Advanced Algebra
Exponential & Logarithmic Functions: Graphing Exponential Functions
The PARENT exponential function : y = B x , B > 0, B ≠ 1
Is a GROWTH function if B >1:
Is a DECAY function if 0<B<1:
Both of these parent curves have a horizontal asymptote on the line y = 0.
Both of these parent curves pass through the point (0, 1).
TRANSFORMATIONS can be applied to the growth and decay functions:
I. Negative Coefficients REFLECT the image:
A.
INSIDE the exponent: y = B − x reflects the graph across the y-axis.
⎛ 1⎞
parent: y = ⎜ ⎟
⎝ 2⎠
parent: y = 2 x
−x
transformed: y = 2
x
⎛ 1⎞
transformed: y = ⎜ ⎟
⎝ 2⎠
−x
The transformed graphs still have an asymptote on y = 0 and go through (0,1).
B.
OUTSIDE the function: y = −B x reflects the graph across the x-axis.
⎛ 1⎞
parent: y = ⎜ ⎟
⎝ 2⎠
parent: y = 2 x
transformed: y = −2
x
x
⎛1⎞
transformed: y = − ⎜ ⎟
⎝2⎠
x
The transformed graphs still have an asymptote on y = 0 and go through (0,-1).
II.
Coefficients ≠ 1 DILATE the image.
A.
OUTSIDE the exponent: y = a • B x stretches/shrinks the graph vertically.
For every point (x, y) on the parent graph, there is a point on the transformed
graph with the y-coordinate multiplied by ‘a’, (x, a•y).
x
x
parent: y = 2
transformed: y = 3 • 2
The transformed graph still has an asymptote on y = 0 but the point (0,1) on the
parent corresponds to the point (0, 3) on the transformed graph.
B.
INSIDE the exponent: y = Bb⋅x stretches/shrinks the graph horizontally.
For every point (x, y) on the parent graph, there is a point on the transformed
graph with the x-coordinate divided by ‘b’, (x/b, y).
x
3x
parent: y = 2
transformed: y = 2
8
6
4
2
-4
-2
2
4
2
The transformed graph still has an asymptote on y = 0 and still has a point at
(0, 1), but other points on the graph have moved.
III.
Constants TRANSLATE the image.
A.
INSIDE the exponent: y = B bx + c shifts graph left (+c) or right (-c).
For every point (x, y) on the parent graph, there is a point on the transformed
graph with the x-coordinate shifted –c/b , (x – c/b, y).
parent: y = 2 x
transformed: y = 2 x +3
The transformed graph still has an asymptote on y = 0 but the point at (0, 1) on
the parent has moved to (-3, 1).
B.
OUTSIDE the exponent: y = B x + d shifts graph up (+d) or down (-d).
For every point (x, y) on the parent graph, there is a point on the transformed
graph with the y-coordinate shifted ‘d’ , (x , y + d).
parent: y = 2 x
transformed: y = 2 x + 3
The transformed graph now has an asymptote on y = 3 and the point at (0, 1) on
the parent has moved to (0, 4).
When a function contains more than one transformation, consider the transformations that
require multiplication and division before the transformations that require addition and
subtraction.
Consider a function with all of the transformations: y = a ⋅ B bx + c + d .
1. Take any point on the parent graph [the easiest being the point (0, 1)] and the
equation of the horizontal asymptote (y = 0), divide the x-coordinate by ‘b’ and
multiply the y-coordinate by ‘a’.
2. Then add ‘–c/b’ to the x-coordinate and add ‘d’ to the y-coordinates.
The parent point (0, 1) becomes (-c/b, 1+d) and the horizontal asymptote becomes y=d.
To graph the function, plot these items and sketch the curve based on the parent (growth
or decay) and any reflections.
+1
Example: y = −3 (2 )
Base = 2 (growth)
a = -3 (reflection)
b=2
c=4
d=1
2x + 4
(0, 1) becomes (-2, 2)
horizontal asymptote becomes y = 1
growth, reflect across x-axis