Sequences and Exponential Functions

Arithmetic Sequences are patterns of numbers where the same amount is added or subtracted each time.

They are important because they model steady, linear growth. For example, if you save the same number of dollars every week, or if a stairway goes up by the same height each step, the pattern can often be described by an arithmetic sequence.

In this lesson, you will learn how to:

• recognize an arithmetic sequence,
• find the common difference,
• write a sequence recursively,
• write a sequence explicitly,
• and find any term in the sequence.

1. What is an arithmetic sequence?

An arithmetic sequence is a sequence in which the difference between consecutive terms is always the same.

This fixed amount is called the common difference.

For example, in the sequence

\(4, 7, 10, 13, 16, \dots\)

each term increases by \(3\). So the common difference is \(d = 3\).

To check whether a sequence is arithmetic, subtract consecutive terms:

• \(7 - 4 = 3\)
• \(10 - 7 = 3\)
• \(13 - 10 = 3\)

Because the difference is always \(3\), this is an arithmetic sequence.

2. Arithmetic growth vs. other kinds of growth

Arithmetic sequences show additive growth. That means you add or subtract the same amount each time.

For example:

• Arithmetic: \(5, 8, 11, 14, 17\) — add \(3\) each time
• Not arithmetic: \(2, 4, 8, 16\) — multiply by \(2\) each time

The second pattern is exponential, not arithmetic, because it grows by multiplying, not by adding.

3. Important vocabulary

• Sequence: an ordered list of numbers
• Term: a number in the sequence
• First term: the starting number, often called \(a_1\)
• Common difference: the amount added or subtracted each time, called \(d\)
• Recursive formula: a rule that uses the term before it
• Explicit formula: a rule that gives any term directly

4. Recursive form

A recursive formula tells you how to get the next term from the previous term.

For an arithmetic sequence, the recursive form is:

$$a_n = a_{n-1} + d \quad \text{for } n \ge 2$$

You must also state the first term:

$$a_1 = \text{first term}$$

Example: For \(4, 7, 10, 13, \dots\)

the recursive rule is

$$a_1 = 4$$

$$a_n = a_{n-1} + 3$$

This means:

• start at \(4\),
• then keep adding \(3\).

5. Explicit form

An explicit formula lets you find any term without listing all the terms before it.

The explicit formula for an arithmetic sequence is:

$$a_n = a_1 + (n-1)d$$

Here:

• \(a_n\) is the \(n\)th term,
• \(a_1\) is the first term,
• \(d\) is the common difference,
• \(n\) is the term number.

This formula works because to get from the first term to the \(n\)th term, you add the common difference \((n-1)\) times.

6. Worked Example 1: Identify the common difference

Consider the sequence:

\(12, 17, 22, 27, 32, \dots\)

Step 1: Subtract consecutive terms.

• \(17 - 12 = 5\)
• \(22 - 17 = 5\)
• \(27 - 22 = 5\)

Step 2: Decide whether the difference stays constant.

Yes, the difference is always \(5\), so this is an arithmetic sequence.

Answer: The common difference is \(d = 5\).

7. Worked Example 2: Write recursive and explicit rules

Write both formulas for the sequence

\(3, 8, 13, 18, 23, \dots\)

Step 1: Find the first term and common difference.

• First term: \(a_1 = 3\)
• Common difference: \(8 - 3 = 5\), so \(d = 5\)

Recursive formula:

$$a_1 = 3$$

$$a_n = a_{n-1} + 5$$

Explicit formula:

Use $$a_n = a_1 + (n-1)d$$

Substitute \(a_1 = 3\) and \(d = 5\):

$$a_n = 3 + (n-1)(5)$$

Simplify:

$$a_n = 3 + 5n - 5$$

$$a_n = 5n - 2$$

Answer:

Recursive: $$a_1 = 3, \quad a_n = a_{n-1} + 5$$

Explicit: $$a_n = 3 + 5(n-1) \quad \text{or} \quad a_n = 5n - 2$$

8. Worked Example 3: Find a term far into the sequence

Find the \(20\)th term of the arithmetic sequence

\(7, 11, 15, 19, \dots\)

Step 1: Identify values.

