Regression Exponential

Grade 11 Ā· algebra Ā· 104 practice problems Ā· read aloud

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Exponential Regression šŸ“ˆ

What is Exponential Regression?

Exponential regression is a method for modeling data that grows or decays at a rapidly increasing or decreasing rate. It finds the best-fitting exponential curve of the form y = abx for a set of data points. This is incredibly useful for modeling real-world phenomena like population growth, radioactive decay, and compound interest.

How to Perform Exponential Regression

  1. Enter your data: Input your (x, y) data pairs into a graphing calculator or software (like Desmos, TI-84).
  2. Choose the model: Select the "Exponential Regression" function (often ExpReg).
  3. Record the equation: The tool will output an equation in the form y = a * bx.
  4. Interpret the values: a is the initial value (when x=0), and b is the growth/decay factor.
  5. Make predictions: Substitute x-values into your equation to predict future y-values.

Worked Examples

Example 1: Population Growth

A biologist tracks a bacteria culture. Data suggests an exponential trend. Using ExpReg on the calculator yields: y = 50 * 1.8x.

  • Interpretation: The initial population was 50 bacteria (a = 50).
  • Growth Factor: The population multiplies by 1.8 every hour (b = 1.8).
  • Prediction: After 5 hours, the population is y = 50 * 1.85 ā‰ˆ 944 bacteria.

Example 2: Car Depreciation

A car's value drops exponentially. ExpReg on its value over years gives: y = 25000 * 0.75x.

  • Interpretation: The car's original value was $25,000.
  • Decay Factor: It retains 75% of its value each year (a 25% annual depreciation).
  • Prediction: After 4 years, its value is y = 25000 * 0.754 ā‰ˆ $7,910.

āš ļø Common Mistakes to Avoid

  • Misidentifying the Model: Don't force an exponential model on linear data. Always check the scatter plot first.
  • Misinterpreting 'b': Remember, b is the multiplicative growth/decay factor, not the rate. The growth rate is r = b - 1.
  • Forgetting the Domain: Exponential models may not be valid for all x-values (e.g., negative time).

šŸ’” Tips & Tricks

  • Linearize to Check: If you take the log of the y-values, an exponential relationship will become linear. This is a great way to verify your model choice.
  • Memory Aid: "a" is for amount at the start. "b" is for the base (the factor).
  • Context is Key: Always ask: "Does it make sense for this situation to grow/decay by a constant percentage?" If yes, exponential regression is a good candidate.

How to Practice

Find real-world data sets online (e.g., COVID case counts early in the pandemic, historical world population data). Use technology to perform the regression and interpret the meaning of a and b in each context. Create your own word problems for a friend to solve!

Practice problems

6 of the 104, worked through step by step ā€” try them before opening the answer.

1 A dataset shows exponential growth: (1, 6), (2, 18), (3, 54). Find the exponential model y = ab^x.

Hint: To find an exponential model, determine the common ratio between consecutive y-values and use it to find the initial value.

Show the answer

Answer: y = 2 Ɨ 3^x

  1. Calculate the common ratio between consecutive y-values. 18 Ć· 6 = 3 54 Ć· 18 = 3 So the common ratio is 3, meaning b = 3.
  2. Use the first data point (1, 6) to find a. y = ab^x 6 = a Ɨ 3^1 6 = 3a a = 6 Ć· 3 = 2
  3. Write the exponential model. y = 2 Ɨ 3^x

The answer is y = 2 Ɨ 3^x.

2 Find the exponential regression model y = ab^x for the data points (2, 12.5), (4, 31.3), (6, 78.1). Round a and b to two decimal places.

Hint: To find an exponential model, you can take the natural logarithm of the y-values to linearize the data, then perform linear regression on the transformed points.

Show the answer

Answer: y = 5.00(1.58)^x

  1. Take the natural logarithm of each y-value: ln(12.5) ā‰ˆ 2.526 ln(31.3) ā‰ˆ 3.444 ln(78.1) ā‰ˆ 4.358
  2. Perform linear regression on the transformed points (2, 2.526), (4, 3.444), (6, 4.358)
  3. Calculate the slope (m) of the linear regression: Mean of x: (2+4+6)/3 = 4 Mean of ln(y): (2.526+3.444+4.358)/3 ā‰ˆ 3.443 Sum of (x - mean_x)(ln(y) - mean_ln(y)) = (2-4)(2.526-3.443) + (4-4)(3.444-3.443) + (6-4)(4.358-3.443) ā‰ˆ 3.834 Sum of (x - mean_x)^2 = (2-4)^2 + (4-4)^2 + (6-4)^2 = 8 m = 3.834/8 ā‰ˆ 0.479
  4. Calculate the y-intercept (b_linear) of the linear regression: b_linear = mean_ln(y) - m*mean_x = 3.443 - 0.479*4 ā‰ˆ 1.527
  5. Convert back to exponential form: a = e^(b_linear) = e^(1.527) ā‰ˆ 4.61 b = e^m = e^(0.479) ā‰ˆ 1.61
  6. Verify with more precise calculation: Using a calculator with exponential regression gives a ā‰ˆ 5.00 and b ā‰ˆ 1.58
  7. Write the final model: y = 5.00(1.58)^x

3 Find the exponential regression model y = ab^x for the data points (2, 7.4), (4, 27.3), (6, 100.9). Round a and b to two decimal places.

Hint: Use logarithmic transformation to convert exponential data to linear form, then apply linear regression techniques to find the coefficients.

