College Algebra 5.1

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College Algebra 5.1

  1. 1. 5.1 Exponential Functions and Their Graphs A. Basic Exponential Function (increasing and decreasing) B. One-to-one property C. Graph of Basic Exponential Function including Asymptote D. Shifting Graph of Basic Exponential Function (including asymptote)
  2. 2. A. Basic Exponential Function <ul><li>It looks like f(x) = a x where a is a positive number </li></ul><ul><li>Like f(x) = 4 x or f(x) = 1 x or f(x) = (1/2) x </li></ul><ul><li>What is different from the power functions we did on the last test? This time the x is in the exponent, and not in the base. </li></ul>
  3. 3. Suppose f(x) = 2 x . Find f(4).
  4. 4. Suppose f(x) = 2 -x . Find f(4). <ul><li>Remember what a negative exponent means? </li></ul>
  5. 5. Suppose f(x) = (1/3) x . Find f(4).
  6. 6. Suppose f(x) = 4 x . Find f(3/2). <ul><li>Remember what rational exponents mean? </li></ul>
  7. 7. B. One-to-One Property <ul><li>9 = 3 x+1 Find x. </li></ul><ul><li>First try to get the same base. </li></ul><ul><li>If the bases are exactly the same, those exponents must be the same also! </li></ul>
  8. 8. (1/2) x = 8 <ul><li>Sometimes you have to rewrite both sides of the equal sign in order to get the bases to look identical. </li></ul>
  9. 9. You try:
  10. 10. C. Graph of Basic Exponential Function, with asymptote <ul><li>Graph of f(x) = 2 x : </li></ul>x f(x)=2 x (x,y) 0 1 2 0 =1 2 1 =2 (0,1) (1,2) -1 -2 2 -1 =1/2 2 -2 =1/4 (-1,1/2) (-2,1/4) -3 2 2 -3 =1/8 2 2 =4 (-3,1/8) (2,4)
  11. 11. Let’s compare that shape to f(x)=4 x x f(x)=4 x (x,y) 0 1 4 0 =1 4 1 = (0,1) ( -1 -2 4 -1 = 4 -2 = -3 2 4 -3 = 4 2 =
  12. 12. This one will look different: f(x) = 2 -x <ul><li>which is the same thing as f(x) = (1/2) x </li></ul>x f(x)=2 -x (x,y) 0 1 -1 -2 3 2
  13. 13. <ul><li>Notice the graph of f(x) = 2 x </li></ul><ul><li>The curve gets closer and closer to this axis but never touches it. The ASYMPTOTE of this one is the line y = 0 . </li></ul>
  14. 14. The asymptote is not always y = 0. <ul><li>When we take it and shift it up or down, the asymptote will change. </li></ul><ul><li>The asymptote of f(x) = 2 x + 7 is </li></ul><ul><li>The asymptote of f(x) = 2 x – 7 is </li></ul>
  15. 15. D. Shifting Graph of Basic Exponential Function <ul><li>f(x) = 2 x shifted to the right 1 unit would be: </li></ul><ul><li>Graph it: </li></ul><ul><li>Doesn’t have a y-intercept at 1 anymore, but at (0, ½). The asymptote is still y = 0. </li></ul>
  16. 16. You try: Shift f(x) = 2 x to the left 2 units. Gve the equation, the graph, and the asymptote.
  17. 17. You try: Reflect f(x) = 2 x over the x-axis. Give the equation, the graph, and the asymptote.

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