Chapter 5Quadratic Equations and Functions

5-1 Warm UpWhat is a quadratic equation? What does the graph look like? Give a real world example of where it is applied.

5-1 Modeling Data with Quadratic FunctionsOBJ: Recognize and use quadratic functions Decide whether to use a linear or a quadratic model

Quadratic FunctionsQuadratic function is a function that can be written in the form f(x)= ax+bx+c, where a 0The graph is a parabolaThe ax is the quadratic termThe bx is the linear termThe c is the constant term

The highest power in a quadratic function is twoA function is linear if the greatest power is one

Tell whether each function is linear or quadraticF(x)= (-x+3)(x-2)Y=(2x+3)(x-4)F(x)=(x+5x)-xY= x(x+3)

Modeling DataLast semester you modeled data that, when looking at the scatter plot, the data seemed to be linearSome data can be modeled better with a quadratic function

Find a quadratic model that fits the weekly sales for the Flubbo Toy Comp

5-1 Wrap UpWhat is a quadratic function?What kinds of situations can a quadratic function model?

5-2 Warm UpList as many things as you can that have the shape of a parabola

5-2 Properties of ParabolasOBJ: Find the min and max value of a quadratic functionGraph a parabola in vertex form

Comparing ParabolasAny object that is tossed or thrown will follow a parabolic path.The highest or lowest point in a parabola is the vertexIt is the vertex that is the maximum or minimum valueIf a is positive the parabola opens up, making the vertex a min pointIf a is negative the parabola opens down, making the vertex a max point

Axis of symmetry divides a parabola into two parts that are mirror images of each otherThe equation of the axis of symmetry is x=(what ever the x coordinate of the vertex is)Two corresponding points are the same distance from the axis of symmetry

Y=ax+bx+c is the general equation of a parabolaIf a>0 the parabola opens upIf a

ExamplesEach tower of the Verrazano Narrows Bridge rises about 650 ft above the center of the roadbed. The length of the main span is 4260 ft. Find the equation of the parabola that could model its main cables. Assume that the vertex of the parabola is at the origin.

Translating ParabolasNot every parabola has its vertex at the originY=a(x-h)+k is the vertex form of a parabolaIt is a translation of y=ax(h,k) are the coordinates of the vertex

Sketch the following graphs

The vertex is ( h, k)Axis of symmetry is x = hIf a > 0 it is a maxIf a < 0 it is a min

Graph, give equation for axis of symmetry and state the vertex.

ExampleSketch the graph of y= -1/2(x-2)+3Sketch the graph of y = 3(x+1)-4

Wrap Up 5-2What does the vertex form of a quadratic function tell you about its graph?

Warm Up 5-3List formulas that you know to use to find answers to problems quickly. (list as many formulas as you can)

5-3 Comparing Vertex and Standard FormsOBJ: Find the vertex of a function written in standard formWrite equations in vertex and standard form

Get into a group of fourTurn to page 211Do part aWhat do you notice about the graphs of each pair of equations?What is true of each pair of equations?Write a formula for the relationship between b and hHow can we modify our formula to show the relationship among a, b, and h. (the last couple of equations)

Standard form of a parabolaWhen a parabola is written as y=ax+bx+c it is standard formThe x coordinate of the vertex can be found by b/(2a)To find the y coordinate by [(b^2-4ac)/4a]

Suppose a toy rocket is launched to its height in meters after t seconds is given by H = -4.9t^2 +20t +1.5. How high is the rocket after one second? How high is the rocket when launched. How high is the rocket after 12 seconds?

ExampleWrite the function y= 2x+10x+7 in vertex formWrite the function y= -x+3x-4 in vertex formWhat is the relationship between the axis of symmetry and the vertex of the parabola?

ExampleAs a graduation gift for a friend, you plan to frame a collage of pictures. You have a 9 ft strip of wood for the frame. What dimensions of the frame give you maximum area of the collage?What is the maximum area for the collage?What is the best name for the geometric shape that gives the maximum area for the frame?Will this shape always give the max area?

Consider this general formula:

A ball is dropped form the top of a 20 meter tall building. Find an equation describing the relation between the height and time. Graph its height h after t seconds. Estimate how much time it takes the ball to fall to the ground. Explain your reasoning

Write y= 3(x-1)+12 in standard formA rancher is constructing a cattle pen by a river. She has a total of 150 ft of fence, and plan to build the pen in the shape of a rectangle. Since the river is very deep, she need only fence three sides of the pen. Find the dimensions of the pens so that it encloses the max area.

