Advanced Algebra Exponential & Logarithmic Functions: Graphing Exponential Functions

The PARENT exponential function : , B > 0, B ≠ 1 y = Bx

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.

parent: y = 2 parent: y =1x

2⎛⎝⎜

⎞⎠⎟

x

transformed: y = 2 transformed: y =− x 12

⎛⎝⎜

⎞⎠⎟−x

The transformed graphs still have an asymptote on y = 0 and go through (0,1).

B. OUTSIDE the function: reflects the graph across the x-axis. y = −B x

parent: y = 2 parent: y =1x

2⎛⎝⎜

⎞⎠⎟

x

transformed: y = − transformed: y = −1

2 x

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: stretches/shrinks the graph vertically. y = a • Bx

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).

parent: y = 2 transformed: y = 3 x • 2 x

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: b xy B ⋅= 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). parent: y = 2 transformed: y = 2 x 3x

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.

-4 -2 2 4

2

2

4

6

8

A. INSIDE the exponent: bx cy B += 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 transformed: x 32xy +=

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: shifts graph up (+d) or down (-d). y = Bx + 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 transformed: y = 2 x 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 ⋅Bbx+ 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.

Example: y = −3 2( )2x+ 4 +1 Base = 2 (growth) a = -3 (reflection) b = 2 c = 4 d = 1 (0, 1) becomes (-2, 2) horizontal asymptote becomes y = 1 growth, reflect across x-axis