. When did Thor s throw hit the ground? 21. When did The Vision s throw hit the ground? Why are there two solutions and which one is right? Sam and Pam had were throwing rocks into a pond. Sam threw from the ground at a speed of 24.5 /. Pam threw from a ledge 4.9 above the ground at a speed of 19.6 /. 22. Write an equation to represent the height of the rock in terms of time when Sam threw. 23. Write an equation to represent the height of the rock in terms of time when Pam threw the rock. 24. When did Sam s throw reach a maximum height and what was that height? 25. When did Pam s throw reach a maximum height and what was that height? 26. When did Sam s throw hit the ground? 27. Approximately when did Pam s throw hit the ground? Why are there two solutions and which one is right? 272