• \(a_1 = 7\)
• \(d = 4\)
• \(n = 20\)

Step 2: Use the explicit formula.

$$a_n = a_1 + (n-1)d$$

$$a_{20} = 7 + (20-1)(4)$$

$$a_{20} = 7 + 19 \cdot 4$$

$$a_{20} = 7 + 76$$

$$a_{20} = 83$$

Answer: The \(20\)th term is \(83\).

This shows why the explicit formula is useful. It helps you jump straight to the term you want.

9. Worked Example 4: Arithmetic sequence with a negative difference

Consider the sequence

\(30, 24, 18, 12, 6, \dots\)

Step 1: Find the common difference.

• \(24 - 30 = -6\)
• \(18 - 24 = -6\)

So \(d = -6\).

Step 2: Write the explicit formula.

Here, \(a_1 = 30\) and \(d = -6\).

$$a_n = a_1 + (n-1)d$$

$$a_n = 30 + (n-1)(-6)$$

$$a_n = 30 - 6(n-1)$$

You could also simplify:

$$a_n = 30 - 6n + 6$$

$$a_n = 36 - 6n$$

Step 3: Find the 6th term.

$$a_6 = 30 + (6-1)(-6)$$

$$a_6 = 30 + 5(-6)$$

$$a_6 = 30 - 30$$

$$a_6 = 0$$

Answer: The explicit formula is $$a_n = 30 - 6(n-1)$$ and the 6th term is \(0\).

10. How to write an arithmetic sequence from a situation

Arithmetic sequences often come from real-life patterns.

Example: A student has \(\$10\) in savings and adds \(\$4\) every week.

The amount of money after each week forms an arithmetic sequence:

\(10, 14, 18, 22, 26, \dots\)

Here:

• first term: \(a_1 = 10\)
• common difference: \(d = 4\)

The recursive formula is

$$a_1 = 10$$

$$a_n = a_{n-1} + 4$$

The explicit formula is

$$a_n = 10 + (n-1)4$$

$$a_n = 10 + 4(n-1)$$

This means the savings grow by the same amount each week, so the pattern is arithmetic.

11. Common mistakes to avoid

• Mixing up adding and multiplying: If the pattern multiplies, it is not arithmetic.
• Forgetting \((n-1)\): The explicit formula is $$a_n = a_1 + (n-1)d$$ not just \(a_1 + nd\).
• Using the wrong common difference: Always subtract consecutive terms carefully.
• Not stating the first term in recursive form: A recursive rule needs a starting point.

12. Quick check for understanding

1. Is \(2, 6, 10, 14, \dots\) arithmetic? What is the common difference?
2. Write a recursive rule for \(9, 12, 15, 18, \dots\).
3. Write an explicit formula for \(5, 9, 13, 17, \dots\).
4. Find the 15th term of \(1, 4, 7, 10, \dots\).

Answers:

1. Yes. The common difference is \(4\).
2. $$a_1 = 9, \quad a_n = a_{n-1} + 3$$
3. $$a_n = 5 + (n-1)4$$
4. $$a_{15} = 1 + (15-1)3 = 1 + 42 = 43$$

13. Summary

An arithmetic sequence is a number pattern with the same common difference between consecutive terms.

You can describe it recursively by telling how to get the next term from the previous one:

$$a_1 = \text{first term}, \quad a_n = a_{n-1} + d$$

You can also describe it explicitly with a formula for any term:

$$a_n = a_1 + (n-1)d$$

If the sequence changes by adding or subtracting the same amount each time, it is arithmetic. Recognizing this pattern helps you model linear growth in math and in real-life situations.

Geometric Sequences are sequences where each term is found by multiplying the previous term by the same number each time.

That multiplying number is called the common ratio. Geometric sequences are important because they show multiplicative change, which is different from arithmetic sequences, where you add or subtract the same amount each time.

For example, in the sequence \(3, 6, 12, 24, 48, \dots\), each term is multiplied by \(2\). So this is a geometric sequence, and the common ratio is \(2\).

