Transformations of Exponential Functions

An exponential function has a variable in the exponent. A basic exponential function can be written as:

y = bˣ

where b > 0 and b ≠ 1.

For example:

y = 2ˣ

Some points on this graph are:

Transformations change the position, orientation, or shape of this basic exponential graph while preserving its exponential nature.

The Parent Exponential Function

The simplest exponential function is often called the parent function:

f(x) = bˣ

For the parent function:

• the y-intercept is (0, 1)
• the horizontal asymptote is y = 0
• the domain is all real numbers
• the range is y > 0

If b > 1, the function shows exponential growth.

If 0 < b < 1, the function shows exponential decay.

Transformations begin with this basic graph and move or reshape it.

General Form of a Transformed Exponential Function

A useful general form is:

y = a · b^(x − h) + k

Each parameter has a different effect:

Understanding these parameters allows us to predict a graph directly from its equation.

Vertical Translations

Adding a number outside the exponential expression moves the graph vertically.

y = bˣ + k

If k > 0, the graph moves up.

If k < 0, the graph moves down.

Example

Start with:

y = 2ˣ

Now consider:

y = 2ˣ + 3

Every y-value increases by 3, so the entire graph moves 3 units upward.

The original horizontal asymptote was:

y = 0

The new horizontal asymptote is:

y = 3

Another Example

y = 2ˣ − 4

The graph moves 4 units downward.

The new horizontal asymptote is:

y = −4

Horizontal Translations

Changes inside the exponent move the graph horizontally.

The general form is:

y = b^(x − h)

The direction can seem backwards at first.

If:

y = 2^(x − 3)

the graph moves 3 units to the right.

If:

y = 2^(x + 3)

the graph moves 3 units to the left.

A useful rule is:

x − h → right

x + h → left

Example

Parent function:

y = 3ˣ

Transformed function:

y = 3^(x − 2)

The entire graph moves 2 units right.

The horizontal asymptote remains:

y = 0

because there has been no vertical translation.

Why Does the Horizontal Direction Look Reversed?

Consider:

y = 2^(x − 3)

In the parent function, the exponent equals zero when:

x = 0

In the transformed function:

x − 3 = 0

so:

x = 3

The point corresponding to the parent function's (0, 1) has therefore moved to:

(3, 1)

This shows that the graph has moved 3 units right.

Vertical Stretches

Multiplying the entire exponential expression by a number greater than 1 creates a vertical stretch.

For example:

y = 3(2ˣ)

Every y-value of the parent function is multiplied by 3.

The graph becomes steeper because its vertical distances from the x-axis are multiplied by 3.

Vertical Compressions

If the multiplier is between 0 and 1, the graph is vertically compressed.

For example:

y = 0.5(2ˣ)

Every y-value is multiplied by 0.5.

At x = 0:

Parent:

y = 2⁰ = 1

Transformed:

y = 0.5(1) = 0.5

The graph is therefore compressed vertically.

Reflections Across the x-Axis

A negative sign outside the exponential expression reflects the graph across the x-axis.

Parent:

y = 2ˣ

Reflection:

y = −2ˣ

Every positive y-value becomes negative.

For example:

The graph is essentially flipped vertically.

Reflections Across the y-Axis

Replacing x with −x reflects the graph across the y-axis.

For example:

y = 2^(−x)

This can also be written as:

y = (1/2)ˣ

The original function represents exponential growth.

After reflection, the graph represents exponential decay.

This illustrates an important relationship between exponential growth and decay.

Multiple Transformations

Several transformations can occur in the same equation.

Consider:

y = 2 · 3^(x − 4) + 5

Compare this with:

y = a · b^(x − h) + k

We can identify:

a = 2

b = 3

h = 4

k = 5

Therefore:

• vertical stretch by a factor of 2
• translation 4 units right
• translation 5 units up
• horizontal asymptote moves from y = 0 to y = 5

Example with Reflection

Consider:

y = −3 · 2^(x + 1) − 4

The transformations from y = 2ˣ are:

• reflection across the x-axis
• vertical stretch by a factor of 3
• translation 1 unit left
• translation 4 units down

The horizontal asymptote is:

y = −4

The negative value of a tells us that the graph has been reflected.

Tracking an Important Point

One useful way to understand transformations is to track a recognizable point.

For:

y = bˣ

the graph contains:

(0, 1)

For:

y = a · b^(x − h) + k

the corresponding point becomes:

(h, a + k)

Example

Consider:

y = 4 · 2^(x − 3) + 1

Here:

a = 4

h = 3

k = 1

So the parent point (0, 1) becomes:

(3, 5)

This gives us a useful point when sketching the transformed graph.

Determining an Equation from a Graph

Sometimes you are given a graph and must determine its equation.

Start with:

y = a · b^(x − h) + k

Then examine the graph systematically.

Find the Horizontal Asymptote

The horizontal asymptote gives k.

If the graph approaches:

y = 4

then:

k = 4

Look for Horizontal Translation

Determine whether the graph has shifted left or right compared with the parent function.

This helps identify h.

Determine Growth or Decay

If the graph rises as x increases, the base may represent exponential growth.

If it decreases toward the asymptote as x increases, it may represent exponential decay or involve a reflection.

Determine the Vertical Scale

Use a known point to determine a.