Suppose a swimming pool 50 m by 20 m is to be built with a walkway around it. IF the walkway is w meters wide, write the total area of the pool and walkway in standard form

Consider thisIf a quarterback tosses a football to a receiver 40 yards downfield, then the ball reaches a maximum height halfway between the passer and the receiver, it will have a equation

ExampleSuppose a defender is 3 yards in front of the receiver. This means the defender is 37 yards from the quarterback. Will he be able deflect or catch the ball?

ExamplesA model rocket is shot at an angle into the air from the launch pad. The height of the rocket when it has traveled horizontally x feet from the launch pad is given by

A 75-foot tree, 10 feet from the launch pad is in the path of the rocket. Will the rocket clear the top of the tree?Estimate the maximum height the that the rocket will reach.

Wrap Up 5-3Describe the similarities and differences between the vertex form and standard form of quadric equations.

Warm Up 5-4Name mathematical operations that are opposites of each other. For example, addition is the opposite of subtraction. Two inverse functions are opposite of each other in the same way.

5-4 Inverses and Square Root FunctionsOBJ: Find the inverse of a function Use square root functions

Consider the functionsF(x)= 2x-8G(x)= (x+8)/2Find F(6) and G(4)F(x) and G(x) are inverses because one function undoes the otherGraph each function on the same coordinate planeFind three coordinates on f(x)Reverse the coordinates and graphWhat do you notice?

DefinitionThe inverse of a relations is the relation obtained by reversing the order of the coordinates of each ordered pair in the relation

RememberIf the graph of a function contains a point (a,b), then the inverse of a function contains the point (b,a)

Example

Inverse Relation TheoremSuppose f is a relation and g is the inverse of f. Then:A rule for g can be found by switching x and yThe graph of g is the reflection image of the graph of f over y=xThe domain of g is the range of f, and the range of g is the domain of f

RememberThe inverse of a relation is always a relationThe inverse of a function is not always a function

ExamplesConsider the function with equation y= 4x-1. Find an equation for its inverse. Graph the function and its inverse on the same coordinate plane. Is the inverse a function?

Consider the function with domain the set of all real numbers and equation y=x^2What is the equation for the inverse? Graph the function and its inverse on the same coordinate plane. Is the inverse a function? Why or Why not?

ExampleGraph the function and its inverse. The write the equation of the inverse

More ExamplesFind the inverses of these functions

Square Root FunctionsY=x is the square root functionThe graph starts at (0,0)The domain is x0The range is y0

ExampleGraph the function and state the domain and range

5-4 Wrap UpWhat can you tell me about a function and its inverse?

Warm Up 5-5Brain storm all the methods you know for solving this equation. Include less efficient methods. We will vote on which you all prefer.

5-5 Quadratic EquationsOBJ: Solve quadratic equations by factoring, finding square roots, and graphing

Zero Product PropertyFor all real numbers a and b. If ab=0, then a=0 or b=0Example(x+3)(x-7)=0(x+3)=0 or (x-7)=0

We can use the Zero Product Property to solve quadratic equations. That is why we learned how to factor.

Important rule for factor quadraticsMake sure the quadratic is = 0 if not add or subtract until all numbers and variables are on the same side.

Examples:Solve each quadratic by factoring

Solve

Solving Quadratics by square rootsWhen equations are in the form of y = ax you can just divide by a, then take the square root. You will have two answers

Solving Quadratic Equations

ExampleSmoke jumpers are firefighters who parachute into areas near forest fires. Jumpers are in free fall from the time they jump from a plane until they open their parachutes. The function y= -16x+1600 gives jumpers height y in feet after x seconds for a jump from 1600 ft. How long is the free fall if the parachute opens at 1000 ft?

Another way to solveGraphing is another way to solve quadratics. The solutions would be at the x intercepts of the parabola

Solve by graphingThe last example and

5-5 Wrap UpDescribe how the zero product property can be used to solve quadratic equations and which method of solving quadratics do you prefer? Why?

