Exponential functions of the form y equals a times b to the x, their tables of values and their graphs. Includes the rising curve when the base exceeds one and the falling curve when it lies between zero and one, the role of a as the y-intercept, why the curve never meets the horizontal axis, and how such a graph differs from a line.
Subject: Algebra 1 · 65 slides · symbolic lesson
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Title
Algebra 1 · Chapter 8 — Exponents and Exponential Functions
Graphs of Exponential Functions
Objectives
Five outcomes, each one you can test yourself on with a pencil and no answer key.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.3 Graphs of Exponential Functions §8.3, pp. 455-460 — the lesson these objectives are drawn from
Warm-up
Lesson 8.2 gave meaning to every integer exponent. This lesson lets the exponent be the variable and draws the result.
Discussion prompt
Evaluate two to the x at x equal to negative two, zero and three. What happens to the outputs as x increases by one each time?
Hint: Use the definitions from the last lesson for the negative input.
Answer:
\[ 2^{-2} = \tfrac{1}{4}, \quad 2^0 = 1, \quad 2^3 = 8 \]
Each step to the right doubles the output rather than adding a fixed amount. A constant multiplier instead of a constant step is what makes the graph a curve rather than a line.
Concept
A function of the form y equals a times b to the x, with b positive and not equal to one, is an exponential function. The variable is the exponent rather than the base, which changes the shape of the graph entirely.
exponential function — A function of the form y equals a times b to the x, where b is positive and not equal to one. The base b is the growth factor and a is the y-intercept.
The base must be positive so that every real exponent gives a real value.
Figure (svg): The two numbers of an exponential function and what each controls
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.3 Graphs of Exponential Functions §8.3, pp. 455-455
Section
Section 1
Concept
To make a table of values, substitute each input into the exponent and evaluate. Negative inputs give reciprocals and a zero input gives one times the coefficient.
Every value of the table is positive when the coefficient is.
Figure (svg): A table of values for an exponential function including negative inputs
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.3 Graphs of Exponential Functions §8.3, pp. 455-455 — Example 1, Evaluate an Exponential Function
Picture it
Six inputs, from negative two to three.
Figure (svg): A table of values for an exponential function including negative inputs
Reading left to right the outputs double, and reading right to left they halve. That constant ratio is the fingerprint of an exponential function.
Worked example
This is Example 1 from the textbook.
\[ \text{Tabulate } \; y = 2^x \; \text{ at } x = -2, -1, 0, 1, 2, 3. \]
Take the negative inputs
Why: Two to the negative two is a quarter; to the negative one is a half.
\[ \frac{1}{4}\text{ and } \frac{1}{2} \]
Take zero
Why: Any nonzero base to the zero power is one.
\[ 1 \]
Take the positive inputs
Why: Two, four and eight.
\[ 2, 4, 8 \]
Check the pattern
Why: Each output is double the one before.
Figure (svg): A table of values for an exponential function including negative inputs
\[ \tfrac{1}{4}, \; \tfrac{1}{2}, \; 1, \; 2, \; 4, \; 8 \]
Verify: check the ratio between consecutive outputs
Why: A half divided by a quarter is two, and eight divided by four is two. The ratio is the same everywhere, which is the exponential analogue of the constant step a linear function had in Lesson 4.2.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.3 Graphs of Exponential Functions §8.3, pp. 455-455
Faded example
Power first, then the coefficient.
Fill in the blanks
3\left(\tfrac212\right)^___ = 3 \cdot ___^2 = 3 \cdot 4 = ___
Why: The negative exponent reciprocates the half to give two, and squaring that gives four. Only then does the coefficient of three multiply in, giving twelve.
Worked example
Guided Practice 1 and 2. One rising and one falling.
\[ \text{Tabulate } \; y = 3^x \; \text{ and } \; y = 2\left(\tfrac{1}{3}\right)^x \; \text{ at the same six inputs.} \]
Take the first at negative inputs
Why: A ninth and a third.
\[ \frac{1}{9}, \frac{1}{3} \]
Continue the first
Why: One, three, nine, twenty-seven.
\[ 1, 3, 9, 27 \]
Take the second at negative inputs
Why: Two times nine and two times three.
\[ 18, 6 \]
Continue the second
Why: Two, two thirds, two ninths, two twenty-sevenths.
