Chapter 7 Quadratic Functions & Applications I

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1 Chapter 7 Quadratic Functions & Applications I In physics the study of the motion of objects is known as Kinematics. In kinematics there are four key measures to be analyzed. These are work, potential, kinetic, and mechanical energy. Work is when a force is applied to an object to cause its displacement. Potential energy is the energy stored in an object at rest. Kinetic energy is the energy of motion, and mechanical energy is the energy possessed by an object due to its motion or position. Each of these forms of energy can be quantified and physicists can calculate various components for each type of energy. For example, kinetic energy is calculated using the formula 1 mv KE,where m is the mass of the object and v is its velocity. Kinetic m energy is measured in Joules. One Joule is equivalent to 1kg. So when s working with the formula, a final answer is stated in Joules. For example, the kinetic energy of a 65kg rollercoaster car traveling at a velocity of 18.3 m/s will be 1.05 x 10 5 Joules. Below is a table of the formulas related to each of the forms of energy mentioned above. Energy Type Formula Units of Measure Potential (gravitational) PE = mgh Joules Kinetic 1 Joules KE mv Mechanical ME = KE + PE Joules Work w Fd cos Joules In the above equations m is the mass of an object, g is gravity (9.8m/s ), v is velocity, h is height, F is force, d is displacement When you examine these equations you should notice that they each have a different form. Potential and mechanical energy are both 1 st degree

2 equations so we know that their graphs will be straight lines. We also see a trigonometric equation in the equation for work. This equation has a nonlinear, wave-like, graph that you will study in a future math course. This leaves us with the equation for kinetic energy. Its equation is a second degree equation. From integrated math 1, we know that all second degree equations are related to the parent function y x, otherwise known as the quadratic function. We also know that the graphs of such functions are nonlinear parabolas. y x In this chapter we will build on the basic information we learned about quadratic equations in integrated math 1. Some of the basics covered last year are summarized in the following sections. To review basics of quadratics from Integrated Math 1 Hawkes Learning Program To help you review the material covered in Integrated Math 1 please visit the following sections chapter 6: sections 6.6, chapter 10: sections 10.1a, 10.

3 Concepts & Skills The concepts and skills covered in this chapter include Properties of Parabolas Transforming Parabolas Identify the Domain & Range of a Parabola Review Factoring & Solving Quadratic Equations Learn the Completing the Square Method for solving Quadratics Calculate the Vertex of the Parabola. Calculate and Recognize that the Vertex of a Parabola is the minimum or maximum value of the parabola Graph a Quadratic using the Vertex, x &y-intercepts, and axis of symmetry. Apply Quadratic Equations to Real Life examples Interpret the meaning of Minimum & Maximum values for a Parabola as they relate to application problems. 7.1a Function (graph) Translations During the last chapters of integrated math 1 you worked with quadratic and absolute value functions. During that time you explored how the different form of the function and the value of particular variables impacted the shape and placement of the graph. Recall, there at that time you learned about vertical and horizontal translation of the two functions. We will continue to work with these translations as well as a new translation a reflection, during this section of the course. For your review, recall that following represent the translations that were studied last year Translation Vertical shift Upward Vertical shift Down Function Form f ( x) c f ( x) c Horizontal shift to the right f ( x h) Horizontal shift to the left f ( x h) Combined vertical & horizontal Shift f ( x h) c

4 INSTANT RECALL! Write the translated form of the absolute value function for each of the following units up. Right.5 units units up 4. Left 6 units units down units right 7. Left 4 units 7.1b Graphing Non-linear Functions As with any function, the fail-safe method for graphing is to setup a function table. Let s face it, sometimes we do not have a calculator at hand. Keys to Graphing Non-linear Functions 1. As the degree of the function increase, so must the number of points you graph! For functions with a degree of or higher use at least six (6) x-values.. Use both positive and negative values to generate coordinates. 3. These are non-linear functions. That means the graphs are not lines. Do not use straight lines to connect coordinates. 4. Polynomial functions (any function with a degree of or higher) have curves that are both SMOOTH (no corners of kinks) and CONTINUOUS (no gaps). This can be seen in the diagram below. What is the parent function of the graph?

