2.1 Stretches of Quadratic Functions

Transcription

1 Unit 2 Graphing Quadratic Functions and Modelling Real World Applications Today's Topic : Graphs of Parabolas Today's Goal : to see what happens to the graph of a parabola when we change the "a" value in the graph y = ax Stretches of Quadratic Functions Complete the following table and graph the 3 functions on the same set of axis. x y = x 2 y = 4x 2 1 y = 2 x What is the relationship between the y values of each parabola? Now on the graphing calculator graph the following sets of functions one at a time. Set 1. Increasing values of "a" in y = ax 2 As the value of "a" increases... Set 2. Decreasing values of "a" in y = ax 2 As the value of "a" get closer to zero... What if the value of "a" was zero? Set 3. Negative values of "a" in y = ax 2 A negative "a" value will...

2 In general : for y = ax 2 Example 1. Graph y = x 2. Then on the same axis, graph y = 3x 2 by comparing it to y = x 2. Example 2. Graph y = x 2. Then 1 on the same axis, graph y = x 2 by comparing it to y = x 2 4. Example 3. The function y = ax 2 passes through (4, 7). What is the value of "a"? Example 4. On Earth, the vertical distance fallen by an object in free fall is given by d(t)=4.9t 2, where t is the time in seconds since the object was dropped and the distance is given in metres. a) How far could an object fall in 4 seconds? b) It is 335m to the Sky Pod (the first viewing deck) of the CN Tower in Toronto. If you were standing below and saw a small chunk of ice fall from the bottom of the Sky Pod, how much time do you have to get out of the way? Homework Page 38 #1 10, 12, 14

3 MCF3M2.2b.notebook August 16, 2016 Today's Topic : quadratic functions Today's Goal : I can relate the equation of a parabola to its graph and I know what part of the equation moves the parabola around the grid. 2.2 Translations of Parabolas In simple terms a translation means a slide around the grid without changing the shape or orientation of what is sliding. The general equation for a transformed parabola is y = (x h) 2 +k Use the graphing calculator to complete the following thoughts... Changing k When I put negative numbers in for k the parabola... When I put positive numbers in for k the parabola... When I put a zero in for k the parabola... Changing h When I put negative numbers in for h the parabola... When I put positive numbers in for h the parabola... When I put a zero in for h the parabola...

4 MCF3M2.2b.notebook August 16, 2016 Example 1. Give the h and k value for each parabola and describe the transformation(s) a) y = (x + 3) b) y = (x 6) 2 2 c) y = x d) y = (x 7) 2 Example 2. What are h and k in y = (x + 2) 2 5. The vertex of y=x 2 was at (0,0), where does it move under the transformation? Sketch the graph. Step 1. Step 2. Homework Page 46 # 1 5, 7, 9, 11

5 MCF3M2.3.notebook August 16, 2016 Today's Topic : Graphing parabolas Today's Goal : I can sketch a graph of a parabola using the transformations from the equation. 1.6 Sketch Graphs Using Transformations y=a(x h) 2 +k Example y = 2(x+3) 2 4 STEP 1. Plot the vertex (h, k). Remember that the sign of h is actually the opposite of the sign that appears in the bracket. STEP 2. Decide whether the parabola opens up or down. If the value of "a" is positive it opens up, if negative it opens down. STEP 3. For a parabola where the "a" value is 1 or 1 (i.e. there is no number in front of the bracket), plot all the other points by moving away from the vertex in the following manner. left/right from vertex up/down = = = =16 : : Or if the value of "a" is something other than 1... left/right from vertex up/down 1 1a 2 4a 3 9a : : n n 2 x a n n 2

6 MCF3M2.3.notebook August 16, 2016 Parabolas features There are 5 main properties of a parabola, and all can be picked out from the equation without having to graph the function... Opening Vertex Axis of Symmetry Max/Min Value Range Opening Vertex Axis of Symmetry Max/Min Value Range Opening Vertex Axis of Symmetry Max/Min Value Range Equation Vertex Opens Axis of Symmetry Max/Min Value Range y = (x+5) 2 7 y = 2(x 1) y = 1/3(x 9) 2 y = 3x y = 1/5x 2 y = 6(x+7) 2 12

7 MCF3M2.3.notebook August 16, 2016 Graph the parabolas from the chart on the previous slide. Homework Page 51 #1 4, 6, 8, 10

8 Topic : Goal : Quadratic Functions I can change a parabola from standard form to vertex form by a process called completing the square. 2.4 Completing the Square The following parabola is in vertex form. Expand the brackets to put it in standard form. y = 2(x 3) 2 +6 What is the one piece of information that you can get from the standard form of a parabola (y = ax 2 + bx + c)? The y intercept is the constant term 'c' when a vertex is in standard form. It is VASTLY more helpful to turn an equation that is given to us in Standard Form, into Vertex form than the other way around. The process we use to do this is referred to as COMPLETING THE SQUARE.