Warm Up 5-6Think about the square root of a negative numberHow do you think you could write the square root of a negative number? What would the square root mean?Please be creative with your responses

5-6 Complex NumbersOBJ: Identify and graph complex numbersAdd, subtract, and multiply complex numbers

Identifying Complex NumbersThe system you use now is called the real number systemReal number system is the rational, irrational, integers, whole, and natural numbersWe will now expand our knowledge to include numbers like -2

Examples

Simplify

The imaginary number i is defined as the principal square root of -1i=-1, and i=-1Other imaginary numbers include -5i, i2, and 2+3iNumbers in the form a+bi form are called Complex Numbers

A+Bi (Complex Numbers)All real numbers are complex numbers where b=05+0i=5An imaginary number is also in the form a+bi, but b00+5i=5i

Simplify each number

Graphing Complex NumbersThey are graphed like regular pointsThe x coordinate is the real number partThe y coordinate is the imaginary number part3+5i would have the point (3,5)

RecallThe absolute value of a real number is its distance from zero on a number lineThe absolute value of a complex number is its distance from the origin on the complex number plane

Formula to find distance

Find |5i||3-4i||-3i||8+6i|

Operations with Complex NumbersTo add or subtract complex numbers, combine the real parts and the imaginary parts separately

Simplify the following expressions(5+7i)+(-2+6i)(8+3i)-(2+4i)

Operations with Complex Numbers(3 +4i) + (7 + 8i) 2i(8 + 5i)(6-5i)+(3 +4i) (5+9i)(2-7i)(1+i)(1-i)

Multiply (5i)(-4i)Multiply (2+3i)(-3+5i)Simplify (3-2i)(-2+4i)(6-5i)(4-3i)(4-9i)(4+9i)

Solving quadratic equations using complex numbersSolve 4k+100=03t+48=05x=-150

Wrap Up 5-6Describe the parts of a complex number and explain what they represent.

Warm Up 5-7(x+7)(x+7)(x+7)x+14x+49How can you determine that this is equivalent to (x+7)

5-7 Completing the SquareOBJ: Solving quadratic equations by completing the squareRewriting quadratic equations in vertex form

Completing the Square

Rewrite the following equations and state the vertex.

Solve the following:x=8x-36x-4x=-85x=6x+8

A local florist is deciding how much money to spend on advertising. The function p(d)=2000 +400d-2d models the profit that the store will makes as a function of the amount of money it spends. How much should the store spend on advertising to maximize its profit?

Real World ExampleSuppose a ball is thrown straight up form a height of 4 feet with an initial velocity of 50 feet per second. What is the maximum height of the ball?

Wrap Up 5-7Explain how to solve quadratic equation by completing the square.

Warm Up 5-8Given 4x+2x+3=0What are the values of a, b, and cFind b, b, 4ac,b-4ac b-4ac2a-b+ b-4ac, -b- b-4ac

5-8 The quadratic formulaOBJ: Solving quadratic equations using the quadratic formulaDetermine types of solutions using the discriminant.

What is a Quadratic EquationA quadratic equation is an equation that can be written in the form

Quadratic FormulaYou can use the quadratic formula to calculate x using a, b, and c. YOU SHOULD MEMORIZE THIS FORMULA

Solve

Solve2x= -6x -72x+4x=-3

Solve 10x^2-13x-3 =0Accounting for a drivers reaction time, the minimal distance in feet it takes for a certain car to stop is approximated by the formula d=.042s^2 + 1.1s+4, where s is the speed in miles per hour. If a car took 200 feet to stop, about how fast was it traveling?``

Discriminant PropertyHas 2 real solutionsHas one real solutionHas 2 complexsolution

Without solving determine how many real solutions the equations have

The amount of power watts generated by a certain electric motor is molded by the equation P(l)=120l-5l where l is the amount of current passing through the motor in amperes (A). How much current should you apply to the motor to produce 600 W of power?

A scoop is a field hockey pass that propels the ball from the ground into the air. Suppose a player makes a scoop that releases the ball with an upward velocity of 34 ft/sec. The function h = -16t+34t models the height h in feet of the ball at time t in seconds. Will the ball ever reach a height of 20ft? If so how many seconds will it take? Will it reach 15 ft? How long will it take?

Lets use the graphing calculatorsx+6x+8=0x+6x+9=0x+6x+10=0

A Way to Sum it Up

Wrap Up 5-8Describe how to use the quadratic formula to solve quadratic equations.