Warm up honors algebra 2 3/7/19 Pick up the handout by the projector and begin working on it  “Exponential Practice” Applying same base exponentials

Reminder on Regressions 1. Perform a linear regression on the following data: 2. Perform a quadratic regression on the following data: x 1 2 3 4 5 6 f(x) 3.45 8.7 13.95 19.2 24.45 29.7 x 1 2 6 11 13 f(x) 3 39 120 170

Calculator reminder!! How to find the curve of best fit when given a data table: Enter data into table on calculator  STAT  EDIT (1)  x-values in L1  y- values in L2 Determine the type of function the data represents - Linear, Quadratic or Exponential

Calculator reminder (cont): STAT  tab over to CALC  select type of function  Enter through 4: LinReg(ax+b) (linear) 5: QuadReg (Quadratic) 0: ExpReg (Exponential) 9: LnReg (Natural Log) Plug the given values into the function formula.

Curve of best fit = Regression Examples: “Find the curve of best fit for this linear data” = “Perform a linear regression” “Perform a quadratic regression” = “Find the curve of best fit”

Linear functions (first degree) First differences are constant!! x -1 1 2 f(x) 5 8 11 First differences are constant, so linear function!! First Differences: +3 +3 +3

Quadratic functions (Second Degree) Second differences are constant!! x -1 1 2 f(x) 3 5 8 First differences are NOT constant!! NOT Linear!! First Differences: +1 +2 +3 Second differences are constant, so Quadratic!! Second Differences: +1 +1

Exponential functions First differences are NOT constant!! NOT Linear!! Ratio of y-values are constant!! x -1 1 2 f(x) 16 24 36 54 Second differences are NOT constant!! NOT Quadratic!! First Differences: +8 +12 +18 Second Differences: +4 +6 Ratios are constant, so Exponential!! Ratios: 24 16 = 36 24 = 54 36 =1.5

Exponential with a constant ratio of 1.5. Determine what type of function the data represents. Find the common difference or ratio. x -1 1 2 3 f(x) 2. 6 4 6 9 13.5 First Differences: +1.3 4 +2 +3 +4.5 Second Differences: +0.66 +1 +1.5 Exponential with a constant ratio of 1.5. Ratios: 4 2. 6 = 6 4 = 9 6 = 13.5 9 =1.5

Linear with a first common difference of 5. Determine what type of function the data represents. Find the common difference or ratio. x -1 1 2 3 f(x) −3 7 12 17 First Differences: +5 +5 +5 +5 Linear with a first common difference of 5.

Quadratic with a second common difference of 1 Determine what type of function the data represents. Find the common difference or ratio. x -1 1 2 3 f(x) 5 8 12 First Differences: +1 +2 +3 +4 Second Differences: +1 +1 +1 Quadratic with a second common difference of 1

Find an exponential model for the data Find an exponential model for the data. Use the model to predict when the tuition at U.T. Austin will be $6000. Enter data in the graphing calculator and use the exponential regression feature (0) Year Tuition 1999-00 $3128 2000-01 $3585 2001-02 $3776 2002-03 $3950 2003-04 $4188 Mention that the r and r^2 show good fit 𝑓 𝑥 =3235.64 1.07 𝑥

Find an exponential model for the data Find an exponential model for the data. Use the model to predict when the tuition at U.T. Austin will be $6000. Graph the function To enter the regression equation as Y1 from the screen, press , choose 5:Statistics, press , scroll to select the EQ menu, and select 1:RegEQ.

The tuition will be about $6000 when t = 9 or 2008–09. Find an exponential model for the data. Use the model to predict when the tuition at U.T. Austin will be $6000. 7500 15 Enter 6000 as Y2. Use the intersection feature. You may need to adjust the dimensions to find the intersection. The tuition will be about $6000 when t = 9 or 2008–09.

Use exponential regression to find a function that models this data Use exponential regression to find a function that models this data. When will the number of bacteria reach 2000? Time (min) 1 2 3 4 5 # 200 248 312 390 489 610 2500 15 The bacteria count at 2000 will happen at approximately 10.3 minutes. 𝑓 𝑥 =199.29 1.25 𝑥

Find a natural log model for the data Find a natural log model for the data. According to the model, when will the global population exceed 9,000,000,000? Global Population Growth Population (billions) Year 1 1800 2 1927 3 1960 4 1974 5 1987 6 1999 𝑓 𝑥 =1824.41+106.48𝑙𝑛𝑥 The population will exceed 9,000,000,000 (x=9) in the year 2058.

Use the logarithmic regression to find a function that models this data. When will the speed reach 8.0 m/s? Time (min) 1 2 3 4 5 6 7 Speed (m/s) 0.5 2.5 3.5 4.3 4.9 5.3 5.6 Use the intersect feature to find y when x is 8. The time it will reach 8.0 m/s is 16.6 min. 𝑓 𝑥 =0.587+2.638𝑙𝑛𝑥