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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.