Worked Example: Finding an Equation

Suppose a graph has:

• horizontal asymptote y = 2
• exponential growth with base 3
• a recognizable transformed point at (1, 6)

Start with:

y = a · 3^(x − 1) + 2

Substitute the point (1, 6):

6 = a · 3^(1 − 1) + 2

6 = a · 3⁰ + 2

6 = a + 2

a = 4

Therefore:

y = 4 · 3^(x − 1) + 2

Comparing Transformations

Consider these functions:

A: y = 2ˣ

B: y = 2ˣ + 4

C: y = 2^(x − 4)

Although both B and C contain the number 4, they represent very different transformations.

Function B moves the graph:

4 units up

Function C moves the graph:

4 units right

This is why it is important to notice whether a number appears inside or outside the exponent.

Transformations and the Horizontal Asymptote

The parent exponential function:

y = bˣ

has horizontal asymptote:

y = 0

For:

y = a · b^(x − h) + k

the horizontal asymptote becomes:

y = k

Neither a horizontal translation nor a vertical stretch changes the y-value of the asymptote.

The vertical translation determines its new position.

Domain and Range

For:

y = a · b^(x − h) + k

the domain remains:

all real numbers

The range depends on the direction of the graph.

If:

a > 0

then:

y > k

If:

a < 0

then:

y < k

For example:

y = 3 · 2ˣ + 5

has range:

y > 5

But:

y = −3 · 2ˣ + 5

has range:

y < 5

Modelling with Transformed Exponential Functions

Transformed exponential functions can represent situations in which exponential growth or decay occurs relative to a starting level or limiting value.

For example, suppose the temperature difference between an object and its surroundings decreases exponentially.

A model might look like:

T = 20 + 60(0.8)ᵗ

Here:

• t represents time
• 20 represents the surrounding temperature
• 60 represents the initial difference between the object and its surroundings
• 0.8 represents exponential decay

As time increases, the exponential part approaches zero.

Therefore:

T approaches 20

The horizontal asymptote has a real-world meaning: it represents the temperature that the object approaches.

Another Modelling Example

Suppose a population is modelled by:

P = 500(1.12)ᵗ + 2000

Here:

500 determines the initial size of the exponential component.

1.12 represents a growth factor of 12% per time period.

2000 vertically translates the exponential model.

At t = 0:

P = 500(1.12)⁰ + 2000

P = 500 + 2000

P = 2500

The transformations therefore affect both the graph and the interpretation of the model.

Common Mistakes

Moving horizontal translations in the wrong direction

For:

y = 2^(x − 3)

the graph moves right, not left.

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Confusing vertical and horizontal translations

y = 2ˣ + 3 means up 3.

y = 2^(x + 3) means left 3.

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Thinking a negative a-value means exponential decay

For:

y = −2ˣ

the negative sign reflects the graph across the x-axis. It does not change the base from growth to decay.

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Forgetting that vertical translations move the asymptote

For:

y = 2ˣ − 7

the horizontal asymptote is:

y = −7

not y = 0.

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Ignoring multiple transformations

A function such as:

y = −4 · 3^(x − 2) + 6

contains several transformations. Analyse each parameter separately.

Key Terms

Exponential function — a function in which the independent variable appears in an exponent.

Parent function — the simplest form of a function used as the starting point for transformations.

Transformation — a change in the position, orientation, or scale of a graph.

Translation — a movement of a graph without changing its basic shape.

Horizontal translation — movement of a graph left or right.

Vertical translation — movement of a graph up or down.

Reflection — a transformation that flips a graph across a line such as the x-axis or y-axis.

Vertical stretch — a transformation that increases vertical distances on a graph by a scale factor greater than 1.

Vertical compression — a transformation that decreases vertical distances by a scale factor between 0 and 1.

Horizontal asymptote — a horizontal line that a graph approaches as x increases or decreases.

Growth factor — the multiplicative factor associated with exponential increase.

Decay factor — a factor between 0 and 1 associated with exponential decrease.

Domain — all allowable input values of a function.

Range — all possible output values of a function.

Key Takeaways

• The parent exponential function can be written as y = bˣ.
• A transformed exponential function can often be written as y = a · b^(x − h) + k.
• h controls horizontal translation.
• k controls vertical translation.
• A positive h in (x − h) moves the graph right.
• Multiplying by a creates a vertical stretch or compression.
• A negative a reflects the graph across the x-axis.
• Replacing x with −x reflects the graph across the y-axis.
• The horizontal asymptote of y = a · b^(x − h) + k is y = k.
• Multiple transformations can occur in the same function.
• Graph features such as the asymptote and known points can be used to determine an equation.
• Transformed exponential functions can model growth or decay around meaningful real-world values.

Check Your Understanding

1. Describe the transformation from y = 2ˣ to y = 2ˣ + 5.
2. Describe the transformation from y = 3ˣ to y = 3^(x − 4).
3. What is the horizontal asymptote of y = 5ˣ − 7?
4. Describe the transformation produced by y = −2ˣ.
5. What transformation occurs from y = 2ˣ to y = 4(2ˣ)?

6. Describe all transformations in:

   y = 3 · 2^(x − 5) + 4

7. Describe all transformations in:

   y = −2 · 5^(x + 3) − 6

8. Determine the horizontal asymptote and range of:

   y = 4 · 3ˣ + 2

9. Determine the horizontal asymptote and range of:

   y = −4 · 3ˣ + 2

10. A transformed exponential function has base 2, horizontal asymptote y = 3, and is shifted 4 units right. Write a possible equation for the function.
11. Explain the difference between y = 2ˣ + 3 and y = 2^(x + 3).
12. A graph has the form y = a · 2^(x − 1) + 4 and passes through (1, 7). Determine a and write the complete equation.