This idea connects to exponential growth and decay. If you keep multiplying by a number greater than \(1\), the terms grow. If you keep multiplying by a number between \(0\) and \(1\), the terms shrink.

How to recognize a geometric sequence

• Look at how one term changes to the next.
• If you multiply by the same number each time, it is geometric.
• If you add or subtract the same number each time, it is arithmetic, not geometric.

To find the common ratio, divide a term by the term before it:

$$r = \frac{\text{next term}}{\text{previous term}}$$

For the sequence \(5, 15, 45, 135, \dots\):

$$\frac{15}{5} = 3, \quad \frac{45}{15} = 3, \quad \frac{135}{45} = 3$$

Since the ratio is always \(3\), the sequence is geometric with common ratio \(r=3\).

Recursive formula for a geometric sequence

A recursive formula tells how to get each term from the one before it.

If the first term is \(a_1\) and the common ratio is \(r\), then:

$$a_n = r\cdot a_{n-1} \quad \text{for } n \ge 2$$

You must also know the first term.

So a recursive geometric sequence is usually written as:

$$a_1 = \text{first term}, \quad a_n = r\cdot a_{n-1}$$

For example, if a sequence starts at \(4\) and each term is multiplied by \(3\), then:

$$a_1 = 4, \quad a_n = 3a_{n-1}$$

This gives the terms:

\(4, 12, 36, 108, \dots\)

Explicit formula for a geometric sequence

An explicit formula lets you find any term directly, without writing all the terms before it.

If the first term is \(a_1\) and the common ratio is \(r\), then the explicit formula is:

$$a_n = a_1r^{n-1}$$

This formula works because:

• The first term has no extra multiplication, so it is just \(a_1\).
• The second term is \(a_1r\).
• The third term is \(a_1r^2\).
• The \(n\)-th term is \(a_1r^{n-1}\).

This is why geometric sequences are closely connected to exponential functions.

Worked Example 1: Identify the common ratio

Sequence: \(2, 10, 50, 250, \dots\)

Step 1: Divide each term by the one before it.

$$\frac{10}{2}=5, \quad \frac{50}{10}=5, \quad \frac{250}{50}=5$$

Step 2: The ratio is always the same, so the sequence is geometric.

Answer: The common ratio is \(5\).

Worked Example 2: Write recursive and explicit formulas

Sequence: \(7, 14, 28, 56, \dots\)

Step 1: Find the first term and common ratio.

First term: \(a_1=7\)

Common ratio:

$$\frac{14}{7}=2$$

So \(r=2\).

Step 2: Write the recursive formula.

$$a_1=7, \quad a_n=2a_{n-1}$$

Step 3: Write the explicit formula.

$$a_n=7\cdot 2^{n-1}$$

Answer:

• Recursive: \(a_1=7, \ a_n=2a_{n-1}\)
• Explicit: \(a_n=7\cdot 2^{n-1}\)

Worked Example 3: Find a specific term

Find the 6th term of the geometric sequence with \(a_1=3\) and \(r=4\).

Use the explicit formula:

$$a_n=a_1r^{n-1}$$

Substitute the values:

$$a_6=3\cdot 4^{6-1}=3\cdot 4^5$$

$$4^5=1024$$

$$a_6=3\cdot 1024=3072$$

Answer: The 6th term is \(3072\).

Worked Example 4: Geometric decay

A bacteria sample is cut in half every hour. It starts with \(160\) bacteria. Write a geometric sequence for the number of bacteria after each hour, and find the number after 4 hours.

Step 1: Identify the first term and ratio.

• First term: \(a_1=160\)
• Cut in half means multiply by \(\frac{1}{2}\)
• So the common ratio is \(r=\frac{1}{2}\)

Step 2: Write the sequence.

\(160, 80, 40, 20, 10, \dots\)

Step 3: Write the explicit formula.

$$a_n=160\left(\frac{1}{2}\right)^{n-1}$$

Step 4: Find the 5th term if the starting amount is term 1 and each new hour gives the next term.