Figure (svg): The solution to Worked example two from guided practice shown as a ladder of expressions, one row per algebraic move
\[ 3^x: \; \tfrac{1}{9}, \tfrac{1}{3}, 1, 3, 9, 27 \]
Verify: check each table's ratio
Why: The first multiplies by three at every step and the second divides by three, which is multiplying by a third. In both cases the ratio is the base, which is exactly what the base of an exponential function records.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.3 Graphs of Exponential Functions §8.3, pp. 455-455
Trap
\[ y = 3\left(\tfrac{1}{2}\right)^x \text{ at } x = -2 \]
Multiply three by a half first, then square the result
Why: Reading left to right suggests doing the multiplication first.
\[ (1.5)^{-2} = \tfrac{4}{9} \quad \text{(wrong)} \]
The exponent applies to the half alone, not to the whole product. The correct value is three times four, which is twelve — a very different answer.
\[ 3\left(\tfrac{1}{2}\right)^{-2} = 3 \cdot 2^2 = 3 \cdot 4 = 12 \]
Evaluate the power first, then multiply by the coefficient
Why: The order of operations puts powers before multiplication.
Writing brackets around the base makes the scope of the exponent visible, which is why the textbook writes it that way.
Sorting
Compare the base with one.
Sort into buckets
Sort each function by whether its outputs grow or shrink as x increases.
The coefficient plays no part in this decision. Only the base decides the direction, and the number one is the dividing line.
Elimination
A negative input to an exponential function.
Eliminate the wrong options
Which value belongs in the table?
Survives elimination: A
Why: Two to the negative three is the reciprocal of eight. Every output of an exponential function with a positive coefficient is positive, which is why the curve never crosses or touches the horizontal axis.
Socratic
The definition excludes negative bases and one.
Discussion prompt
Explain why an exponential function requires a positive base, and why a base of exactly one is excluded too. Then say what the graph would look like if the base were one.
Hint: Try a negative base at a fractional exponent.
Answer:
With a negative base, some exponents give no real value at all — the square root of negative two, which is what a negative base to the one half would mean, is not a real number. Restricting the base to positive values keeps the function defined for every real input, which is what makes a smooth curve possible.
A base of one gives one to any power, which is one, so the function would be the constant y equals a — a horizontal line rather than a curve. Excluding it keeps the definition to the functions that genuinely grow or shrink, which is why it appears alongside the positivity condition.
Section
Section 2
Concept
Plot the points from the table and join them with a smooth curve. With a base greater than one the curve rises steeply to the right and approaches the horizontal axis to the left.
Values between the plotted points are real, which is why the curve is smooth.
Figure (svg): The graph of y equals two to the x, rising steeply to the right
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.3 Graphs of Exponential Functions §8.3, pp. 456-456 — Example 2, Graph an Exponential Function when b is greater than 1
Picture it
Rising right, hugging the axis left.
Figure (svg): The graph of y equals two to the x, rising steeply to the right
The y-intercept of one is the value at x equal to zero, which is the coefficient — here an invisible one. Everything to the left of the axis lies between zero and one.
Worked example
This is Example 2 from the textbook.
\[ \text{Graph } \; y = 2^x \; \text{ from the table, and evaluate it at } x = 1.5. \]
List the six points
Why: From (-2, 1/4) up to (3, 8).
Plot them
Why: The fractional ones sit just above the axis on the left.
Join with a smooth curve
Why: Not with straight segments.
Evaluate at 1.5
Why: A calculator gives about 2.83.
\[ 2.83 \]
Figure (svg): The graph of y equals two to the x, rising steeply to the right
\[ 2^{1.5} \approx 2.83 \]
Verify: check that 2.83 fits the curve
Why: It lies between two and four, which are the values at one and two — exactly where the curve passes at x equal to one and a half. That the function has values at non-integer inputs is what makes the smooth curve the right picture.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.3 Graphs of Exponential Functions §8.3, pp. 456-456
Prediction
The function is y equals two to the x.
