Questions? Standard

Questions? Honors

Questions? Honors

Math 2 Warm Up In the Algebra 2 Practice Workbook, Practice 5-8 (p. 70) Solve: #50-64 even, 24, 30, 39 Discriminant: #1, 4, 7, 49

Questions?

Unit 4: “Quadratic Functions” Solving Quadratic Equations: By Graphing Objective: To solve quadratic equations and systems of equations that contain at least one quadratic functions by graphing. When the graph of a function intersects the x-axis, the y-value of the function is 0. The solutions of the quadratic equation ax2 + bx + c = 0 are the x-intercepts of the graph of y = ax2 + bx + c. The solutions are also called “zeros of the function” or “roots of the function”.

Solve Quadratic Equations by Graphing Solution ax2 + bx + c = 0

Solve Quadratic Equations by Graphing Step 1: Quadratic equation must equal 0! ax2 + bx + c = 0 Step 2: Press [Y=]. Enter the quadratic function into Y1. Enter 0 into Y2. Press [Graph]. MAKE SURE BOTH X-INTERCEPTS ARE ON SCREEN! “ZOOM OUT” OR “ZOOM IN” IF NEEDED! Step 3: Find the intersection of y = ax2 + bx + c and y = 0. Press [2nd] [Trace]. Select [5: Intersection]. First Curve? Press [Enter], Second Curve? Press [Enter], Guess? Using left and right arrows, move the cursor to one of the x-intercepts then press [Enter] a 3rd time. Step 4: Repeat Step 3 for the second x-intercept!

Solve by Graphing 𝒙 𝟐 + 6x + 4 = 0

Solve by Graphing 𝟐𝒙 𝟐 + 4x – 7 = 0

Solve by Graphing − 𝟏 𝟐 𝒙 𝟐 +𝟐𝒙+𝟗 = 0

Solve by Graphing 𝟑𝒙 𝟐 + 5x = 20

Solve by Graphing 𝟓𝒙 𝟐 =𝟏𝟗𝒙 −𝟕

Solve by Graphing* 𝟏 𝟑 𝒙 𝟐 +𝟕𝒙 −𝟖 = 0 Off Normal screen

Solve by Graphing* −𝟑 𝒙 𝟐 + 5x – 4 = 0 No Solution

Solve by Graphing* 𝒙 𝟐 − 𝟖𝒙 + 16 = 0 Zoom In Move Cursor or Zoom Box 𝒙 𝟐 − 𝟖𝒙 + 16 = 0 Zoom In Move Cursor or Zoom Box On x-axis! VERTEX

Solve by Graphing* 𝟖 𝒙 𝟐 −𝟏𝟎𝒙+𝟑 = 0 Zoom In Move Cursor or Zoom Box

Solve a System with a Quadratic Equation A system of equations is a set of two or more equations that use the same variables. A solution of a system of equations is the set of values for the variables that makes ALL the equations true. On a graph: the solutions of a system is any point (x, y) where ALL the graphs of the functions in the system intersect. Solutions Solutions

Solve a System with a Quadratic Equation 𝒚= 𝒙 𝟐 + 2x − 𝟒 𝒚=−𝒙+𝟑

Solve a System with a Quadratic Equation 𝒚=𝟐𝒙 𝟐 + x 𝒚= 𝟒 𝟑 𝒙+𝟒

Solve a System with a Quadratic Equation 𝒚= 𝒙 𝟐 + 𝟒𝒙 + 𝟕 𝒚=−𝟐𝒙

Solve a System with a Quadratic Equation 𝒚= 𝒙 𝟐 −𝟔x + 𝟏𝟎 𝒚=𝟏

Solve a System with Quadratic Equations 𝒚= 𝒙 𝟐 − 𝟔𝒙+𝟓 𝒚=−𝟐𝒙 𝟐 +𝟓𝒙

Solve a System with Quadratic Equations 𝒚=𝟐𝒙 𝟐 + 𝟕𝒙 𝐲= 𝟏 𝟒 𝒙 𝟐 −𝟓𝒙−𝟗

Solve a System with Quadratic Equations 𝒚=−𝒙 𝟐 + 𝟖𝒙−𝟓 𝐲=− 𝟑 𝟓 𝒙 𝟐 +𝟒

Honors Math 2 Assignment In the Algebra 2 textbook, pp. 266-268 #20-31, 54-56, 67-69 Solve each quadratic equation or system by graphing on the calculator. Round answers to the nearest hundredth. TURN IN WHEN FINISHED!

Math 2 Assignment pp. 266-268 #20-31, 54-56, 67-69 In the Algebra 2 textbook, pp. 266-268 #20-31, 54-56, 67-69 Solve each quadratic equation or system by graphing on the calculator. Round answers to the nearest hundredth. TURN IN WHEN FINISHED!