After 4 hours, we are at the 5th term.

$$a_5=160\left(\frac{1}{2}\right)^4$$

$$a_5=160\cdot \frac{1}{16}=10$$

Answer: After 4 hours, there are \(10\) bacteria.

Arithmetic vs. Geometric Sequences

• Arithmetic: add or subtract the same number each time.
• Geometric: multiply by the same number each time.

Compare these:

• Arithmetic: \(4, 7, 10, 13, \dots\) because you add \(3\)
• Geometric: \(4, 12, 36, 108, \dots\) because you multiply by \(3\)

This difference matters because arithmetic sequences model constant difference, while geometric sequences model constant ratio.

Important things to remember

• The common ratio can be a whole number, fraction, or negative number.
• If the ratio is negative, the signs of the terms alternate.
• If \(r>1\), the sequence grows.
• If \(0<r<1\), the sequence shrinks.
• If \(r=1\), every term stays the same.

For example, the sequence \(5, -10, 20, -40, \dots\) is geometric because each term is multiplied by \(-2\).

Common mistakes

• Confusing adding with multiplying.
• Forgetting that the explicit formula uses \(n-1\), not just \(n\).
• Using subtraction to check a geometric sequence instead of division.
• Mixing up the first term \(a_1\) with the common ratio \(r\).

Quick check

1. Is \(6, 18, 54, 162, \dots\) geometric? Yes, because the common ratio is \(3\).
2. What is the explicit formula if \(a_1=5\) and \(r=2\)?
   $$a_n=5\cdot 2^{n-1}$$
3. What is the 4th term of \(2, 6, 18, 54, \dots\)?
   The 4th term is \(54\).

Summary

A geometric sequence is a sequence with a constant common ratio. You can describe it recursively with a first term and a rule that multiplies by the ratio, or explicitly with the formula $$a_n=a_1r^{n-1}$$. Geometric sequences are a way to model repeated multiplication, so they are closely connected to exponential growth and decay.

Exponential Function Structure helps us understand patterns that grow or shrink by the same factor over equal steps.

In 9th Grade, this is important because it helps you tell the difference between arithmetic patterns, which change by adding or subtracting the same amount, and exponential patterns, which change by multiplying by the same number.

A common exponential function looks like this:

$$y=a(b)^x$$

Each part of the function has a job:

• 0 is the starting amount, or the value of the function when \(x=0\).
• 1 is the base, or growth/decay factor.
• 2 tells how many times the factor is applied.

To understand the structure of an exponential function, focus on the two main numbers: the initial value \(a\) and the base \(b\).

1. The initial value \(a\)

The number \(a\) is the amount you start with. It is the value of \(y\) when \(x=0\), because:

$$y=a(b)^0=a(1)=a$$

So if a function is \(y=5(2)^x\), the starting value is 5.

If a function is \(y=12(0.8)^x\), the starting value is 12.

2. The base \(b\)

The base tells how the function changes each time \(x\) increases by 1.

• If \(b>1\), the function shows exponential growth.
• If \(0<b<1\), the function shows exponential decay.

This is one of the most important ideas in exponential function structure.

Why? Because the base tells whether the values get larger or smaller over time.

For example:

• In \(y=3(2)^x\), the base is 2. Since \(2>1\), this is growth.
• In \(y=7(0.5)^x\), the base is 0.5. Since \(0<0.5<1\), this is decay.

How exponential patterns differ from arithmetic patterns

An arithmetic pattern adds or subtracts the same amount each time.

For example, in the sequence 4, 7, 10, 13, each term increases by 3.

An exponential pattern multiplies by the same factor each time.

For example, in the sequence 4, 8, 16, 32, each term is multiplied by 2.

So when you look at an exponential function, ask yourself:

• What is the starting value?
• What number is the pattern multiplied by each step?
• Is the base greater than 1 or between 0 and 1?

Recognizing growth and decay

Here is a quick way to decide whether \(y=a(b)^x\) is growth or decay:

• If \(b=2\), \(b=3\), or \(b=1.1\), it is growth.
• If \(b=0.9\), \(b=0.4\), or \(b=\frac{1}{2}\), it is decay.

Notice that a decimal less than 1 means the output becomes a fraction of the previous amount.

