Chapter 10Chapter 10

Quadratic and Exponential Quadratic and Exponential FunctionsFunctions

10.1: Graphing Quadratic 10.1: Graphing Quadratic FunctionsFunctions

A Quadratic Function is an equation A Quadratic Function is an equation like the ones we worked with in like the ones we worked with in Chapter 9.Chapter 9.

y = axy = ax22 + bx + c + bx + c

The graph of a quadratic function is The graph of a quadratic function is called a called a parabolaparabola

Parts of ParabolaParts of Parabola

Vertex: the highest or lowest point Vertex: the highest or lowest point of the parabolaof the parabola– Vertex @ bottom means MinimumVertex @ bottom means Minimum– Vertex @ top means MaximumVertex @ top means Maximum

Axis of Symmetry: the imaginary Axis of Symmetry: the imaginary line that “splits” the parabolaline that “splits” the parabola

Minimum VertexMinimum Vertex

bottom at the isVertex

Symmetry of Axis

Maximum VertexMaximum Vertex

topat theVertex

Symmetry of Axis

State the axis of symmetry and the vertex of each State the axis of symmetry and the vertex of each equation.equation.

xx22 + 2x – 3 = 0 + 2x – 3 = 0

1. Graph in Calculator1. Graph in Calculator

2. Press “22. Press “2ndnd” “TRACE”” “TRACE”

3. Choose “Minimum” or “Maximum”3. Choose “Minimum” or “Maximum”

4. Use arrows to go left & Press “ENTER”4. Use arrows to go left & Press “ENTER”

5. Use arrows to go right & Press “ENTER”-2 times5. Use arrows to go right & Press “ENTER”-2 times

6. Round to the nearest whole #s6. Round to the nearest whole #s

7. Answer: (-1, -4)7. Answer: (-1, -4)

The Axis of Symmetry is the same as The Axis of Symmetry is the same as the x-coordinate. the x-coordinate.

For this problem the answer is 1For this problem the answer is 1

10.2 & 10.4: Solving Quadratic 10.2 & 10.4: Solving Quadratic EquationsEquations

We are going to talk We are going to talk about 3 ways to solve.about 3 ways to solve.

First, what does it First, what does it mean to solve?mean to solve?

To find the roots or To find the roots or zeros of the equation.zeros of the equation.

This means where the This means where the parabola crosses the parabola crosses the x-axis.x-axis.

One RootOne Root

spot. oneat

axis- xesOnly touch

No RootsNo Roots

axis-xnot touch Does

Solve by GraphingSolve by GraphingExample: yExample: y22 + 3y – 10 = 0 + 3y – 10 = 0

Type in your calculator in YType in your calculator in Y11==

In YIn Y22= put 0= put 0

Press GRAPHPress GRAPH

Press TRACE and follow the parabola to Press TRACE and follow the parabola to get as close to one of the roots as get as close to one of the roots as possible.possible.

Press 2Press 2ndnd, TRACE, 5, ENTER (3x), TRACE, 5, ENTER (3x)

Do same at other root (if there is one)Do same at other root (if there is one)

Answer: -5 & 2Answer: -5 & 2

Solve Using Quadratic FormulaSolve Using Quadratic Formula

Solve ySolve y22 + 3y – 10 = 0 + 3y – 10 = 0

1-Find A, B, and C1-Find A, B, and C– A=1, B=3, C=-10A=1, B=3, C=-10

2-Plug in and Solve2-Plug in and Solve

Solve Using Equation SolverSolve Using Equation SolverPress MATHPress MATH

Choose 0Choose 0

You see 0= (if not, press up)You see 0= (if not, press up)

Type equation beside 0=Type equation beside 0=

Press ENTERPress ENTER

By x= type in -20By x= type in -20

ALPHA, ENTER (this gives you one root)ALPHA, ENTER (this gives you one root)

By x= type in 20By x= type in 20

ALPHA, ENTER (this gives you the other ALPHA, ENTER (this gives you the other root)root)

DiscriminantDiscriminantThe discriminant is the expression The discriminant is the expression under the radical sign…bunder the radical sign…b22 – 4ac – 4acSo the discriminant of ySo the discriminant of y22 + 3y – 10 = 0 is…+ 3y – 10 = 0 is…(3)(3)22 – 4(1)(-10) or 49 – 4(1)(-10) or 49What does this mean?What does this mean?Negative discriminant=no real rootsNegative discriminant=no real roots0 discriminant=1 real root0 discriminant=1 real rootPositive discriminant=2 real rootsPositive discriminant=2 real roots

Your TurnYour Turn

xx22 + 7x + 6 = 0 + 7x + 6 = 0

Use the discriminant to find how Use the discriminant to find how many roots it will have.many roots it will have.

It has 2It has 2

What are they?What are they?

