Exponential Functions

2. Exponential Functions

3. Basic Exponential Function f(x)=acbx exponential function in base c asymptote: x-axis a affects the vertical stretch a is also the initial value a can also flip the function b affects the horizontal stretch b is the period

4. Role of the parameter “a” and “c” Where parameter b must always be positive!!! if c > 1 if 0 < c < 1 a > 0 a < 0

5. Examples For each of the following, determine the base and indicate if the function is increasing or decreasing. Base = 3/4 Base = 3/2 Base = 3/2 Decreasing Decreasing Increasing

6. Exponential Functions f(x)=acb(x-h) + k h shifts the initial value to the right or left k shifts the initial value up or down k is the horizontal asymptote

7. Exponential Functions The rule f(x)=acb(x-h) + k can be re-written in the form: f(x)=acx + k where the function passes through the point (0, a + k) and has y = k as its asymptote

8. Example Convert to the form f(x)=acx + k f(x) = 5(3)2(x+1) + 7 f(x) = 5(32) (x+1) + 7 f(x) = 5(9)(x+1) + 7 f(x) = 5(9x.91)+ 7 f(x) = 45(9x)+ 7 f(x) = 45(9)x+ 7

9. Example Convert to the form f(x)=acx + k f(x) = -3(2)3x+1 + 4 f(x) = -3(2)3(x+1/3) + 4 f(x) = -3(8)(x+1/3) + 4 f(x) = -3(8x.81/3)+ 4 f(x) = -3(8x.2)+ 4 f(x) = -6(8x)+ 4 f(x) = -6(8)x+ 4

10. Example If the point A(2,9) belongs to the graph of an exponential function of the form y = cx, determine the rule 9 = c2 3 = c y = 3x

11. Example If the points A(1,4) and B(2,12) belong to the graph of an exponential function of the form y = acx, determine the rule 4 = ac1 12 = ac2 4 = a 12 = a c1 c2 4 = 12 c1 c2 4c2 = 12 c1 4c = 12 c = 3

12. Exponential Functions f(x)=acb(x-h) + k FINDING THE ZEROS Plug in y=0 and solve for x! y=5(3)x-2 – 15 0=5(3)x-2 – 15 15=5(3)x-2 15/5=(3)(x-2) 3=(3)(x-2) 1 = x-2 3=x

13. Euler's Number • A special irrational base of the exponential function • has a value of 2.71828… • abbreviated as e Examples: y = ex y = 3(e)x-1 - 2 • y = ex

14. Example √e(e2x-1)(ex)=√e3