For example, multiplying by 0.5 means taking half each time. Multiplying by 0.8 means keeping 80% each time.

Looking at values in a table

You can also recognize exponential structure in a table.

Suppose we have this function:

$$y=2(3)^x$$

If we substitute values of \(x\):

• When \(x=0\), \(y=2(3)^0=2\)
• When \(x=1\), \(y=2(3)^1=6\)
• When \(x=2\), \(y=2(3)^2=18\)
• When \(x=3\), \(y=2(3)^3=54\)

The outputs are 2, 6, 18, 54.

These are not increasing by the same amount, but they are being multiplied by 3 each time. That is exponential growth.

Now look at:

$$y=10(0.5)^x$$

• When \(x=0\), \(y=10\)
• When \(x=1\), \(y=5\)
• When \(x=2\), \(y=2.5\)
• When \(x=3\), \(y=1.25\)

The outputs are cut in half each time, so this is exponential decay.

Worked Example 1: Identify the starting value and base

Consider the function:

$$y=4(5)^x$$

Step 1: Find the starting value.

The number in front is \(a=4\), so the starting value is 4.

Step 2: Find the base.

The base is \(b=5\).

Step 3: Decide if it is growth or decay.

Since \(5>1\), the function shows exponential growth.

Answer: Starting value = 4, base = 5, growth model.

Worked Example 2: Identify decay

Consider the function:

$$y=9(0.2)^x$$

Step 1: Find the starting value.

\(a=9\), so the function starts at 9.

Step 2: Find the base.

\(b=0.2\).

Step 3: Decide if it is growth or decay.

Because \(0<0.2<1\), the function is an exponential decay model.

Answer: Starting value = 9, base = 0.2, decay model.

Worked Example 3: Use a table to find the factor

A pattern has values:

• \(x=0\), \(y=6\)
• \(x=1\), \(y=12\)
• \(x=2\), \(y=24\)
• \(x=3\), \(y=48\)

Step 1: Find the starting value.

When \(x=0\), \(y=6\), so \(a=6\).

Step 2: Find the multiplying factor.

$$\frac{12}{6}=2,\quad \frac{24}{12}=2,\quad \frac{48}{24}=2$$

The values are multiplied by 2 each time, so \(b=2\).

Step 3: Write the function.

$$y=6(2)^x$$

Step 4: Classify it.

Since \(2>1\), it is exponential growth.

Worked Example 4: Decide whether a function is exponential growth or decay

Look at the function:

$$y=15\left(\frac{3}{4}\right)^x$$

Step 1: Identify the base.

The base is \(\frac{3}{4}=0.75\).

Step 2: Compare the base to 1.

Since \(0<0.75<1\), this is exponential decay.

Step 3: Identify the starting value.

The starting value is 15.

Answer: Starting value = 15, base = \(\frac{3}{4}\), decay model.

Common mistakes to avoid

• Mixing up \(a\) and \(b\): In \(y=a(b)^x\), the number in front is the starting value. The number inside parentheses is the base.
• Thinking all increasing patterns are exponential: A pattern is only exponential if it is multiplied by the same factor each time.
• Forgetting that decimals less than 1 mean decay: A base like 0.6 or 0.95 gives decay, not growth.
• Looking for equal differences instead of equal factors: Exponential patterns use multiplication, not repeated addition.

Helpful checklist

When you see an equation in the form \(y=a(b)^x\), ask:

1. What is the starting value \(a\)?
2. What is the base \(b\)?
3. Is \(b>1\) or is \(0<b<1\)?
4. Does the pattern represent growth or decay?

Brief Summary

An exponential function has the form $$y=a(b)^x$$ where \(a\) is the starting value and \(b\) is the multiplying factor.

If \(b>1\), the function shows exponential growth. If \(0<b<1\), the function shows exponential decay.

To recognize exponential structure, look for a pattern that changes by the same factor each step, not the same difference.

Graphing Exponential Models

In 9th Grade Maths, you learn that some patterns grow by adding the same amount each time, while others grow by multiplying by the same factor each time. When a pattern changes by multiplication, it can often be modeled by an exponential function.