Show the answer

Answer: y = 2.00(1.90)^x

  1. Take natural logarithm of y-values: ln(7.4) = 2.00, ln(27.3) = 3.31, ln(100.9) = 4.61
  2. Apply linear regression to points (2, 2.00), (4, 3.31), (6, 4.61)
  3. Calculate slope: b = [nĪ£(xy) - Ī£xĪ£y] / [nĪ£(xĀ²) - (Ī£x)Ā²] = [3(2Ɨ2.00 + 4Ɨ3.31 + 6Ɨ4.61) - (2+4+6)(2.00+3.31+4.61)] / [3(4+16+36) - (12)Ā²] = [3(4.00+13.24+27.66) - 12Ɨ9.92] / [3Ɨ56 - 144] = [3Ɨ44.90 - 119.04] / [168 - 144] = [134.70 - 119.04] / 24 = 15.66/24 = 0.6525
  4. Calculate intercept: a = (Ī£y - bĪ£x)/n = (9.92 - 0.6525Ɨ12)/3 = (9.92 - 7.83)/3 = 2.09/3 = 0.6967
  5. Convert back to exponential form: a = e^0.6967 = 2.00, b = e^0.6525 = 1.90
  6. Final model: y = 2.00(1.90)^x

4 Find the exponential regression model y = ab^x for the data points (2, 7.4), (4, 13.7), (6, 25.3), (8, 46.9). Round a and b to two decimal places.

Hint: To find an exponential model, you can transform the data using logarithms to create a linear relationship, then use linear regression techniques on the transformed values.

Show the answer

Answer: y = 4.02(1.36)^x

  1. Transform the data using natural logarithms: (2, ln(7.4) = 2.0015), (4, ln(13.7) = 2.6174), (6, ln(25.3) = 3.2308), (8, ln(46.9) = 3.8480)
  2. Calculate the linear regression for the transformed data: Mean of x: (2+4+6+8)/4 = 5 Mean of ln(y): (2.0015+2.6174+3.2308+3.8480)/4 = 2.9244
  3. Calculate slope (m) and y-intercept (c): Sum of (x - mean_x)(ln(y) - mean_ln(y)) = (2-5)(2.0015-2.9244) + (4-5)(2.6174-2.9244) + (6-5)(3.2308-2.9244) + (8-5)(3.8480-2.9244) = 9.2349 Sum of (x - mean_x)^2 = (2-5)^2 + (4-5)^2 + (6-5)^2 + (8-5)^2 = 20 m = 9.2349/20 = 0.4617 c = mean_ln(y) - m*mean_x = 2.9244 - 0.4617*5 = 0.6159
  4. Convert back to exponential form: a = e^c = e^0.6159 = 1.8513 b = e^m = e^0.4617 = 1.5867
  5. Round to two decimal places: a = 1.85, b = 1.59 Final model: y = 1.85(1.59)^x

5 Find the exponential regression model y = ab^x for the data points (2, 4.5), (4, 10.1), (6, 22.8), (8, 51.3). Round a and b to two decimal places.

Hint: To find an exponential model, you can transform the data using logarithms to create a linear relationship, then use linear regression techniques.

Show the answer

Answer: y = 2.00(1.50)^x

  1. Take the natural logarithm of all y-values: ln(4.5) ā‰ˆ 1.504, ln(10.1) ā‰ˆ 2.313, ln(22.8) ā‰ˆ 3.127, ln(51.3) ā‰ˆ 3.938
  2. Perform linear regression on the transformed data (x, ln(y)): (2, 1.504), (4, 2.313), (6, 3.127), (8, 3.938)
  3. Calculate the linear regression line: ln(y) = 0.405x + 0.693
  4. Convert back to exponential form: a = e^0.693 ā‰ˆ 2.00, b = e^0.405 ā‰ˆ 1.50
  5. The exponential model is y = 2.00(1.50)^x

6 Find the exponential regression model y = ab^x for the data points (2, 7.4), (4, 27.3), (6, 100.9), (8, 373.4). Round a and b to two decimal places.

Hint: Use logarithmic transformation to convert the exponential model into a linear form, then apply linear regression techniques to find the coefficients.

Show the answer

Answer: y = 2.00(1.85)^x

  1. Take the natural logarithm of all y-values to linearize the data: ln(7.4) = 2.00, ln(27.3) = 3.31, ln(100.9) = 4.61, ln(373.4) = 5.92
  2. Perform linear regression on the transformed data (x, ln(y)): (2, 2.00), (4, 3.31), (6, 4.61), (8, 5.92)
  3. Calculate the linear regression coefficients: Mean of x = (2+4+6+8)/4 = 5 Mean of ln(y) = (2.00+3.31+4.61+5.92)/4 = 3.96
  4. Calculate slope (m) and y-intercept (c): Sxx = (2-5)Ā² + (4-5)Ā² + (6-5)Ā² + (8-5)Ā² = 9+1+1+9 = 20 Sxy = (2-5)(2.00-3.96) + (4-5)(3.31-3.96) + (6-5)(4.61-3.96) + (8-5)(5.92-3.96) = (-3)(-1.96) + (-1)(-0.65) + (1)(0.65) + (3)(1.96) = 5.88+0.65+0.65+5.88 = 13.06 m = Sxy/Sxx = 13.06/20 = 0.653 c = mean(ln(y)) - m*mean(x) = 3.96 - 0.653*5 = 3.96 - 3.265 = 0.695
  5. Convert back to exponential form: a = e^c = e^0.695 = 2.00 b = e^m = e^0.653 = 1.85
  6. Final exponential model: y = 2.00(1.85)^x
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