4.2 Graphs of Exponential Functions

The Values

Example

Fill in the following table:

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

g(x) =( 1

3

)x

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

913 1 3 9

g(x) =( 1

3

)x9 3 1 1

319

The Values

Example

Fill in the following table:

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

g(x) =( 1

3

)x

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

913 1 3 9

g(x) =( 1

3

)x9 3 1 1

319

Plotting the curves

•

•

•••

f (x) = 3x

•••

•

•

g(x) =( 1

3

)x

Plotting the curves

•

•

•••

f (x) = 3x

•••

•

•

g(x) =( 1

3

)x

Plotting the curves

•

•

•••

f (x) = 3x

•••

•

•

g(x) =( 1

3

)x

Plotting the curves

•

•

•••

f (x) = 3x

•••

•

•

g(x) =( 1

3

)x

The effects of the parameter a

f (x) = 1 · 2x

f (x) = 2 · 2x

The effects of the parameter a

f (x) = 1 · 2xf (x) = 2 · 2x

The effects of the parameter a

f (x) = 1 · 2x

f (x) = 12 · 2x

The effects of the parameter a

f (x) = 1 · 2xf (x) = 1

2 · 2x

The effects of negatives

f (x) = 2x

f (x) = −2x

The effects of negatives

f (x) = 2xf (x) = −2x

The effects negatives

f (x) = 2x

f (x) = 2−x

The effects negatives

f (x) = 2xf (x) = 2−x

The effects of the parameter b

f (x) = 2x

f (x) = 3xf (x) = 4x

f (x) =( 1

2

)xf (x) =

( 13

)x

The effects of the parameter b

f (x) = 2xf (x) = 3x

f (x) = 4xf (x) =

( 12

)xf (x) =

( 13

)x

The effects of the parameter b

f (x) = 2xf (x) = 3xf (x) = 4x

f (x) =( 1

2

)xf (x) =

( 13

)x

The effects of the parameter b

f (x) = 2xf (x) = 3xf (x) = 4x

f (x) =( 1

2

)x

f (x) =( 1

3

)x

The effects of the parameter b

f (x) = 2xf (x) = 3xf (x) = 4x

f (x) =( 1

2

)xf (x) =

( 13

)x

End Behavior

f (x) = 2x

What happens to the curve as x→ ∞?

What happens to the curve as x→ −∞?

End Behavior

f (x) = 2x

What happens to the curve as x→ ∞?What happens to the curve as x→ −∞?

End Behavior

f (x) = 2x

What happens to the curve as x→ ∞?What happens to the curve as x→ −∞?

End Behavior

f (x) = 2−x

What happens to the curve as x→ ∞?

What happens to the curve as x→ −∞?

One end will always approach infinity. The horizontalasymptote will change, though, based on the transformation ...

End Behavior

f (x) = 2−x

What happens to the curve as x→ ∞?What happens to the curve as x→ −∞?

One end will always approach infinity. The horizontalasymptote will change, though, based on the transformation ...

End Behavior

f (x) = 2−x

What happens to the curve as x→ ∞?What happens to the curve as x→ −∞?

One end will always approach infinity. The horizontalasymptote will change, though, based on the transformation ...

End Behavior

f (x) = 2−x

What happens to the curve as x→ ∞?What happens to the curve as x→ −∞?

One end will always approach infinity. The horizontalasymptote will change, though, based on the transformation ...

Sketching Exponential Curves

We have already seen the effect of negatives and in changingsome parameters.

f (x) = abx−h + k

What effect does the parameter h have on our curve?

What effect does the parameter k have on the curve?

Sketching Exponential Curves

We have already seen the effect of negatives and in changingsome parameters.

f (x) = abx−h + k

What effect does the parameter h have on our curve?

What effect does the parameter k have on the curve?

Sketching Exponential Curves

We have already seen the effect of negatives and in changingsome parameters.

f (x) = abx−h + k

What effect does the parameter h have on our curve?

What effect does the parameter k have on the curve?