This lesson will show you how to graph exponential models and how to identify important features like the domain, range, y-intercept, and horizontal asymptote.

An exponential model often looks like this:

In this form:

•  is the starting value, because when \(x=0\), the function becomes \(y=a\).

• \(b\) is the growth or decay factor.

• \(x\) is the input, often representing time or number of steps.

If \(b>1\), the graph shows exponential growth.

If \(0<b<1\), the graph shows exponential decay.

For example:

• \(y=2(3)^x\) is growth because 3 is greater than 1.

• \(y=5\left(\frac{1}{2}\right)^x\) is decay because \(\frac{1}{2}\) is between 0 and 1.

What exponential graphs look like

Exponential graphs are not straight lines. They are curved.

• In exponential growth, the graph rises slowly at first and then more quickly.

• In exponential decay, the graph falls quickly at first and then levels off.

Most basic exponential graphs have a horizontal asymptote at:

A horizontal asymptote is a line that the graph gets closer and closer to, but does not cross in the basic models you graph in this lesson.

Key features to identify

1. y-intercept

   The y-intercept happens when \(x=0\).

   For \(y=a(b)^x\):

   $$y=a(b)^0=a(1)=a$$

   So the y-intercept is always:

   $$ (0,a) $$

2. Domain

   The domain is the set of all possible \(x\)-values.

   For exponential functions, the domain is usually all real numbers.

   $$(-\infty,\infty)$$

3. Range

   The range is the set of all possible \(y\)-values.

   For basic exponential models with \(a>0\), the graph stays above the x-axis, so:

   $$ (0,\infty) $$

   This means \(y\) is always positive.

4. Horizontal asymptote

   For the basic form \(y=a(b)^x\), the horizontal asymptote is:

   $$y=0$$

Steps for graphing an exponential model

1. Identify whether the function shows growth or decay by looking at \(b\).

2. Find the y-intercept by substituting \(x=0\).

3. Make a table of a few values, such as \(x=-2,-1,0,1,2\).

4. Plot the points carefully.

5. Draw a smooth curve through the points.

6. Show the horizontal asymptote, usually \(y=0\).

Worked Example 1: Basic exponential growth

Graph:

Step 1: Decide growth or decay

Since the base is 2, and \(2>1\), this is exponential growth.

Step 2: Make a table

Step 3: Plot the points

Plot:

• \((-2,\frac{1}{4})\)

• \((-1,\frac{1}{2})\)

• \((0,1)\)

• \((1,2)\)

• \((2,4)\)

Step 4: Describe the graph

• It rises from left to right.

• It crosses the y-axis at \((0,1)\).

• It gets very close to the x-axis on the left.

• The horizontal asymptote is \(y=0\).

Key features

• Domain: \(( -\infty, \infty )\)

• Range: \((0,\infty)\)

• y-intercept: \((0,1)\)

• Horizontal asymptote: \(y=0\)

Worked Example 2: Exponential decay

Graph:

Step 1: Decide growth or decay

Since \(\frac{1}{2}\) is between 0 and 1, this is exponential decay.

Step 2: Find the y-intercept

So the y-intercept is:

Step 3: Make a table

Step 4: Describe the graph

• It falls from left to right.

• It crosses the y-axis at \((0,3)\).

• As \(x\) gets larger, the graph gets closer to 0.

• The horizontal asymptote is \(y=0\).

Key features

• Domain: \(( -\infty, \infty )\)

• Range: \((0,\infty)\)

• y-intercept: \((0,3)\)

• Horizontal asymptote: \(y=0\)

Worked Example 3: Using the starting value

Graph:

This function has a starting value of 5 and doubles each time \(x\) increases by 1.

Step 1: Find the y-intercept

So the y-intercept is:

Step 2: Make a table

Step 3: Graph and describe

• The graph shows growth because the base is 2.

• It starts at 5 on the y-axis.

• It increases faster and faster as \(x\) increases.

• It approaches \(y=0\) on the left side.

