MAC 1140 – Section 4.1 – Exponential Expressions and Functions

A basic polynomial (algebraic) function is of the form f(x) = xn where the variable is the base.

Examples: f(x) = x2 g(x) = x-3

A basic exponential (transcendental) function is of the form f(x) = ax where the variable is the exponent. Examples: f(x) = 3x g(x) = 4-x

Example: Given f(x) = 2x evaluate:

1. f(-3) 2. f(-2) 3. f() 4. f(3)

Complete the chart.

x / -3 / -2 / -1 / -1/2 / 0 / 1/2 / 1 / 2 / 3
f(x)=2x
g(x)=2-x

Sketch a graph of your data. Label each graph.

What happens to f(x) as x ® ¥ ?

What happens to f(x) as x ® -¥?

What happens to g(x) as x ® ¥ ?

What happens to g(x) as x ® -¥?

Determine each of the following. f(x) g(x)

Domain:

Range:

y-intercept:

x-intercept:

Equation of the horizontal asymptote:

Is the function increasing or decreasing?

Use your calculator to graph. Sketch and label the graphs below:

On graph a: y1 = 2x y2 = 4x y3 = 8x On graph b: y1 = 2-x y2 = 4-x

What affect does the increased base What affect does the “negative x” have on graph?

have on the shape of the graph?

a. b.


Transformations of graph of f(x) = bx where b>1.

a. An extra term (added or subtracted) moves graph up if added; down if subtracted.

b. A number added or subtracted to the x-value moves graph left if added; right if subtracted.

c.  A “-“ in front of bx reflects graph in x-axis—function is decreasing.

d. A “-“ in front of x reflects graph in y-axis----function is decreasing.

Sketch the graphs below using transformations: Identify the domain, range, and horizontal asymptote. Is it increasing or decreasing?

1. y = 2x + 3 2. y = 2x - 5 3. y = -3 + 2x

4. y = 2(x + 1) 5. y = 2(x – 1) 6. y = -2(x – 5)

7. y = 2(x + 1) – 4 8. y = 3 - 2(x + 3) 9. y = 3(x-3) – 4

Rewrite y = (1/2)x by first rewriting 1/2 using a negative exponent [] .

Note that the graphs of y = (1/2)x and y = ______are the same.

Graph using transformations: Identify the domain, range, and horizontal asymptote. Is it increasing or decreasing?

10. y = 2(- x) – 4 11. y = 3 + 2-x 12. y = 3 – 2-x

Replace x with x +2 in the equation: y = 2-x Replace x with x – 4 in the equation: y = 2-x

13. y = 3-(x +4) 14. y = 3 (2-x) 15. y = 24-x

(Must factor out “–“ first) (Must factor out “–“ first)

:

Solve each equation by using a graphic calculator. Put the left side in for y1 and the right side in for y2.

16. 2x = 5 17. 8x+2 = 4 18. 3x =7.25 19. 42x – 1 = 6 20. 7x – 3 = -3.58

Page 272: 1 – 9 odds, 13, 15, 21, 31 – 35 all, 37, plus the following problems on the calculator.

1. 5x = 2 2. 33x + 4 = 7 3. 5-x – 2 = - 3.44 4. 3 + 2x = 6.25 (074)