Graphing exponential and logarithmic functions Exponential function

Accelerated Precalculus Graphing exponential and logarithmic functions

Name: ____________________________

f ( x)  b x , where b  0, b  1, and x is any real number.

Exponential function with base b:

The graph of the basic exponential function f ( x)  b x has the following characteristics: 1. 2. 3. 4.

Reference point: Range: y-intercept: Horizontal asymptote:

(0, 1) (0, ) (0, 1) y0 (0,1)

In addition, the domain is all real numbers. For exponential functions of the form

f ( x)  ab xh  k 1. 2. 3. 4.

or

f ( x)  ab( xh)  k

a  0 reflects about the x-axis  x reflects about the y-axis h translates the graph left or right (horizontal shift) k translates the graph and the horizontal asymptote up or down (vertical shift)

Examples: Ex. 1: For the following exponential function, find the (1) horizontal asymptote, (2) range, (3) y-intercept, and (4) reference point. Sketch the graph.

f ( x)  2 x1 1. 2. 3. 4.

Because k  0 , the horizontal asymptote is y  0. Because a  1  0 , the graph lies above the horizontal asymptote and the range is (0, ) . f (0)  201  2 , so the y-intercept is (0, 2) . To find the reference point: a. Set the expression in the exponent equal to 0 and solve for x: x 1  0

x  1 b. Sub in this x-value to find the y-coordinate: y  211

y  20 y 1 c. The reference point is (1, 1). 5. Sketch.

Ex. 2: For the following exponential function, find the (1) horizontal asymptote, (2) range, (3) y-intercept, and (4) reference point. Sketch the graph.

f ( x)  e2 x  2 1. Because k  2 , the horizontal asymptote is _______________. 2. Because a  1  0 , the graph lies _______________ the horizontal asymptote and the range is _______________. 3. The y-intercept is _______________ or approximately _______________.

4. The reference point is _______________.

5. Sketch.

Logarithmic function with base b: f ( x)  logb x , where b  0, b  1, and x  0. The graph of the basic logarithmic function f ( x)  logb x has the following characteristics: 1. Reference point: (1, 0) 2. Domain: (0, ) 3. x-intercept: (1, 0) 4. Vertical asymptote: x0 (1, 0) In addition, the range is all real numbers. For logarithmic functions of the form f ( x)  a logb ( x  h)  k or f ( x)  a logb [( x  h)]  k 1. 2. 3. 4.

a  0 reflects about the x-axis  x reflects about the y-axis h translates the graph and the vertical asymptote left or right (horizontal shift) k translates the graph up or down (vertical shift)

Examples: Ex. 1: For the following logarithmic function, find the (1) vertical asymptote, (2) domain, (3) x-intercept, and (4) reference point. Sketch the graph. f ( x)  log( x  3)

1. To find the vertical asymptote, set the expression in the parentheses equal to 0 and solve for x: x  3  0 so x  3 2. For the domain, the expression in the parentheses must be greater than 0: x  3  0 so x  3 3. To find the x-intercept, let y  0 and solve for x: 0  log( x  3) 100  x  3 1 x3 x  2 4. To find the reference point: a. Set the expression in the parentheses equal to 1 and solve for x: x  3 1 x  2

5. Sketch.

b. Sub in this x-value to find the y-coordinate: y  log(2  3) y  log1 y0

Ex. 2: For the following logarithmic function, find the (1) vertical asymptote, (2) domain, (3) x-intercept, and (4) reference point. Sketch the graph. y   ln(2  x)  1

1. Find the vertical asymptote, ___________________:

2. Find the domain, ____________________________:

3. Find the x-intercept, ______________ or approximately ______________:

4. Find the reference point, _______________________:

5. Sketch.