Unit 4: Part 1 Graphing Quadratic Functions

Transcription

1 Name: Block: Unit : Part 1 Graphing Quadratic Functions Da 1 Graphing in Verte Form & Intro to Quadratic Regression Da Graphing in Intercept Form Da 3 Da Da 5 Da Graphing in Standard Form Review Graphing & Analzing Quadratics Review: Graphing & Analzing Quadratic Functions Quiz Graphing & Analzing Quadratic Functions

2 Tentative Schedule of Upcoming Classes Da 1 Da Da 3 Da Da 5 Da B Mon 10/19 Verte Form A Tues 10/0 B Wed 10/1 Intercept Form A Thurs 10/ B Fri 10/3 Standard Form A Mon 10/ B Tues 10/7 Skills Review #3 Due & Skills Check A Wed 10/ OTTW Quadratics Project workda B Thurs 10/9 Review for Quiz: Graphing & Analzing A Fri 10/30 Quadratic Functions* B Wed 11/ Quiz: Graphing & Analzing A Thurs 11/5 Quadratic Functions *OTTW Quadratics Project is due on Quiz Review da! Absent? See Ms. Huelsman AS SOON AS POSSIBLE to get work and an help ou need. Notes are alwas posted online on the calendar. (If links are not cooperative, tr changing to list mode) You ma also Ms. Huelsman at Kelse.huelsman@lcps.org with an questions! Need Help? Ms. Huelsman and Mu Alpha Theta are available to help Monda, Tuesda, Thursda, and Frida mornings in L50 starting at :10. Ms. Huelsman is in L0 on Wednesda mornings. Need to make up a test/quiz? Math Make Up Room schedule is posted around the math hallwa & in Ms. Huelsman s classroom

3 Da 1 Notes: Introducing the QUADRATIC function! In Verte Form: = a ( h) + k Welcome to our second function famil the QUADRATIC FUNCTION f() = (the parent function) What are some characteristics that ou notice? 3 1 What is different between this function and the absolute value function? Wh? (Look at the table!) ALL quadratic functions have ke features that we care about: 1. Verte a point. Ais of smmetr Equation of a line 3. Min or ma A point. X-intercepts A point 5. Y-intercepts A point. Increasing and Decreasing Intervals An interval 7. End behavior As, f() As, f(). Domain An interval 9. Range An interval

4 Verte form of a quadratic function: f () = a ( h) + k Eplain the difference between an absolute value function and a quadratic function when ou are looking at the equations. Given the quadratic function = a ( - h) + k ** a is NOT slope like it was in abs val! ** If a > 0, does the graph open up or down? If a < 0, does the graph open up or down? If a > 1, does the graph have a vertical stretch or vertical shrink? If 0 < a < 1, does the graph have a vertical stretch or vertical shrink? What is the verte? What does the parameter k control? What does the parameter h control? Write an equation of a quadratic function with a verte at (-, 5) that opens down and has vertical shrink. Complete the table below without our calculator: 1 3 Function Direction Dilation Verte Domain Range = - ( + ) Stretch + 3 Up Shrink Down Standard 1 = ( ) Stretch + 5 Up Shrink Down Standard = ( +1) = 1 Up Down Up Down Stretch Shrink Standard Stretch Shrink Standard How can ou tell if a verte is a ma or min without graphing? How did we find stretch or shrink for absolute value? Wh can't we use a as slope for quadratic functions?

5 How do ou graph without a calculator? 1. Find our verte.. Place our verte in the middle of the table of values. 3. Fill in the -values that surround the verte ( below, above).. Plug in -values to find the -values for our remaining points. 1. = - X Y. = ½ X Y Is this stretched or shrunk? Is this stretched or shrunk? = ( - 5) - 7. = - ( + ) + X Y X Y 10 10

6 Let s review all the characteristics of our graphs and how to find an inverse of a given function. 7. Graph: = ( + ) +. Is the inverse of #7 a function? using our calculator Eplain: Graph the inverse of # Domain: Zeroes: Range: Y-intercept: Increasing: Decreasing: As, f ( ) As, f ( ) Finding the equation of a graphed function: Step 1: What is the general form of the parent graph? Step : Put (h, k) the verte into our equation. 5 Step 3: Substitute another point into the equation for & Solve for a. 5 Step : Write the final equation with a.

7 We can also do this on our calculator using the regression feature: 1. STAT EDIT. Enter values in L1 and values in L 3. STAT CALC. OPTION : QUADRATIC REGRESSION a = b = c = 5 Final equation: 5 Now ou tr Given the graph, write the quadratic equation in verte form for each of the following without using a calculator: The verte is. The verte is. Equation: Equation:

8 Da Notes: Quadratic Functions in Intercept Form = a( p)( q) ALWAYS, SOMETIMES, NEVER? Tell whether each statement is alwas, sometimes, or never true. Let s review quadratic functions: 1. The graph of a quadratic function is a V shape.. The range of a quadratic function is the set of all real numbers. 3. The graph of a quadratic function contains the point (0, 0).. The verte of a parabola occurs at the minimum value of the function. 5. A quadratic function has two real solutions.. If a quadratic function s verte is on the -ais, then it has eactl one solution. 7. The inverse of a quadratic function is also a function. Is this reall a quadratic? Graph these with our calculator and see. 1. = ( + 3)( - 1) (p =, q = ) Verif algebraicall b multipling: How do we know this is a quadratic now? 10. = ( - 1)( - ) (p =, q = ) Verif algebraicall: What patterns do ou notice in this equation tpe? 10