5 Outside of graphing by hand, the best way to graph a function is with a calculator. It is fast, and very accurate more than the human hand. Make sure however to adjust the domain and range of the viewing window to make sure you are seeing the full graph. To review the general shape of the graphs of the parent functions refer back to chapter of the integrated math 1 text. You Try! a) The following is a table of values for the function f(x)=x. Copy and complete the table. X y ) Plot the points for the pairs of values given in the table by constructing a proper coordinate plane. Pay attention to the scale of the domain and range BEFORE you draw your plane! ) Do the points you have plotted suggest a line or a curve? 3) Complete the graph by drawing a smooth curve through the points you have plotted. 4) Does the graph have a line of symmetry? If so, at what point would it cut the curve? Is the point a high or low point on the curve? What is the equation for this line of symmetry? 5) Notice that the curve does not lie below the x-axis. If you were to draw the curve for x=-10 to 10, do you agree that all the points of this curve would still lie above the x-axis? What if x is drawn from -100 to 100? 6) From the graph, find the values of y when x = 1.5 and x = ) From the graph, find the values for x when y = 8 b) The following is a table of values for the function f(x)=1+x-3x. Copy and complete the table. X y ) Plot the points for the pairs of values given in the table by constructing a proper coordinate plane. ) Do the points you have plotted suggest a line or a curve?

6 3) Complete the graph by drawing a smooth curve through the points you have plotted. 4) What is the equation for this line of symmetry? High point or low point? 5) From your graph, find the value of y when x =.8 6) From your graph, find the value of x when y = -8 c) Using your calculator, graph the function f(x)= x +x-8. 1) Does this graph have a high or low point? Using the calculator, can you find the coordinates of this point on the graph? ) State the domain and range of the function in interval notation. 3) Where are the x-intercepts of the graph? 4) What is the y-intercept? 5) What is the axis of symmetry? 6) Can you express where the graph is decreasing? Increasing? (hint: use the domain as your guide) 7.a Forms of a Quadratic Function By definition, any second degree expression is a quadratic function. Below is a table which summarizes the forms in which we typically see quadratics in mathematics and subsequently real life. Form ax ax +bx +c ax +bx ax +c a(x-h) +k (x+p)(x-q) Name Parent Function Standard Form Binomial Binomial Vertex Form Factored or Intercept Form

7 Exploration Use your graphing calculator to explore the following graphs of the above forms of the quadratic equations a) f(x) = ax if a= -, a=-1, a=-0.5, a=0.5, a=1, and a=. What impact does a have on the graph? b) f(x) = (x-h) with h=-4, h=-, h=, and h=4. How does the changing value of h impact the graph? c) f(x)=x +k with k=-4, k=-, k=, and k=4. How does the value of k impact the graph? d) f(x)=(x-3)(x+) What do you observe with this graph? 7.b Reflections in the Coordinate Plane Any time a function can be mirrored (reflected) around either the x-axis or y-axis, a reflection occurs. As you saw in example a of the exploration above, when a=- the graph reflected down versus when a =, the graph opened up. X-axis reflection for the parent quadratic function can be seen in the graph below In general reflections in the coordinate axes of the graph of y=f(x) are represented as follows 1. Reflections in the x-axis: y=-f(x). Reflection in the y-axis y = f(-x)

8 Graphically we can see these represented using the function Parent y x x-axis reflection y-axis reflection

9 Generally, when dealing with quadratic functions we only concern ourselves with x-axis reflection because of the symmetric nature of the parabola. It is hard to see y-axis symmetry when the two halves of the graph are symmetric to one-another! The same can be said for the absolute value function. All other functions in the function family however, can easily show y-axis reflection. 7.3a Methods for Solving Quadratics Because there are different forms of a quadratic, there are also different methods for solving them. Remember, the solution to a quadratic may be called, solution, answers, roots, x-intercepts, or zeros of the problem but what is most important is that you need to remember that there are always two answers for every quadratic equation! Methods for Solving Quadratics 1. Square Root Method. Solve by Factoring 3. Quadratic Formula 4. Completing the Square IMPORTANT TO REMEMBER: The solutions of a quadratic equation are the x-intercepts of the quadratics graph. The graph of a quadratic function is called a Parabola. Your primary resource for quadratic functions in this chapter is your Hawkes Learning Program. Hawkes Learning Program In this chapter we will cover the following sections chapter 6: section 6.7, chapter 10: sections 10.1b, 10.3, 10.5