Predict first
What is the y-intercept?
Correct: 1, the value at x equal to 0.
\[ 2^0 = 1 \;\Longrightarrow\; (0, 1) \]
Why: Any nonzero base to the zero power is one, so the curve passes through (0, 1). The base of two is the growth factor rather than the intercept, and the curve approaching the horizontal axis on the left says nothing about where it crosses the vertical one.
Worked example
Two features are visible immediately.
\[ \text{Describe the } y\text{-intercept and the left-hand behaviour of } \; y = 2^x. \]
Find the y-intercept
Why: At x equal to zero the value is one.
\[ (0, 1) \]
Look to the left
Why: The outputs are a half, a quarter, an eighth and so on.
Say what they approach
Why: They get closer to zero without reaching it.
Say why they never reach it
Why: Every value is a reciprocal, which is never zero.
Figure (svg): An exponential curve approaching the horizontal axis without reaching it
\[ (0, 1); \; y \to 0 \text{ as } x \text{ decreases} \]
Verify: compute a value far to the left
Why: At x equal to negative ten the value is one over one thousand and twenty-four, which is tiny and positive. However far left you go the value shrinks and stays above zero, which is what approaching without reaching means.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.3 Graphs of Exponential Functions §8.3, pp. 456-456
Error analysis
The student plotted the points of y equals two to the x and joined them.
Annotate
On: \( \begin{aligned} &\text{plotted } (-2, \tfrac{1}{4}) \text{ through } (3, 8) \\ &\text{joined them with straight segments} \\ &\text{stopped the graph at } x = -2 \end{aligned} \)
The textbook's Study Tip says that later courses give two to the x a precise meaning at every real input, and that a smooth curve is how it is represented here.
Faded example
Intercept and left-hand behaviour.
Fill in the blanks
The graph of y = 2^x has y-intercept 1 and approaches the horizontal axis as x becomes very negative.
Why: The intercept is the value at zero, which is one. Moving left halves the output repeatedly, so the values shrink towards zero without ever reaching it.
Elimination
Six points from a table of an exponential function.
Eliminate the wrong options
How do you complete the graph?
Survives elimination: A
Why: The function has a value at every real input, so the graph is a smooth unbroken curve continuing in both directions. Option B is the natural instinct from plotting tables and it misrepresents every value between the points.
Socratic
It gets extremely close.
Discussion prompt
Explain why the graph of two to the x never meets the horizontal axis, using the definition of a negative exponent. Then say what it would take for an exponential graph to cross the axis.
Hint: Ask what value would have to be produced.
Answer:
Touching the axis would need an output of zero, so two to the x would have to be zero for some x. Every negative exponent gives a reciprocal, and one divided by any number is never zero — so no input produces zero, and the curve stays strictly above the axis however far left it goes.
Crossing would require negative outputs as well, which cannot happen while the base and the coefficient are positive. A negative coefficient would flip the whole curve below the axis, and it still would not cross — the graph would approach the axis from below instead. Nothing of this form ever meets the axis.
Section
Section 3
Concept
With a base between zero and one the outputs shrink as the input grows, so the curve falls to the right and approaches the horizontal axis on that side instead.
Everything else about the shape is a mirror image.
Figure (svg): The graph of a decreasing exponential function
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.3 Graphs of Exponential Functions §8.3, pp. 456-456 — Example 3, Graph an Exponential Function when b is between 0 and 1
Picture it
Intercept three, halving to the right.
Figure (svg): The graph of a decreasing exponential function
The curve is the mirror image of a rising one, reflected in the vertical axis. That is exactly what replacing a base by its reciprocal does.