9 What is NICE about INTERCEPT form? What was NICE about VERTEX form? How will we find the verte and ais of smmetr given this form? Graphing in intercept form: 1. Find & graph the X-intercepts.. Find & graph the verte. 3. Connect the points to make the parabola. 3. = ( )( ). f() = -½( + )( ) -intercepts:, Verte: -intercepts:, Verte: Domain: Range: -intercept: Increasing Interval: Decreasing Interval: Ma or min? As, f ( ) As, f ( ) 10 Domain: Range: -intercept: Increasing Interval: Decreasing Interval: Ma or min? As, f ( ) As, f ( ) 10

10 Sketch the graph of a quadratic function that has at least one solution of =0. 10 How would ou graph the following function? = ( 3) = -3( ). = ( )( + ) -intercepts:, Verte: -intercepts:, Verte: Domain: Range: -intercept: Increasing Interval: Decreasing Interval: Ma or min? As, f ( ) As, f ( ) 10 Domain: Range: -intercept: Increasing Interval: Decreasing Interval: Ma or min? As, f ( ) As, f ( ) What happens if we give # vertical stretch or shrink? New quadratic in verte form Did changing "a" affect the intercepts?

11 More Practice with Regression Find the linear and quadratic curve of best fit for the following data, rounding coefficients to 3 decimal places. Which regression is BEST? How do we know which one is better? X 10 1 Y How do I know if I found the BEST Curve? Turn on Diagnostics with CATALOG->DIAGNOSTICS ON **Look at the R value** The R value tells ou how good of a fit the data is. (1 means perfect fit.) 1. Enter our values: STAT choose 1: EDIT Tpe values into L1 Tpe values into L. Choose our function tpe: STAT CALC : LinReg and 5: QuadReg Your Calculator should read LinReg (a+b) 3. Enter our lists: L1 and L are chosen b default. (Keep hitting "enter" until ou hit "Calculate"). Write our equations below. Round to 3 decimals. Linear equation: R : Quadratic equation: R : How can we get an idea about whether data is more linear or more quadratic? Logicall: How is the data increasing? Visuall: To see our scatter plot: STAT PLOT 1 turn on, ZOOM 9) ZOOM puts it back in standard mode What is true about a LINEAR relationship? What is true about a QUADRATIC relationship?

12 Da 3 Notes: Quadratic Functions in Standard Form = a + b + c What was AWESOME about the VERTEX form of a quadratic? What was AWESOME about the INTERCEPT form of a quadratic? Do ou see an helpful information in the STANDARD FORM of a quadratic? What will be a little bit more challenging? Standard Form: = a + b + c Summar of STANDARD FORM Verte has -coordinate. (How will ou know if this is a min or a ma?) Find the -coordinate of the verte b plugging the value of the verte into the equation. b b The verte is the ordered pair, f ( ). a a The ais of smmetr is = What happens at the -intercept? Then the -intercept is. So, the point (0, ) is on the parabola. If a is positive,. If a is negative,. The solutions to the quadratic equation are the -intercepts. What can we do to find these when we are given standard form?

13 Steps for Graphing: = = + 1 Step 1: Find the verte: (, ) Formula: = b a Plug into the function to find. Step : Complete a table of values X Y X Y Place Verte in middle. Fill in -values. Pick -values on one side of verte to plug in. Use smmetr to fill in the remaining values. Step 3: Graph our points and connect intercept: (, ) -intercept: (, ) Ais of Smmetr

14 Let s review all the characteristics of our graphs 3. = = verte: verte: X Y X Y Domain: Range: Increasing: Decreasing: Zeroes: Y-intercept: As, f ( ) As, f ( ) 10 Domain: Range: Increasing: Decreasing: Zeroes: Y-intercept: As, f ( ) As, f ( )

15 Practice with Regression The following table shows the results of an eperiment testing the maimum weight (in tons) supported b ice that is inches thick. (thickness of ice in inches) a) Does this data seem more linear or more quadratic? Wh? (weight in tons) b) Find the curve of best fit. Round numbers to 3 decimal places. c) How much weight can be supported b ice that is " thick? Hint: is weight or? Is thickness of ice or? d) How much weight can be supported b ice that is 3 feet thick? Hint: is weight or? Is thickness of ice or? e) Estimate the thickness of ice required to support a weight of 30 tons. Hint: is weight or? Is thickness of ice or? What are we looking for?

16 Da Notes: Regression Finding the Line or Curve of Best Fit Usuall we are given an EQUATION, and we find points on that function. REGRESSION is the process of finding an equation when we are given POINTS 1. The table below shows the number (in thousands) of alternative fueled cars in the United States, ears after Make a scatter plot using the data. X Y Does the scatter plot increase or decrease? What shape does the data seem to make? How man alternative-fueled cars were there in 005? DRAW a line that would fit this data To find the line of best fit using LINEAR REGRESSION 1. Enter our values: STAT choose 1: EDIT Tpe values into L1 Tpe values into L To see our scatter plot: STAT PLOT->1 turn on, ZOOM->9) ZOOM-> puts it back in standard mode. Choose our function tpe: STAT CALC : LinReg Your Calculator should read LinReg (a+b) 3. Enter our lists: L1 and L are chosen b default. (Keep hitting "enter" until ou hit "Calculate"). Write our equation. Round to 3 decimals. Now, CALCULATE how man alternative-fueled cars there were in 005 using the LINE OF BEST FIT (linear regression)