10 7.3b Applications of Quadratic Equations As was stated at the beginning of this chapter, quadratics evolved out of the need to model real life situations with mathematics -in this case, situations that are non-linear. The following are some problems that apply quadratic reasoning to such situations. Referring back to the problems related to physics, let s model the kinetic energy of a car that has a mass of 75kg as it travels with varying velocity. Recall, 1 mv KE is a second degree (therefore, quadratic) equation. Logically we know that the car cannot have a negative velocity so one of the things you need to recognize from the start is that you are only going to be dealing with the first quadrant of the graph. Second, since velocity of the car is not constant we need to recognize that the graph will open up because as velocity increases so does the kinetic energy and at different points in the cars journey its level of kinetic energy will depend on its velocity. The graph of this can be seen below. In this model, x represents velocity. The corresponding table of values for this model is as follows for velocity of 0-80 m/s

11 To use the kinetic energy equation in this way provides us with good information about the vehicle. We know the graph has a minimum value at a velocity of zero (that s where it has its most Potential Energy) and increases as the velocity increases. What if we turn this problem around however and ask the question, If a car has kinetic energy of 30,000 J and a mass of 75kg, what is its velocity? We can solve this quadratic for the value! 30,000 (75/ ) v 30, v 30,000 / v v v 8.86m / s v For this simple physic example we have modeled kinetic energy both visually and then algebraically You Try! a) Using your calculator, generate the kinetic energy graph for a rolling marble if it has a mass of 10g. b) In the example above, what is the velocity of the marble if it has the following kinetic energy 115J, 4500J, and 615J

12 It is important to note here that conceptually, the above problems only reflect the relationship of kinetic energy to mass and velocity on a continued application of velocity. We know in fact that during a cars journey it will slow down and stop and speed-up at different points which would directly change the shape of the graph for the actual kinetic energy incurred by the car during the trip. The model above is in its simplest form showing the relationship that as velocity increases on a continued basis so does kinetic energy. Here are some other examples showing where quadratics can apply in life. Work the problem for the required information. Cost Analysis 1. A power drink manufacturer has a daily production cost that can be Revenue modeled with the following C( x) 60, x 0.065x where C is the total cost (in dollars) to produce the power drink and x is the number of units produced each day. What is the number of units that need to be produced each day that will yield the manufacturer a minimum cost?. Microsoft has estimated that the total revenue R (in thousands of dollars) earned from selling hand-held video games is given by p R 5p 100 p where p is the price per unit in dollars. Remember revenue is just the amount of money they take in from the sale of the units. It is not the profit for the hand-held unit. Profit = Revenue Cost! a) Find the revenue when price per unit = $0, $5, and $35 b) Find the unit price that will give Microsoft the maximum revenue c) What is the maximum revenue? d) Explain what is happening to Microsoft revenue as the unit price for the video game goes from $7 to $3.

13 Height of a Ball 3. The height, y, in feet, of a punted football is approximated by the 9 3 following y 16 x x where x is the horizontal distance (in 05 5 feet) from where the football is punted. a) Graph the path of the punted football b) How high is the football when it is punted? Why? c) What is the highest point the punted football reaches? d) How far does the football travel before it lands on the ground? Assume no one is there to receive the punt. Consumption Analysis 4. According to the US Department of Agriculture, the annual per capita consumption C of cigarettes by American adults between 1960 and 003 can be modeled by C t t 1.3t where t is the year, with t=0 corresponding to a) Use your calculator to graph this model. What is an appropriate viewing domain and range for this model? b) In what year did the US experience a maximum in the consumption of cigarettes? How many where consumed? c) In 1966, federal law required tobacco companies to post the infamous Warning: The Surgeon General has determined that Cigarette Smoking May be Hazardous to your Health. on every package of cigarettes. Based on the date and the model above, do you think the warning had any impact on the consumption of cigarettes in the US? Why or Why not? d) What was the consumption of cigarettes in 000? e) In the year 000, the US population (age 18 and older) was 09,117,000. Of those, 48,306,000 were smokers. What was the average annual consumption of cigarettes per smoker in 000? What was the average daily consumption per smoker in 000?

14 Web & Video Links (to review Quadratic Basics) uadratic-equations-in-standard-form 3. a-algebra-i--quadratic-roots 4. mple-quadratic-equation 5. pplying-the-quadratic-formula 6. pplication-problem-with-quadratic-formula