Worked example
This is Example 3 from the textbook.
\[ \text{Graph } \; y = 3\left(\tfrac{1}{2}\right)^x \; \text{ at } x = -2 \text{ through } 3. \]
Evaluate at negative two
Why: The half reciprocates to two, squared is four, times three.
\[ 12 \]
Continue the table
Why: Six, three, three halves, three quarters, three eighths.
Plot and join smoothly
Why: The curve falls steeply then flattens.
Read the features
Why: Intercept three, approaching the axis on the right.
\[ (0, 3) \]
Figure (svg): The graph of a decreasing exponential function
\[ 12, \; 6, \; 3, \; \tfrac{3}{2}, \; \tfrac{3}{4}, \; \tfrac{3}{8} \]
Verify: check the ratio between consecutive outputs
Why: Six divided by twelve is a half, and three eighths divided by three quarters is a half. The ratio is the base at every step, exactly as it was for the rising curve — only its size relative to one has changed.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.3 Graphs of Exponential Functions §8.3, pp. 456-456
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| b greater than 1 | b between 0 and 1 | |
|---|---|---|
| The curve | rises to the right | falls to the right |
| Approaches the axis | on the left | on the right |
| The y-intercept | the coefficient a | the coefficient a |
Only the first two rows differ. The intercept is the coefficient in both cases, because the base disappears at an exponent of zero whatever it is.
Worked example
One base and its reciprocal.
\[ \text{Compare the graphs of } \; y = 2^x \; \text{ and } \; y = \left(\tfrac{1}{2}\right)^x. \]
Tabulate both at the same inputs
Why: One rises through 1, 2, 4; the other falls through 1, 1/2, 1/4.
Compare the intercepts
Why: Both pass through (0, 1).
Compare the directions
Why: One rises to the right and the other falls.
Name the relationship
Why: Each is the other reflected in the vertical axis.
Figure (svg): Two columns contrasting a linear function with an exponential one
\[ \left(\tfrac{1}{2}\right)^x = 2^{-x} \]
Verify: check the algebra behind the reflection
Why: A half to the x is the same as two to the negative x, by the definition of a negative exponent. Replacing x by negative x is exactly what reflects a graph in the vertical axis, so the picture and the algebra agree.
Trap
\[ y = 3\left(\tfrac{1}{2}\right)^x \]
Expect the curve to go below the axis as x grows, since the values are shrinking
Why: Shrinking suggests eventually passing zero and continuing.
The outputs are three halves, three quarters, three eighths — always positive and always smaller. Halving a positive number repeatedly never reaches zero, let alone passes it.
The curve approaches the horizontal axis from above and stays above it forever.
Compute a value far to the right and look at its sign
Why: At x equal to ten the value is three over one thousand and twenty-four, tiny and positive.
Repeated halving is the clearest picture of approaching zero without arriving, and it is worth doing on a calculator once.
Faded example
Reciprocate, then raise, then multiply.
Fill in the blanks
3\left(\tfrac123/4\right)^___ = 3 \cdot 2^2 = ___ \qquad 3\left(\tfrac______\right)^___ = 3 \cdot \tfrac______ = ___
Why: A negative input reciprocates the fractional base into something greater than one, giving a large value, while a positive input keeps it small. That contrast is what makes the curve fall from left to right.
Prediction
The function is y equals 4 times 0.8 to the x.
Predict first
What does the graph do as x increases?
Correct: It falls, approaching the axis from above.
\[ 4(0.8)^{10} \approx 0.43 \quad \text{small and positive} \]
Why: The base of 0.8 is between zero and one, so each step multiplies by a proper fraction and the outputs shrink. The coefficient sets the starting height rather than the direction. And repeated multiplication by a positive number never reaches zero, so the curve stays above the axis — approaching it slowly, because 0.8 is close to one.
Socratic
The two curves are mirror images.
Discussion prompt
Explain why the graph of a half to the x is the graph of two to the x reflected in the vertical axis. Then say what happens to the graph of a base and its reciprocal in general.
Hint: Rewrite the fractional base using a negative exponent.
Answer:
A half to the x is one over two to the x, which by Lesson 8.2 is two to the negative x. Replacing x by negative x in a function swaps the left and right halves of its graph, which is precisely a reflection in the vertical axis.