-6 & -1-6 & -1

10.5: Exponential Functions10.5: Exponential FunctionsExponential Functions have a Exponential Functions have a variable for an exponentvariable for an exponent– Ex: y = 3Ex: y = 3xx

You will need to be able to do 4 You will need to be able to do 4 things.things.– GraphGraph– State the y-interceptState the y-intercept– State the value of yState the value of y– Identify Exponential BehaviorIdentify Exponential Behavior

GraphGraph

Make a table and plug in the points Make a table and plug in the points -2, -1, 0, 1, & 2-2, -1, 0, 1, & 2

Example:Example:

Graph y = 3Graph y = 3xx

Plot the Points & ConnectPlot the Points & Connect

Graph with CalculatorGraph with CalculatorType it in your calculatorType it in your calculator

Ex: y = 3Ex: y = 3xx

In yIn y11= type in 3= type in 3xx

Press GRAPHPress GRAPH

Press ZOOM & 0Press ZOOM & 0

State the y-interceptState the y-intercept

1-Put 0 in for X and Solve1-Put 0 in for X and Solve

OROR

2-Graph the Equation2-Graph the Equation

Press 2Press 2ndnd, Graph, Graph

Find x = 0Find x = 0

What is y?What is y?

Find the Value of yFind the Value of y

Type it in Calculator and press EnterType it in Calculator and press Enter

Ex: y = 8Ex: y = 80.80.8

y = 5.3y = 5.3

Identify Exponential BehaviorIdentify Exponential BehaviorDoes the following set of data display Does the following set of data display exponential behavior? Why or why exponential behavior? Why or why not?not?

YesYesBecause the x-values increase the Because the x-values increase the same each time and the y-values same each time and the y-values have a common factor of 6have a common factor of 6**Has to have a common factor. **Has to have a common factor. Common difference will not work.Common difference will not work.

Graph to IdentifyGraph to Identify

1- Press STAT1- Press STAT

2- Choose 1: Edit…2- Choose 1: Edit…

3- Put x-values in L3- Put x-values in L11 and y-values in L and y-values in L22

4- Press 24- Press 2ndnd Y= Y=

5- Choose #1 and Choose On5- Choose #1 and Choose On

6- At Type: Choose the 26- At Type: Choose the 2ndnd graph graph

7- Press GRAPH7- Press GRAPH

10.6: Growth and Decay10.6: Growth and Decay

The Equation for Exponential Growth The Equation for Exponential Growth is….is….

y = C(1 + r)y = C(1 + r)tt

– y is the final amounty is the final amount– C is the initial amountC is the initial amount– r is the rate of change (a decimal)r is the rate of change (a decimal)– t is the amount of timet is the amount of time

ExampleExampleIn 1971, there were 294,105 females In 1971, there were 294,105 females participating in high school sports. participating in high school sports. Since then, that number has Since then, that number has increased an average of 8.5% per increased an average of 8.5% per year. According to this, how many year. According to this, how many females participated in high school females participated in high school sports in 2001?sports in 2001?

C is 294,105C is 294,105

r is .085r is .085

t is 30t is 30

So…So…

y = 294,105(1 + .085)y = 294,105(1 + .085)3030

Answer: 3,399,340Answer: 3,399,340

Compound InterestCompound InterestCompound interest is a special Compound interest is a special application of Exponential Growthapplication of Exponential GrowthThe Equation is…The Equation is…

A is the amount of the investmentA is the amount of the investmentP is the principal amountP is the principal amountr is the annual rate of interestr is the annual rate of interestn is the number of times the interest n is the number of times the interest is compounded each yearis compounded each yeart is the amount of time (# of years)t is the amount of time (# of years)

nt

n

rPA

1

ExampleExampleDetermine the amount of an Determine the amount of an investment if $500 is invested at an investment if $500 is invested at an interest rate of 5.75% compounded interest rate of 5.75% compounded monthly for 25 years.monthly for 25 years.P is $500P is $500r is .0575r is .0575n is 12n is 12t is 25t is 25So…So…A = 500(1 + .0575/12) A = 500(1 + .0575/12) (12)(25)(12)(25)

Answer: $2097.86Answer: $2097.86

Exponential DecayExponential Decay

The Equation for Exponential Decay The Equation for Exponential Decay is…is…

y = C(1 - r)y = C(1 - r)tt

y is the final amounty is the final amount

C is the initial amountC is the initial amount

r is the rate of decay (as a decimal)r is the rate of decay (as a decimal)

t is the amount of timet is the amount of time

ExampleExampleIn 1950, the use of coal by residential In 1950, the use of coal by residential and commercial users was 114.6 and commercial users was 114.6 million tons. Many businesses now million tons. Many businesses now use cleaner sources of energy. As a use cleaner sources of energy. As a result, the use of coal has decreased result, the use of coal has decreased by 6.6% per year. Estimate the by 6.6% per year. Estimate the amount of coal that will be used in amount of coal that will be used in 2015.2015.C is 114.6C is 114.6r is .066r is .066t is 65t is 65So…So…y = 114.6(1 - .066)y = 114.6(1 - .066)6565

Answer: 1.35 million tonsAnswer: 1.35 million tons