Quadratic and Exponential Functions Chapter 10 Quadratic and Exponential Functions

10.1: Graphing Quadratic Functions A Quadratic Function is an equation like the ones we worked with in Chapter 9. y = ax2 + bx + c The graph of a quadratic function is called a parabola

Parts of Parabola Vertex: the highest or lowest point of the parabola Vertex @ bottom means Minimum Vertex @ top means Maximum Axis of Symmetry: the imaginary line that “splits” the parabola

Minimum Vertex

Maximum Vertex

State the axis of symmetry and the vertex of each equation. x2 + 2x – 3 = 0 1. Graph in Calculator 2. Press “2nd” “TRACE” 3. Choose “Minimum” or “Maximum” 4. Use arrows to go left & Press “ENTER” 5. Use arrows to go right & Press “ENTER”-2 times 6. Round to the nearest whole #s 7. Answer: (-1, -4)

The Axis of Symmetry is the same as the x-coordinate. For this problem the answer is 1

10.2 & 10.4: Solving Quadratic Equations We are going to talk about 3 ways to solve. First, what does it mean to solve? To find the roots or zeros of the equation. This means where the parabola crosses the x-axis.

One Root

No Roots

Solve by Graphing Example: y2 + 3y – 10 = 0 Type in your calculator in Y1= In Y2= put 0 Press GRAPH Press TRACE and follow the parabola to get as close to one of the roots as possible. Press 2nd, TRACE, 5, ENTER (3x) Do same at other root (if there is one) Answer: -5 & 2

Solve Using Quadratic Formula Solve y2 + 3y – 10 = 0 1-Find A, B, and C A=1, B=3, C=-10 2-Plug in and Solve

Solve Using Equation Solver Press MATH Choose 0 You see 0= (if not, press up) Type equation beside 0= Press ENTER By x= type in -20 ALPHA, ENTER (this gives you one root) By x= type in 20 ALPHA, ENTER (this gives you the other root)

Discriminant The discriminant is the expression under the radical sign…b2 – 4ac So the discriminant of y2 + 3y – 10 = 0 is… (3)2 – 4(1)(-10) or 49 What does this mean? Negative discriminant=no real roots 0 discriminant=1 real root Positive discriminant=2 real roots

Your Turn x2 + 7x + 6 = 0 Use the discriminant to find how many roots it will have. It has 2 What are they? -6 & -1

10.5: Exponential Functions Exponential Functions have a variable for an exponent Ex: y = 3x You will need to be able to do 4 things. Graph State the y-intercept State the value of y Identify Exponential Behavior

Graph Make a table and plug in the points -2, -1, 0, 1, & 2 Example: Graph y = 3x

Plot the Points & Connect

Graph with Calculator Type it in your calculator Ex: y = 3x In y1= type in 3x Press GRAPH Press ZOOM & 0

State the y-intercept 1-Put 0 in for X and Solve OR 2-Graph the Equation Press 2nd, Graph Find x = 0 What is y?

Find the Value of y Type it in Calculator and press Enter Ex: y = 80.8

Identify Exponential Behavior Does the following set of data display exponential behavior? Why or why not? Yes Because the x-values increase the same each time and the y-values have a common factor of 6 **Has to have a common factor. Common difference will not work.

Graph to Identify 1- Press STAT 2- Choose 1: Edit… 3- Put x-values in L1 and y-values in L2 4- Press 2nd Y= 5- Choose #1 and Choose On 6- At Type: Choose the 2nd graph 7- Press GRAPH

10.6: Growth and Decay The Equation for Exponential Growth is…. y = C(1 + r)t y is the final amount C is the initial amount r is the rate of change (a decimal) t is the amount of time

Example In 1971, there were 294,105 females participating in high school sports. Since then, that number has increased an average of 8.5% per year. According to this, how many females participated in high school sports in 2001? C is 294,105 r is .085 t is 30 So… y = 294,105(1 + .085)30 Answer: 3,399,340

Compound Interest Compound interest is a special application of Exponential Growth The Equation is… A is the amount of the investment P is the principal amount r is the annual rate of interest n is the number of times the interest is compounded each year t is the amount of time (# of years)

Example Determine the amount of an investment if $500 is invested at an interest rate of 5.75% compounded monthly for 25 years. P is $500 r is .0575 n is 12 t is 25 So… A = 500(1 + .0575/12) (12)(25) Answer: $2097.86

Exponential Decay The Equation for Exponential Decay is… y = C(1 - r)t y is the final amount C is the initial amount r is the rate of decay (as a decimal) t is the amount of time

Example In 1950, the use of coal by residential and commercial users was 114.6 million tons. Many businesses now use cleaner sources of energy. As a result, the use of coal has decreased by 6.6% per year. Estimate the amount of coal that will be used in 2015. C is 114.6 r is .066 t is 65 So… y = 114.6(1 - .066)65 Answer: 1.35 million tons