Sketching Exponential Curves

Example

Graph f (x) = 2 + 2 · 3x

before we go to the graph, we want to think about a few things:

y-intercept? f (0) = 2 + 2 · 30 = 4, so (0, 4)as x→ ∞, f (x)→ ∞as x→ −∞, f (x)→ 2

Sketching Exponential Curves

Example

Graph f (x) = 2 + 2 · 3x

before we go to the graph, we want to think about a few things:

y-intercept? f (0) = 2 + 2 · 30 = 4, so (0, 4)as x→ ∞, f (x)→ ∞as x→ −∞, f (x)→ 2

Sketching Exponential Curves

Example

Graph f (x) = 2 + 2 · 3x

before we go to the graph, we want to think about a few things:

y-intercept?

f (0) = 2 + 2 · 30 = 4, so (0, 4)as x→ ∞, f (x)→ ∞as x→ −∞, f (x)→ 2

Sketching Exponential Curves

Example

Graph f (x) = 2 + 2 · 3x

before we go to the graph, we want to think about a few things:

y-intercept? f (0) = 2 + 2 · 30 = 4, so (0, 4)

as x→ ∞, f (x)→ ∞as x→ −∞, f (x)→ 2

Sketching Exponential Curves

Example

Graph f (x) = 2 + 2 · 3x

before we go to the graph, we want to think about a few things:

y-intercept? f (0) = 2 + 2 · 30 = 4, so (0, 4)as x→ ∞, f (x)→

∞as x→ −∞, f (x)→ 2

Sketching Exponential Curves

Example

Graph f (x) = 2 + 2 · 3x

before we go to the graph, we want to think about a few things:

y-intercept? f (0) = 2 + 2 · 30 = 4, so (0, 4)as x→ ∞, f (x)→ ∞

as x→ −∞, f (x)→ 2

Sketching Exponential Curves

Example

Graph f (x) = 2 + 2 · 3x

before we go to the graph, we want to think about a few things:

y-intercept? f (0) = 2 + 2 · 30 = 4, so (0, 4)as x→ ∞, f (x)→ ∞as x→ −∞, f (x)→

2

Sketching Exponential Curves

Example

Graph f (x) = 2 + 2 · 3x

before we go to the graph, we want to think about a few things:

y-intercept? f (0) = 2 + 2 · 30 = 4, so (0, 4)as x→ ∞, f (x)→ ∞as x→ −∞, f (x)→ 2

Sketching Exponential Curves

Sketching Exponential Curves

Sketching Exponential Curves

Sketching Exponential Curves

Sketching Exponential Curves

Example

Graph f (x) = 3− 12

x.

Are we expecting an increasing or decreasingfunction?y-intercept? f (0) = 3− 1

20= 2, so (0, 2)

as x→ ∞, f (x)→ 3as x→ −∞, f (x)→−∞

Sketching Exponential Curves

Example

Graph f (x) = 3− 12

x.

Are we expecting an increasing or decreasingfunction?

y-intercept? f (0) = 3− 12

0= 2, so (0, 2)

as x→ ∞, f (x)→ 3as x→ −∞, f (x)→−∞

Sketching Exponential Curves

Example

Graph f (x) = 3− 12

x.

Are we expecting an increasing or decreasingfunction?y-intercept?

f (0) = 3− 12

0= 2, so (0, 2)

as x→ ∞, f (x)→ 3as x→ −∞, f (x)→−∞

Sketching Exponential Curves

Example

Graph f (x) = 3− 12

x.

Are we expecting an increasing or decreasingfunction?y-intercept? f (0) = 3− 1

20= 2, so (0, 2)

as x→ ∞, f (x)→ 3as x→ −∞, f (x)→−∞

Sketching Exponential Curves

Example

Graph f (x) = 3− 12

x.

Are we expecting an increasing or decreasingfunction?y-intercept? f (0) = 3− 1

20= 2, so (0, 2)

as x→ ∞, f (x)→

3as x→ −∞, f (x)→−∞

Sketching Exponential Curves

Example

Graph f (x) = 3− 12

x.

Are we expecting an increasing or decreasingfunction?y-intercept? f (0) = 3− 1

20= 2, so (0, 2)

as x→ ∞, f (x)→ 3

as x→ −∞, f (x)→−∞

Sketching Exponential Curves

Example

Graph f (x) = 3− 12

x.

Are we expecting an increasing or decreasingfunction?y-intercept? f (0) = 3− 1

20= 2, so (0, 2)

as x→ ∞, f (x)→ 3as x→ −∞, f (x)→

−∞

Sketching Exponential Curves

Example

Graph f (x) = 3− 12

x.

Are we expecting an increasing or decreasingfunction?y-intercept? f (0) = 3− 1

20= 2, so (0, 2)

as x→ ∞, f (x)→ 3as x→ −∞, f (x)→−∞

Sketching Exponential Curves

Sketching Exponential Curves

Sketching Exponential Curves

Sketching Exponential Curves