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Exponential Functions Exploration and Equations Solving Drill

This lesson covers exponential functions, exploring graph behavior, solving equations, exponent rules, and predicting population growth. It includes examples of exponential regression calculation and practical applications like cost estimation and compound interest formulas. A comprehensive study guide for students.

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Exponential Functions Exploration and Equations Solving Drill

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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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