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Memorize the following!. 2. 2. Three forms of a quadratic equation. Standard: y = ax + bx + c Vertex: y = a(x – h) + k Intercept: y = a(x – p)(x – q) X coordinate of vertex: -b 2a. VERTEX FORM. THE PARENT FUNCTION of the quadratic family. Characteristics: - PowerPoint PPT Presentation

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Memorize the following!

Memorize the following!Three forms of a quadratic equation.Standard: y = ax + bx + cVertex: y = a(x h) + kIntercept: y = a(x p)(x q)X coordinate of vertex:-b2a

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VERTEX FORMIf |a | > 1, parabola narrowsIf |a | < 1, parabola widensIf a < 0, parabola opens downTHE PARENT FUNCTIONof the quadratic familyCharacteristics:shape is parabolic and symmetricaxis of symmetry is verticalhas a maximum or a minimum valueVertex is (h, k)

Graphing from vertex form

Plot the vertex ( h, k )Determine if the parabola opens up or downDraw the axis of symmetry as a dashed vertical lineChoose an x value on one side of the vertex, Substitute this value in, solve for y and plot the point.Plot the reflection image of the point.Repeat steps 5 and 6 for an additional pointSketch the parabola.STANDARD FORMEffect of a is the same. If a is positive the parabola opens up, if negative, down.Axis of symmetry is the vertical line x = -b2aTo complete graph, choose an x value near the vertex, calculate its corresponding y value, plot the point, then reflect it over the line of symmetry.The y-intercept is (0, c). Plot this point, then reflect it over the line of symmetry.- or-Graphing from standard formPlot the vertex, the x coordinate is found using Find the y coordinate by plugging the x value into the original function and solving for y. -Determine if the parabola opens up or down-Sketch the axis of symmetry using a dashed line.-Choose a value for x, plug it into the equation and solve for y. Plot this point-Plot the reflection image of this point-Choose another x value and repeat the last 2 steps-Sketch the parabola

INTERCEPT FORMEffect of a is the same.

The x-intercepts are p and q.The axis of symmetry goes through the x-axis at the midpoint of the segment defined by p and q.The path of a baseball after it is hit is modeled by the function

h is the height of the baseball in feet and d is the distance in feet the baseball is from home plate. What is the maximum height reached by the baseball? How far is the baseball from home plate when it reaches its maximum height?

QUADRATIC FUNCTIONSThe Golden Gate Bridge in San Francisco has two towers that rise 500 feet above the road and are connected by suspension cable as shown. Each cable forms a parabola with the equation + 8

What is the distance between the two towers if the cable supports are 500 feet above the road?

What is the height above the road of a cable at its lowest point?Although a football field appears to be flat, its surface is actually shaped like a parabola so that rain runs off to either side. The cross section of a synthetic field can be modeled by the equation

+ 1.5

Where x and y are measured in feet.

What is the fields width?What is the maximum height of the fields surface?A kernel of popcorn contains water that expands when the kernel is heated, causing it to pop. The equations below give the popping volume, y (in cubic cm/gram) of popcorn with the moisture content as x (as percent of the popcorns weight).

For hot-air popping, what moisture content maximizes popping volume?What is the maximum volume?For hot-oil popping, what moisture content maximizes popping volume?What is the maximum volume?Writing Quadratic Functions in Standard Form

Multiply using FOILcombine like termsdistribute the -1

You have 400 feet of fencing with which you need to enclose a rectangular field along a straight river bank. (There is no need to fence along the river.)

Let x represent the width of the field and write an equation to represent the area of the field.

Find the dimensions of the field with the maximum area.

Warm-up

Anne and Linda went out to lunch. When the bill came Anne realized she had only 8 dollars in her wallet. Linda told her not to worry because she had enough money to cover three fifths of the bill. With Annes eight dollars and Lindas money they could pay. How much was the lunch bill? How much did Linda contribute?

Factor the following expressions.

x2 x - 6 x2 -3x

Factoring

Factor out the GCF

Look for special patterns such as the difference of two squares, perfect square trinomialFOIL to check your answers!Undo FOIL!

Factor out any GCF first!

Difference of Squares

Perfect Square Trinomials

Solving quadratic equations by factoring

- Set the equation equal to zero .(Youre making y zero because youre finding the x intercepts.)

-Factor the equation completely.

-Set each factored portion equal to zero and solve for the variable.

