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Lesson 1.3 Exponential Functions

Lesson 1.3 Exponential Functions. Part I HW: page 26: 1-20 For 19, just use the calculator to determine an exponential model, rather than completing a and b. Drill: Solve each equation. x 3 + 9 = 17 2y 2 + 2 = 10 ½z 3 - 8 = 24.

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Lesson 1.3 Exponential Functions

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  1. Lesson 1.3Exponential Functions Part I HW: page 26: 1-20 For 19, just use the calculator to determine an exponential model, rather than completing a and b.

  2. Drill: Solve each equation • x3 + 9 = 17 • 2y2 + 2 = 10 • ½z3 - 8 = 24

  3. Exploration on the graphing calculator:You have 15 minutes to complete this on a separate piece of paper. • Graph the function f(x) = ax for a = 2, 3, 5. Use the window [-5, 5] by [-2, 5]. • For what values of x is it true that 2x < 3x < 5x ? • For what values of x is it true that 2x > 3x > 5x ? • For what values of x is it true that 2x = 3x = 5x ? • Graph the function y = (1/a)x for a = 2, 3, 5 • Repeat parts 2-4 for the function in part 5.

  4. Exponent Rules • Product of Powers Postulate • ax ● ay = ax+y • Power of a Power Postulate • (ax)y = axy • Power of a Product Postulate • (ab)x = axbx • Quotient of Powers Postulate • (ax /ay ) = ax-y • Power of a Quotient Postulate • (a/b)x = ax / bx • Zero Exponent Theorem • a0 = 1

  5. Exponential Function • Let ‘a’ by a positive real number other than 1. The function f(x) = ax is the exponential function with base a. • Graph the function f(x) = 2(3x) – 4. State domain and range. -3.8 -3.3 -2 2 14 • Domain: all real numbers • (-∞,∞) • Range: y > 4 • (-4, ∞)

  6. Finding Zeros (x-intercepts) • Find the zeros of f(x) = 5x– 2.5 • Let y1 = f(x) • Let y2 = 0 • Graph (standard window is fine…ZOOM 6) • 2nd TRACE, 5, ENTER, ENTER, ENTER • In the case of multiple zeros, you will need to move the cursor towards the other zero(s) before hitting ENTER, ENTER, ENTER

  7. Rewriting Bases • Rewrite 4x with a base of 2 • 4 = 22 • So 4x = (22)x • Leaving 4x = 22x • Rewrite (1/64)x with a base of 4 • 64 = 43 • So (1/64) = 4-3 • (1/64)x = (4-3)x • = 4-3x

  8. Exponential Growth vs. Decay: y = k(a)x, k>0 Growth Decay 0<a<1 Domain: (-∞,∞) Range: (0,∞) y-intercept is (0,k) As x increases, for 0<a<1, f(x) decreases, approaching zero The x-axis (y = 0) is the asymptote • a>1 • Domain: (-∞,∞) • Range: (0,∞) • y-intercept is (0,k) • As x increases, for a > 1, f(x) also increases without bound • The x-axis (y = 0) is the asymptote

  9. Predicting Population • In 1995, the US population was estimated at 264,000,000 people and was predicted to grow about 0.9% a year for the near future. • With these assumptions, state a formula for the US population x years after 1995. B. From the formula, estimate the population in 2010.

  10. Cost of a Penn State Education/Semester (Tuition Only) for PA Residents • x = years after 1991 and y = cost of tuition • Pick any two points on your curve. • Step 1: Using the formula y = kax, form a system: • Step 2: Divide the equations (higher power/lower power) to find a: • Step 3: Substitute a into EITHER equation to find k: • 4: Rewrite, substituting a and k:

  11. Using the calculator… 1) Enter data into STAT, 1:edit, L1 = x, L2 = y • Plot on the calculator: STATPLOT, Type: scatterplot (1st one), Xlist = L1, YList = L2 • ZOOM, 9:Zoomstat • Exponential regression: STAT, →Calc, 0:ExpReg • To put into Y=: VARS, →Y-Vars, 1:Function, 1:Y1 • 9:ZoomStat to see line with points

  12. Drill Find the exponential regression by hand using any two points. (Let x = year after 1990) Find the exponential regression on the calculator. Predict the population for Virginia in 2011

  13. Half-Life • Formula: A = A0(.5)t/h • A = final amount after t years. • h = half life time period • A0 = original amount • A certain substance has a half-life of 24 years. If a sample of 80 grams is being observed, how much will remain in 50 years? • A = final amount after 50 years. • h = 24 years • A0 = 80 grams • A = 80(.5)50/24 = 18.88 grams

  14. compound interest formula: A = P (1 +r/n)nt • A = final amount • P = original amount • R = interest rate • N = number of compounding periods • T = time • A bank is currently offering a certificate of deposit paying 5.25% interest compounded quarterly. Find the value of the CD after two years if $1000 is invested. • A = final amount • P = 1000 • R = 5.25 = .0525 • N = 4 • T = 2 • A = 1000 (1 +.0525/4)4*2 • = $1109.95

  15. If the interest were compounded continuously, the amount would approach the irrational number e » 2.718281828…. e Continuously Compounded Interest A = Pert P = principal, r = rate, t = years

  16. Suppose you invest $100 at 4.5% interest, compounded continuously, for 5 years. Calculate how much will be in the account. Compare this to an account compounded monthly.

  17. Doubling Time • Determine how long it will take for an investment of $P to triple if you compound continuously at a rate of 3.7% • A = Pert • 3P = Pe.037t • 3 = e.037t • Let y1 = 3 and y2 = e.037t • Intersect (You will need to change windows): t = 29.69 years

  18. Group activity: copy and complete the tables below. Hand in at the end of class! example Hw: p. 27: 21-32 p. 28: 39-46 p. 2-: 1-4

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