9 Let's start by remembering how to expand the square of a bracket. (x + 5) 2 (x 8) 2 So if I told you that the following trinomials came from squaring a binomial, could you tell me what term is missing? x x + = (x ) 2 x 2 12x + = (x ) 2 x 2 20x + = (x ) 2 x x + = (x ) 2

10 How to complete the square, when a=1 Example 2. Change y = x 2 + 8x 12 into vertex form. We want to try and turn this into a squared bracket. The problem is that the " 12" is not the correct constant term for this to be a perfect square. STEP 1. First separate the constant term (which is not what we want) from the rest of the equation, by placing brackets around the first two terms. STEP 2. y = x 2 + 8x 12 Jump down a couple of steps. If this bracket were going to be a perfect square of a binomial, what would the binomial be? What constant is missing from the bracket to make it a perfect square trinomial? y = (x 2 + 8x ) 12 = = (x ) 2 STEP 3. You can't just add something into the equation or you change it. Whatever you put in, you must also take back out to restore the balance of the equation. Now pull the bracket in, and simplify the constants on the end. y = (x 2 + 8x ) 12 = =(x ) 2

11 Example 3. Try the following a) y = x 2 6x + 18 b) y = x x c) y = x 2 40x + 2

12 How to complete the square, when a 1 Example 4. Change y = 4x x + 8 STEP 1. Start the same way, by putting brackets around the first two terms. We know that in vertex form the x doesn't have a coefficient, so divide the coefficient out of the brackets. y = 4x x + 8 STEP 2. Jump down a couple of steps. If this bracket were going to be a perfect square of a binomial, what would the binomial be? What constant is missing from the bracket to make it a perfect square trinomial? y = 4(x 2 6x ) + 8 = = 4(x ) 2 STEP 3. You can't just add something into the equation or you change it. Whatever you put in, you must also take back out to restore the balance of the equation. Now pull the bracket in. BUT this time when you pull the bracket past the correction term, remember that the number in front of the brackets belongs to it as well. To pull the bracket in, you must MULTIPLY the extra term by the front number. y = 4(x 2 6x + 9 ) 12 = =(x ) 2

13 Example 5. Try the following a) y = 2x x + 43 b) y = 3x 2 18x + 13 c) y = x 2 2x Homework Page 132 #(1, 3, 4, 7, 9, 11)eop

14 MCF3M2.5.notebook August 17, 2016 Topic : Graphing Quadratic Functions by Completing the Square Goal : I can successfully complete the square and graph the quadratic function using the information from the completed square format. 2.5 Graphing Quadratic Functions There are 3 forms of a quadratic function: Let us look at y = x 2 + 8x Pros and cons for each form... Standard form: Factored form: Vertex form:

15 MCF3M2.5.notebook August 17, 2016 Today's goal is to go from standard form and put the equation in vertex form and graph it. Example: y = 2x x + 32 y = 0.5x 2 + 3x 5

16 MCF3M2.6.notebook August 17, 2016 Topic : Graphing by Factoring Goal : I can successfully factor and quadratic function from standard form and use the information to graph the quadratic function. 2.6 Graphing by Factoring Pros and cons... Process...

17 MCF3M2.6.notebook August 17, 2016 Graph the following by factoring: y = 0.5x 2 6x +16 y = 6x 2 7x 3

18 Topic : Real World Applications of Quadratic Functions Goal : I can determine what strategy to use (factoring, completing the square) to determine the maximum/minimum value of a function or determine the x and y intercepts, depending on the requirements of the question at hand. 2.7 Real World Applications of Quadratics What can we find and how do we find it! x intercepts y intercept vertex Initial value Axis of Symmetry Example: A model submarine is used in a swimming pool. Data collected by these crazy students uses the floor of the pool as the reference point and the submarine ascends to the surface and then returns to the floor following the function d(t) = 0.5t t 50, where d(t) is depth in centimeters and t is time in seconds. Answer the following. A) What is the initial depth of the submarine? B) How long will it take the submarine to reach the surface? C) When is the depth 35 cm?

19 Example: Zach hits a baseball and the Fox computer determines the flight path is modelled by h(d) = 0.15d d +0.8, where h(d) represents the height and d represents the horizontal distance, both measured in metres. Answer the following: A) Determine the maximum height of the ball. B) When does the ball reach its maximum height? C) How far from Zach will the ball land? D) At what height did Zach hit the ball from?

20 MCF3M2.8.notebook August 17, 2016 Topic : Modelling Quadratic Functions with Technology Goal : I can determine the maximum/minimum value of a function or determine the x and y intercepts, depending on the requirements of the question at hand by using technology to model the question at hand. 2.8 Modelling Quadratics Modelling data has become easier and easier thanks to technology. For this example I will be using DESMOS. What information do we need? Example: A baseball follows a quadratic path, where h is the height of the baseball, in metres, and d is the horizontal distance, in metres, that the baseball travels after it is batted. Use the given table of data to answer the following: d h A) Make a scatter plot of the data. B) Estimate the coordinates of the vertex C) Use technology to create an algebraic model for the data. D) Determine the exact value of the vertex E) How far did the ball travel in the air?