Key features

• Domain: \(( -\infty, \infty )\)

• Range: \((0,\infty)\)

• y-intercept: \((0,5)\)

• Horizontal asymptote: \(y=0\)

How exponential graphs are different from arithmetic patterns

It is important to connect graphs to patterns.

• An arithmetic pattern adds the same amount each time. Its graph is a straight line.

• An exponential pattern multiplies by the same factor each time. Its graph is curved.

For example:

• Arithmetic: 2, 5, 8, 11, 14 adds 3 each time.

• Exponential: 2, 6, 18, 54, 162 multiplies by 3 each time.

When you graph an exponential model, look for a curve that changes more and more quickly or slows toward 0.

Common mistakes to avoid

• Mixing up growth and decay: If the base is bigger than 1, it is growth. If the base is between 0 and 1, it is decay.

• Forgetting the y-intercept: Always plug in \(x=0\).

• Drawing a straight line: Exponential graphs are curved, not linear.

• Saying the graph touches the asymptote: It gets closer and closer to the asymptote.

• Using the wrong range: For these basic models, the graph stays above 0, so the range is \((0,\infty)\).

Quick check questions

1. Is \(y=4(3)^x\) growth or decay?

2. What is the y-intercept of \(y=7\left(\frac{1}{4}\right)^x\)?

3. What is the horizontal asymptote of \(y=2^x\)?

4. What is the domain of \(y=5(2)^x\)?

Answers

1. Growth, because 3 is greater than 1.

2. \((0,7)\)

3. \(y=0\)

4. \(( -\infty, \infty )\)

Summary

An exponential model has the form \(y=a(b)^x\). The value of \(a\) gives the y-intercept, and the value of \(b\) tells whether the graph shows growth or decay.

To graph an exponential function, make a table of values, plot the points, and draw a smooth curve. For basic exponential models, the domain is all real numbers, the range is positive values, the y-intercept is \((0,a)\), and the horizontal asymptote is \(y=0\).

Comparing Linear and Exponential Growth

In math, many situations involve quantities that grow over time. Two important kinds of growth are linear growth and exponential growth.

Linear growth means a quantity increases by the same amount each step. Exponential growth means a quantity increases by the same factor or same percent each step.

This lesson will help you understand how these two types of growth compare, how to recognize them in tables and equations, and why an exponential quantity will eventually grow larger than any linear quantity if you continue far enough.

1. What is linear growth?

A linear pattern changes by adding or subtracting a constant amount each time. This constant amount is called the common difference.

A linear sequence might look like:

Each term increases by 3, so this is linear growth.

A linear function is often written as:

Here:

• \(m\) is the rate of change, or how much the output increases when the input increases by 1.
• \(b\) is the starting value.

For example, in

the quantity grows by 4 each step.

2. What is exponential growth?

An exponential pattern changes by multiplying by the same number each time. This number is called the common ratio.

An exponential sequence might look like:

Each term is multiplied by 2, so this is exponential growth.

An exponential function is often written as:

Here:

• \(a\) is the starting value when \(x=0\).
• \(b\) is the growth factor.

For example, in

the starting value is 5, and the quantity doubles each time \(x\) increases by 1.

3. The key difference: adding versus multiplying

This is the most important idea:

• Linear growth adds the same amount again and again.
• Exponential growth multiplies by the same factor again and again.

At first, linear growth can seem faster, especially if it starts with a bigger value. But exponential growth speeds up more and more because each new increase is based on a larger amount than before.

That is why exponential growth will eventually surpass linear growth.

4. Comparing values in a table

One good way to compare linear and exponential growth is to make a table of values.

Suppose we compare:

Now compute some values.

At first, the linear function is larger. But by \(x=7\), the exponential function has become larger.

This shows the big idea: even if exponential growth starts smaller, it can eventually overtake linear growth.

5. Why does exponential growth eventually win?

In a linear function, the amount added each time stays constant. For example, if a function grows by 10 each step, then the increases are:

In an exponential function, the increases themselves get bigger and bigger. For example, for \(2^x\):

• From \(1\) to \(2\), it increases by 1.
• From \(2\) to \(4\), it increases by 2.
• From \(4\) to \(8\), it increases by 4.
• From \(8\) to \(16\), it increases by 8.
• From \(16\) to \(32\), it increases by 16.