Round answers to nearest hundredth. Math 2 Warm Up Part 1 Finish pp. 266-268 #20-31, 54-56, 67-69 Round answers to the nearest hundredth. Turn in TEST REVIEW FOLDER when finished. Part 2 In the Algebra 2 Practice Workbook, Practice 5-5 (p. 64) #4, 14, 33, 45, 50, 57, 59, 68, 69, 70 Solve by Graphing! Round answers to nearest hundredth.

Questions?

Unit 4: “Quadratic Functions” Modeling Data with Quadratic Equations Objective: To model a set of data with a quadratic function. Graph: Graph: (-5, 6), (-4, 1), (-2, -3) (-3, -4), (2, 1), (3, 8) (-1, -2), (0, 1) Left Graph has info to find quad eq using y=a(x-h)2 + k Right Graph does not! Can not identify vertex. So method today… 𝒚=𝒂 (𝒙−𝒉) 𝟐 + 𝒌

Finding a Quadratic Model Substitution Method Substitute the values of x and y into 𝒚=𝒂 𝒙 𝟐 +𝒃𝒙+𝒄 Given: (-3, -4), (2, 1), (3, 8) −𝟒=𝒂 −𝟑 𝟐 +𝒃 −𝟑 +𝒄 𝟗𝒂−𝟑𝒃+𝒄=−𝟒 1=𝒂 𝟐 𝟐 +𝒃 𝟐 +𝒄 𝟒𝒂+𝟐𝒃+𝒄=𝟐 𝟖=𝒂 𝟑 𝟐 +𝒃 𝟑 +𝒄 𝟗𝒂+𝟑𝒃+𝒄=𝟖 ONLY works for 3 data points!

Finding a Quadratic Model Regression Method 1) Turn on plot: Press [2nd] [Y=]. Press [ENTER] to select Plot 1. Highlight “On”, Press [ENTER].  2) Turn on diagnostic: Press [2nd] [0] for catalog. Scroll down to DiagonsticOn. Press [ENTER] to select. Press [ENTER] again to activate.

Finding a Quadratic Model Regression Method 3) Enter data values: Press [STAT] then [ENTER] for EDIT menu, Enter x-values in L1, Enter y-values in L2. Clear Lists (if needed): Highlight L1 or L2 at top. Press [CLEAR], [ENTER]. 4) Graph a scatter plot: Press [ZOOM] Select [9: ZoomStat]

Finding a Quadratic Model Regression Method 5) Find quadratic equation to fit data: Press [STAT], over to CALC menu, Select [5: QuadReg] Press [ENTER] 4 times, then Calculate [ENTER] . Write quadratic equation using the values of a, b, and c rounded to the nearest thousandths (if needed). Write down the R2 value!

𝑹 𝟐 is a statistical measure of how close the data values are to the “fitted” regression model. the value of R2 is between 0 and 1 (0 ≤ R2 ≤ 1). If R2  = 1, then ALL the data points “fit” the regression equation. If R2 = 0, then NONE of the data points “fit” the regression equation. The higher the R2 value, the better the regression equation “fits” the data.

Find a quadratic equation to model the values in the table. x y -3 -4 2 1 3 8 Y=

Find a quadratic equation to model the values in the table. x y 2 3 13 4 29 R2 = 1

Find a quadratic equation to model the values in the table. x y -5 -18 7 2 -9 R2 = 1

Find a quadratic equation to model the values in the table. x y -2 27 1 10 5 -6 7 8 R2=0.9084709368

Apply! The table shows data about the wavelength (in meters) and the wave speed (in meters per second) of deep water ocean waves. Find a quadratic equation to model the data. Find the wave speed of a deep water wave that has a wavelength of 6 meters. Wavelength (m) Wave Speed (m/s) 3 6 5 16 7 31 8 40 b) Solve for y

Apply! The table at the right shows the height of a column of water (in millimeters) as it drains from its container. Find a quadratic equation to model the data. Find the water level at 25 seconds. Find the water level at 90 seconds. Find the water level at 3 minutes. Is the answer for d) reasonable?

Apply! The table gives the number of millions of US cell phone subscribers since 1998. Find a quadratic equation to model the data. In what year will the numbers of subscribers exceed 170,000,000? Year Subscribers (in millions) 1998 69 1999 86 2000 109 2001 128 2002 140 b) Solve for x

Apply! A study compared the speed (in miles per hour) and the average fuel economy (in mile per gallon) for a midsize car. The data is shown in the table at the right. Find a quadratic model to represent the data. Find the speed that maximizes the fuel economy. Speed (miles per hour) Fuel Economy (miles per gal) 20 25.5 30 29.1 40 30.0 50 30.2 60 28.8 70 25.7 b) Find the vertex

Honors Math 2 Assignment In the Algebra 2 textbook, pp. 237-238 #16-22, 30, 31, 38 Write down the R² value for each equation! TURN IN WHEN FINISHED!

Write down the R² value for each equation! Math 2 Assignment In the Algebra 2 textbook, pp. 237-238 #16-22, 30, 31, 38 Write down the R² value for each equation! TURN IN WHEN FINISHED!