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1 Chapter 6 Quadratic Functions Determine the characteristics of quadratic functions Sketch Quadratics Solve problems modelled b Quadratics 6.1Quadratic Functions A quadratic function is of the form where a, b and c are. Also Eample: The graph of a quadratic function has a ver distinct shape called a 1 P a g e

2 Uses of Quadratics and Parabolas Projectile Motion Anthing that is thrown that has some horizontal motion. Jumping on a bike, skis, snowboards, skidoos, etc. Running off a diving board. Arrows, or bullets that are shot. Throwing footballs or baseballs, or kicking a soccer ball. When ou take. The concave on a. Recall 106 science P a g e

3 Parts of a Parabola Place where parabola crosses. Place where parabola crosses. of a parabola The that passes through the 3 P a g e

4 Quadratic functions and Parabolas An relation that can be represented b a parabola can be modelled b a. A quadratic function must have a term has its highest. The general equation for a quadratic is: Note: In tet this is called as well! is the quadratic (or squared term) is the linear term is the constant term In a b c is called the. 4 P a g e

5 Eample: Which of the following is a quadratic function? If it is not state wh. If it is state which direction it opens ( 1)( 3) Tet Page 34 #1,, 4, 5, 6 5 P a g e

6 Putting Disguised Quadratics into General Form So, wh do we need to be able to put quadratics into the general form? So we can determine the values of a, b, and c. These values tell us about the of the parabola. For eample, what does a tell us?. What else do these values tell us? To determine this lets complete the following activit. 6 P a g e

7 Effects of a on the parabola First investigate the effect of changing the value of a 1. What happens to the direction of the opening of the quadratic if a < 0 and a > 0?. A) If the quadratic opens upward, is the verte a maimum or minimum point? B) What if the quadratic opens downward? 3. Is the shape of the parabola effected b the parameter a? In other words are some graphs?. 4. What happens to the -intercepts as the value of a is changed? 5. What is the impact on the graph if a = 0? 7 P a g e

8 Effects of b on the parabola 6. What is the effect of parameter b in = a + b + c? 7. Is the parabola s line of smmetr changing? Effects of c on the parabola 8. What is the effect of parameter c in =a +b + c?. 9. How can ou identif the -intercept from the equation in general form? 10. Is the line of smmetr affected b the parameter c? 8 P a g e

9 Sketching Parabolas For each of the following complete the table of values and sketch the graph on the grids provided P a g e

10 Verte: Ais of Smmetr: Domain: Range: Verte: Ais of Smmetr: Domain: Range: P a g e -

11 Another wa of finding verte Consider the last problem: What are the values of : a = b = c = Calculate : How does this value relate to the verte? The -value of the verte is found b putting the value into the function. Eamples: 1. Find the verte of the following: A) (#3 in the sketches) B) P a g e

12 . Find the maimum or minimum -value for the following. A) B) Find the Range and the ais of smmetr for the following. A) B) P a g e

13 4. The height of a soccer ball kicked into the air is given b the function: 5 0 Determine the maimum height of the ball and the time when this occurred. 5. The path of the snowboarded below is given b the equation: A) Determine the boarders ma height. 5.B) Determine the domain and range the boarder s jump. 13 P a g e

14 Finding verte from the average of -values The -coordinate of the verte ( ) can also be found b taking average of the -values for an two points on the parabola that has the. Table of values Given points A) (3, 5) and (7,5) B) (1, ), (0, 5), (3, ) 14 P a g e

15 6.3 Drawing more accurate parabolas A quick sketch of a parabola can be made if ou know the and the the parabola. Eample: Sketch the graph of = +-3 The sketch the graph of = +-3 can be made more if we know more. The etra points that are tpicall used are the places where the function. To find the we set and solve for Remember that for the function =a +b + c the is 15 P a g e

16 Lets sketch the parabola again using the Is there another point that we can plot based from the -intercept?. To find the we set and solve for When finding the of a function ou are actuall finding the for the function. ie. You are finding that make the function. To solve for we will need to the quadratic. You will be required to use developed in Mathematics 101 to determine the. 16 P a g e

17 SO, how do we factor Product and Sum Method You must find two numbers such that the product of the two numbers equals, the. The sum of the two numbers must equal, the. So now what? To find the zeros from the of the function we use the The states that: if the of two real numbers is, then or of the numbers must be. Find the zeros of Lets sketch the parabola again using the as well 17 P a g e

18 Eamples For each of the following find A) the verte B) the -intercept C) the -intercepts Then use this information to draw a graph of the function P a g e

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21 101 Factoring Trinomials of the form a +b+c Warm UP Factor: 5 6 When the leading coefficient of a trinomial is not 1 the. When factoring the trinomial a +b+c, we will find two numbers that multipl to give the product and will have a sum of Factor P a g e

22 Find the - intercepts of Note** Before starting to factor a trinomial, alwas check to see if ou can remove a. III Factor 8h +0h+18 P a g e