The increases keep getting larger, so the exponential function eventually catches up and passes the linear function.

6. Recursive patterns

Sometimes sequences are described recursively, which means each term is based on the one before it.

A linear recursive pattern uses addition:

This means the sequence starts at 4 and adds 3 each time:

An exponential recursive pattern uses multiplication:

This means the sequence starts at 4 and multiplies by 2 each time:

When comparing recursive patterns, ask:

• Are we adding the same amount each time? Then it is linear.
• Are we multiplying by the same number each time? Then it is exponential.

7. Worked Example 1: Identify the type of growth

Decide whether each sequence is linear or exponential.

(a) \(6, 11, 16, 21, 26\)

Check the differences:

The same amount, 5, is added each time. So this is linear.

(b) \(3, 9, 27, 81\)

Check the ratios:

The same factor, 3, is used each time. So this is exponential.

8. Worked Example 2: Compare a linear and an exponential function

Compare:

Find values for both functions.

At \(x=2\), the linear value is larger: \(18>9\).

At \(x=3\), the exponential value is larger: \(27>25\).

So the exponential function surpasses the linear function at \(x=3\).

9. Worked Example 3: Compare recursive sequences

Sequence A is defined by:

Sequence B is defined by:

Write the first few terms.

Sequence A:

Sequence B:

At first:

• Term 1: equal
• Term 2: B is slightly larger
• Later terms: B becomes much larger

Sequence A grows linearly because it adds 4 each time. Sequence B grows exponentially because it doubles each time.

This example clearly shows how exponential growth quickly becomes much greater than linear growth.

10. Worked Example 4: A real-world style problem

Two students are saving money.

• Maria starts with \(\$20\) and saves \(\$15\) each week.
• Jalen starts with \(\$1\), and his money doubles each week.

We can model their savings after \(x\) weeks as:

Let us compare values.

After 7 weeks, Jalen has more money than Maria.

Maria’s savings grow steadily by the same amount. Jalen’s savings start very small, but doubling each week causes the amount to grow very quickly. This is exponential growth passing linear growth.

11. Important idea: “Eventually” does not mean “immediately”

When we say exponential growth eventually surpasses linear growth, we mean that if the pattern continues long enough, the exponential quantity will become greater.

This does not mean exponential growth is always bigger at the beginning.

In fact, many exponential functions start smaller than linear functions. The important idea is what happens over time.

12. How to compare linear and exponential growth step by step

1. Identify each pattern as linear or exponential.
2. For linear patterns, look for a constant difference.
3. For exponential patterns, look for a constant ratio.
4. Make a table of values if needed.
5. Compare the outputs.
6. Look for when the exponential values become greater than the linear values.

13. Common mistakes to avoid

• Confusing addition and multiplication. A pattern that adds 5 each time is linear, not exponential.
• Looking only at the first few terms. Exponential growth may not seem larger at first.
• Forgetting the starting value matters at the beginning. A linear function can start bigger, but exponential growth can still overtake it later.
• Assuming all fast growth is exponential. To be exponential, the quantity must multiply by the same factor each step.

14. Quick check questions

Try these on your own:

1. Is the sequence \(4, 8, 12, 16, 20\) linear or exponential?
2. Is the sequence \(2, 6, 18, 54\) linear or exponential?
3. Which will eventually be larger: \(5x+100\) or \(2^x\)?
4. If one quantity adds 12 each day and another multiplies by 2 each day, which one will eventually grow faster?

Answers:

1. Linear
2. Exponential
3. \(2^x\)
4. The one that multiplies by 2 each day

15. Summary

Linear growth means adding the same amount each time. Exponential growth means multiplying by the same factor each time.

Linear growth is steady, but exponential growth speeds up because its increases get bigger and bigger. Even if an exponential pattern starts smaller than a linear one, it will eventually surpass the linear pattern if you continue far enough.

When comparing the two, use differences, ratios, equations, recursive rules, or tables of values to decide what type of growth you have and when one becomes greater than the other.