The same holds for any base and its reciprocal: b to the x and one over b to the x are always reflections of each other. So every falling exponential is a rising one seen backwards, which is why the two cases need only one set of ideas between them.
Section
Section 4
Concept
In y equals a times b to the x, the coefficient a is the y-intercept and the base b is the factor the output is multiplied by at each unit step.
Both numbers can be read off the equation without computing anything.
Figure (svg): The two numbers of an exponential function and what each controls
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.3 Graphs of Exponential Functions §8.3, pp. 455-456 — the form of an exponential function and the graphs in Examples 2 and 3
Picture it
Where it starts, and what it does.
Figure (svg): The two numbers of an exponential function and what each controls
This is the exponential analogue of slope-intercept form. One number places the graph and the other decides its behaviour, exactly as in Lesson 4.7.
Worked example
No computation needed.
\[ \text{For } \; y = 2^x, \quad y = 3\left(\tfrac{1}{2}\right)^x, \quad y = 5(1.5)^x, \text{ give the intercept and direction.} \]
Take the first
Why: The coefficient is an invisible one and the base exceeds one.
\[ (0, 1),\text{ rising} \]
Take the second
Why: Coefficient three, base a half.
\[ (0, 3),\text{ falling} \]
Take the third
Why: Coefficient five, base one and a half.
\[ (0, 5),\text{ rising} \]
Note what decided each
Why: The coefficient gave the intercept; the base gave the direction.
Figure (svg): The two numbers of an exponential function and what each controls
\[ (0, 1) \uparrow, \quad (0, 3) \downarrow, \quad (0, 5) \uparrow \]
Verify: confirm one intercept by substituting
Why: At x equal to zero the second gives three times one, which is three. The base vanished because any base to the zero power is one, which is why the intercept is always the coefficient regardless of the base.
Matching
Coefficient and base.
Match the pairs
Why: The coefficient is the intercept in every case, and the base decides the direction by whether it exceeds one. The two features are read off independently, which makes sketching such a graph quick.
Worked example
The base is visible in a table as well as in the equation.
\[ \text{A table gives } 5, 15, 45, 135. \text{ What exponential function produced it?} \]
Find the ratio
Why: Fifteen over five is three, and forty-five over fifteen is three.
\[ \text{ratio } 3 \]
Identify the base
Why: The constant ratio is the base.
\[ b = 3 \]
Identify the coefficient
Why: The first value, at x equal to zero, is five.
\[ a = 5 \]
Write the function
Why: Five times three to the x.
\[ y = 5 x 3 ^{x} \]
Figure (svg): A table of values for an exponential function including negative inputs
\[ y = 5 \cdot 3^x \]
Verify: check the last entry
Why: Five times three cubed is five times twenty-seven, which is one hundred and thirty-five — matching the table. Recovering a function from its ratio is the exponential version of finding a line's slope from a table in Lesson 4.2.
Trap
\[ y = 3\left(\tfrac{1}{2}\right)^x \]
Say the y-intercept is a half, since that is the number in the function
Why: The base is the most prominent number, so it looks like the important one.
Substituting zero gives three times one, which is three. The base is what multiplies at each step, and it disappears entirely at an exponent of zero.
\[ y(0) = 3\left(\tfrac{1}{2}\right)^0 = 3 \cdot 1 = 3 \]
Substitute zero to find the intercept, or read the coefficient
Why: Any base to the zero power is one, so only the coefficient survives.
The two numbers do different jobs, and substituting zero is the check that separates them.
Faded example
The ratio between consecutive outputs.
Fill in the blanks
\text3 5, 15, 45: \quad \dfrac33 = ___, \text___ ___ \text___ y = 5 \cdot ___^x
Why: The ratio between consecutive outputs is constant and equals the base, exactly as the constant difference gave the slope of a line. The coefficient is the value at x equal to zero.
Elimination
Substitute zero to check.
Eliminate the wrong options
Which one?