23 IV Solve 6k -11k-35 = 0 V Factor 4g +11g+6 3 P a g e

24 VI Solve 3s -13s-10 = 0 VII Factor P a g e

25 Practice for Assignment 6. Sketch the graph of the following b finding the intercepts and verte of parabola. A) 6 7 B) P a g e

26 C) ( )( 4) D) 1 ( )( 4) 3 6 P a g e

27 Note: The last two equations in our notes and on Assignment 6. were epressed in a special form called the of the quadratic. Factored form of a quadratic Where r and s are the of the function are of the parabola. is the. Eample Sketch the graph of a parabola that passes through the points (-3,0) and (4, 0) Make more parabolas that are different from the first one, but still have How man parabolas do ou think are possible? Eplain 7 P a g e

28 Note: The goal is for ou to recognize that a of parabolas are possible when the. For eample the factored form represents the famil of parabolas that we have drawn through the points (-3,0) and (4, 0) When provided with an, however, ou can narrow down the formula for the quadratic equation. In order to determine the multiplier in the factored form = a( +3)( - 4), ou need to choose point on the parabola and use substitution. Select one of the parabolas that ou have drawn and determine the leading coefficient a. Write the factored form for our parabola. Epand the factored form to epress the quadratic in standard form. 8 P a g e

29 Eample: Determine the quadratic function, in standard form, with factors ( + 3) and ( - 5) and a -intercept of -5. Sketch the parabola 9 P a g e

30 A) Find the equations of the parabola graphed below: B) 30 P a g e

31 Each form of a quadratic has its own characteristics, and its own benefits. If the quadratic is written in general or standard form,, ou can determine: the and the direction of the of the parabola directl from the equation. the -coordinate of the b using If the equation is written in factored form, ou can determine: the of the graph and the direction of the of the parabola. the -coordinate of the verte b taking the Both of these forms required to find the. There is one more form of the quadratic which enables ou to determine the verte called the, 31 P a g e

32 Verte Form of a Quadratic Verte Form : a is the. If a is positive the parabola opens If a is negative the parabola opens The point is the of the parabola. Eample: What is the verte of the following? A) ( 1) 3 B) ( ) 3 5 Sketch: 1 1 What is the verte? Which wa is the graph opened? 3 P a g e

33 Sketch: 1 3 What is the verte? Which wa is the graph opened? What is the equation of the ais of smmetr? What is the ma/min -value? What is the domain and range? 33 P a g e

34 Finding Equations of Parabolas What is the verte? Write the verte form for this parabola How do we find a? What is the equation of the ais of smmetr? What is the ma/min -value? What is the domain/range? What is the equation s general form? 34 P a g e

35 Quiz Find the 3 different equation forms for this graph. 35 P a g e

36 Word Problems 1. A ball is thrown from an initial height of 1 m and follows a parabolic path. After seconds, the ball reaches a maimum height of 1 m. Algebraicall determine the quadratic function that models the path followed b the ball, and use it to determine the approimate height of the ball at 3 seconds. A)How is the shape of the graph connected to the situation? B) What do the coordinates of the verte represent? C) What do the and -intercepts represent? D) Wh isn t the domain all real numbers in this situation? 36 P a g e

37 . The goalkeeper kicked the soccer ball from the ground. It reached a maimum height of 4. m after. seconds. The ball was in the air for 4.4 s. A) Determine the quadratic function that models the height of the ball above the ground. B) How high is the ball after 4 s? C) What is the domain and range of this function? 37 P a g e

38 3. A quarterback throws the ball from an initial height of 6 feet. It is caught b the receiver 50 feet awa, at a height of 6 feet. The ball reaches a maimum height of 0 feet during its flight. Determine the quadratic function which models this situation and state the domain and range. Distinguishing between a function and a function that represents a real world situation 4.A) State the domain and range for the function f() = P a g e

39 4B) Suppose the function h(t) = -0.15t + 6t represents the height of a ball, in metres, above the ground as a function of time, in seconds. State the domain and range. 5. The path of a model rocket can be described b the quadratic function = - - 1, where represents the height of the rocket, in metres, at time seconds after takeoff. A) Identif the maimum height reached b the rocket and determine the time at which the rocket reached its maimum height. B) State the domain and range. 39 P a g e

40 Other Word Problems 6. You have 600 meters of fencing and a large field. You want to make a rectangular enclosure split into two equal lots. What dimensions would ield an enclosure with the largest area? 7. John has ordered 40 feet of fencing to build a dog enclosure. One side of his house will be one of the sides of the enclosure. Determine the dimensions that will give his dogs the maimum area to run around in. 40 P a g e

41 Will it Fit? 8. A two lane highwa runs through a tunnel that is framed b a parabolic arch, which is 0 m wide. The roof of the tunnel, measured 4 m from the right base is 4 m above the ground. Can a truck that is 4 m wide and 6. high pass through the tunnel? 41 P a g e

### Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form

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