Zero Product PropertyIf A and B are real numbers or algebraic expressions, If AB = 0 then A = 0 or B = 0

In Quadratic equations the x intercepts are also called -solutions-roots-zeros of the function

When solving, instructions might say:-Solve the equation.-Find the functions zeros.-Write the quadratic function in intercept form and find the functions zeros. You are making a coffee table with a glass top surrounded by a cherry border. The glass is 3 ft. by 3 ft.. You want the cherry borer to be a uniform width. You have 7 square feet of cherry. What should the width of the border be?

The dimensions of a rectangular garden were 5 meters by 12 meters. Each dimension was increased by the same amount. The garden then had an area of 120 square meters. Find the dimensions of the new garden.

A graphic artist is designing a poster that consists of a rectangular print with a uniform border. The print is to be twice as tall as it is wide. The border is to be 3 inches wide. If the area of the poster is to be 680 square inches find the dimensions of the print.

You have made a rectangular stained glass window that is 2 feet by 4 feet. You have 7 square feet of clear glass to create a border of uniform width around the window. What should the width of the border be?

Warm-up Solve the following equations by factoring.

16x2 = 8x -1

y = 25x2 -4

15 = 3x2 -12x

5x2 = 30x

Factor the following expressions.

X2 + 2x + 1

X2 4x + 4

4x2 +12x + 9

Solve the following equations.

X2 -10x + 25 = 0x2 16x = -64

warm-upIf x2 = 16 what is the value of x?

Product Property Quotient Property

A square-root expression is simplified if:No radicand has a perfect square factor other than 1There is no radicand in the denominatorSimplify

1245 6 . 2698

Rationalizing the denominator-the process of eliminating a radical from the denominator.

Solving quadratic equations by taking square roots.Rewrite the equation so the squared term is isolated on one side of the equal sign.Take the square root of each side.Remember there will be a positive and a negative root.Ex.

You can use square roots to solve some quadratic equations.

-you can use this method when there is no linear term

Ex. 2x2 + 1 = 171/3(x + 5)2 = 7

Remember + and - !!!!!!

The height h(in feet) of the object t seconds after it is dropped can be modeled by the function:

h = -16t2 + ho

(ho is the objects initial height)A stunt man working on the set of a movie is to fall out of a window 100 feet above the ground. For the stunt mans safety, an air cushion 26 feet wide by 30 feet long by 9 feet high is positioned on the ground below the window.For how many seconds will the stunt man fall before he reaches the cushion?A movie camera operating at a speed of 24 frames per second records the stunt mans fall. How many frames of film show the stunt man falling?

From 1990 to 1993 the number of truck registrations (in millions) in the United States can be approximated by the model R = .29t2 + 45 where t is the number of years since 1990. During which year were approximately 46.16 million trucks registered?

In 1992 the average income I(in dollars) for a doctor aged x years could be modeled by:I = -425x2 + 42,500x 761,000For what ages did the average income for a doctor exceed $250,000?

The aspect ratio of a TV screen is the ratio of the screens width to its height. For most TVs the aspect ratio is 4:3. What are the width and height of the screen for a 27 inch TV? Use the Pythagorean Theorem to solve.

QUADRATICSTHERES AN APP FOR THAT!The tallest building in the United States is in Chicago, Illinois. It is 1450 tall.a) How long would it take a penny to drop from the top of this building?b) How fast would the penny be travelling when it hits the ground if the speed is given by s = 32t where t is the number of seconds since the penny was dropped?WHAT IF THE REAL NUMBER SYSTEM ISNT SUFFICIENT?WE INVENT SOME MORE NUMBERS!IMAGINARY NUMBERSii is the imaginary unit it is equal to the square root of -1

i = = 6i=i

A complex number in standard form - a + bi

Ex. 2 + 3i

A pure imaginary number 2i

Adding and Subtracting Complex NumbersThe imaginary number i acts like a variable but when you get substitute -1

Multiplying Complex Numbers

Substitute -1 for i squaredComplex conjugates example (5 + 3i) and (5 3i)

The product of complex conjugates is always a real number.

Dividing Complex Numbers

QUIZ 5.4YET ANOTHER WAY TO SOLVEQUADRATIC EQUATIONSCompleting the square-Use inverse operations to move the constant term to the other side of the equal sign-Divide the coefficient of the linear term in half and square it, add it to both sides of the equal sign-Factor the now perfect square trinomial-Solve by taking the square root of both sidesComplete the expression to make it a square:Rewrite each expression as the square of a binomia