Survives elimination: A
Why: The coefficient is the intercept, so four times any base to the zero power gives four. Option C is the instructive one: a base of four does not make the intercept four, because the base vanishes at an exponent of zero.
Socratic
Whatever the base happens to be.
Discussion prompt
Explain why the y-intercept of an exponential function is its coefficient regardless of the base. Then compare this with the role of b in slope-intercept form.
Hint: Substitute zero and use Lesson 8.2.
Answer:
At x equal to zero the power becomes b to the zero, which is one for every allowed base. So the whole expression collapses to a times one, which is a — the base has no influence at all at that single input.
It is the exact analogue of the constant term in slope-intercept form, which Lesson 4.7 showed is the y-intercept because the x-term vanishes at zero. In both forms one number places the graph vertically and the other controls its behaviour, and in both cases the placing number is the one that survives at x equal to zero.
Section
Section 5
Concept
A linear function adds a constant at each step and a exponential one multiplies by a constant. That single difference produces a straight line in one case and a curve in the other.
A table can be classified by testing differences and ratios.
Figure (svg): Two columns contrasting a linear function with an exponential one
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.3 Graphs of Exponential Functions §8.3, pp. 455-460 — the contrast with the linear functions of Chapters 4 and 5
Picture it
One constant, applied two ways.
Figure (svg): Two columns contrasting a linear function with an exponential one
Both kinds of function are governed by a constant behaviour and two numbers. Whether that constant is added or multiplied is what separates them.
Worked example
Test differences, then ratios.
\[ \text{Classify } \; 3, 7, 11, 15 \; \text{ and } \; 3, 6, 12, 24. \]
Test the first for differences
Why: Four each time.
Classify it
Why: A constant difference means linear.
Test the second for differences
Why: Three, six, twelve — not constant.
Test it for ratios
Why: Two each time.
Figure (svg): Two columns contrasting a linear function with an exponential one
\[ y = 4x + 3 \qquad y = 3 \cdot 2^x \]
Verify: write both functions and check an entry
Why: The linear model gives four times three plus three, which is fifteen, and the exponential gives three times eight, which is twenty-four. Both match their tables' last entries, confirming the classification.
Sorting
Test differences first, then ratios.
Sort into buckets
Sort each table by the kind of function that produced it.
Two of the tables decrease, one linearly and one exponentially. Decreasing does not by itself say which kind you have — the test is whether the differences or the ratios are constant.
Worked example
The exponential starts behind and does not stay there.
\[ \text{Compare } \; y = 10x \; \text{ and } \; y = 2^x \; \text{ at } x = 1, 5, 10 \text{ and } 20. \]
At x equal to 1
Why: Ten against two.
At x equal to 5
Why: Fifty against thirty-two.
At x equal to 10
Why: One hundred against one thousand and twenty-four.
At x equal to 20
Why: Two hundred against over a million.
Figure (svg): The solution to Worked example which grows faster in the end shown as a ladder of expressions, one row per algebraic move
\[ 2^{20} \approx 1\,048\,576 \text{ against } 200 \]
Verify: say why the overtaking is inevitable
Why: The linear function adds ten each step while the exponential doubles, and doubling eventually beats adding any fixed amount however large. That is why exponential growth is described as fast even when it starts slowly.
Trap
At x equal to 1 and 5, the linear function is larger, so it grows faster.
Conclude that ten x outgrows two to the x
Why: The first few values genuinely favour the linear function.
By x equal to ten the exponential is ten times larger, and by twenty it is thousands of times larger. Early values say nothing about eventual behaviour.
Compare the behaviour rather than the values: adding against multiplying
Why: A constant multiplier always eventually beats a constant addend.
Extending the table far enough is the concrete way to see it, and the crossover is usually further out than people expect.
Faded example
Differences, then ratios.
Fill in the blanks
For 3, 6, 12, 24 the differences are 3, 6, 12, which are not constant, and the ratios are all 2, so the table is exponential.
Why: Testing the differences rules out a linear model and testing the ratios identifies the base. Doing them in that order is efficient, since a constant difference settles it immediately when there is one.
Hypothesis
Predict before you check.
Predict first
Will y = 2^x eventually exceed y = 1000x?
Correct: Yes, since doubling eventually beats adding any fixed amount.
\[ 2^{20} \approx 1\,048\,576 \quad \text{against} \quad 1000(20) = 20\,000 \]
Why: At x equal to ten the exponential is only about a thousand against ten thousand, and at x equal to twenty it is over a million against twenty thousand. A constant multiplier compounds while a constant addend does not, so the exponential always wins eventually — the coefficient only delays the crossover.
Socratic
A constant difference gave a straight line.
Discussion prompt
Explain why multiplying by a constant at each step produces a curve rather than a line. Then say what happens to the steepness of an exponential graph as you move right.
Hint: Compare the size of consecutive steps.
Answer:
A constant difference means every step rises by the same amount, so the steepness never changes and the graph is straight. A constant ratio means each step rises by a multiple of the current value, so as the value grows the steps grow too — the steepness increases and the graph bends upwards.
So an exponential graph's slope is not constant but proportional to its height. Far to the right it becomes almost vertical, and far to the left almost horizontal as it flattens towards the axis. That changing steepness is exactly what Lesson 4.5 said a curve has and a line does not.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Linear | Exponential | |
|---|---|---|
| Form | y = mx + b | y = ab^x |
| Constant feature | the difference between outputs | the ratio between outputs |
| Graph | a straight line | a curve approaching an axis |
The second row is the test to apply to a table, and the third is what that test predicts about the picture.
Pattern
Whether the base is above or below one, the same five moves cover it.
Steps one and two take a glance and predict the whole shape, so the table becomes a confirmation rather than a discovery.
OpenStax Intermediate Algebra 2e, §10.2 Evaluate and Graph Exponential Functions §10.2
Check
Substitute zero.
Check your understanding
What is the y-intercept of y = 7(2)^x?
Answer: A
Why: At x equal to zero the power is two to the zero, which is one, so the value is seven times one. The coefficient is always the intercept whatever the base.
Check
Compare the base with one.
Check your understanding
Which function has a graph that falls to the right?
Answer: A
Why: The base of 0.4 is between zero and one, so each step multiplies by a proper fraction and the outputs shrink. The coefficient of five sets the starting height rather than the direction.
Check
Test the differences and the ratios.
Check your understanding
The table 2, 6, 18, 54 was produced by which kind of function?
Answer: A
Why: The differences are four, twelve and thirty-six, which are not constant, and the ratios are all three. So the function is exponential with a base of three and a coefficient of two.
Real world
This is Example 5's situation. The number of shipwrecks recorded in the northern Gulf of Mexico between 1680 and 1980 is modelled by an exponential function of the number of years since 1680.
Discussion prompt
Say what the coefficient and the base of such a model would represent, describe the shape of its graph, and say why an exponential model rather than a linear one might fit shipwreck counts.
Hint: Think about what grows in proportion to itself.
Answer:
The coefficient would be the recorded number of wrecks in the first period, and the base the factor by which that count multiplies in each subsequent period. A base above one gives a curve rising ever more steeply to the right.
An exponential model fits because the number of wrecks depends on the amount of shipping, and shipping traffic itself grew by a percentage each decade rather than by a fixed number of vessels. When a quantity's growth is proportional to its current size, the result is exponential rather than linear.
The model should be read with the same caution as any other. It describes the period it was fitted to, and running it forwards indefinitely would predict impossible numbers — exponential models always eventually predict more than the world can supply.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
Does the graph of y = 2^x ever touch the x-axis?
Correct: No, it approaches the axis but never reaches it.
\[ 2^{-20} = \dfrac{1}{1\,048\,576} > 0 \]
Why: Touching would require an output of zero, and every negative exponent gives a reciprocal, which is never zero. At x equal to negative ten the value is about a thousandth, and at negative twenty about a millionth — always smaller and always positive. The first option confuses the y-intercept with an x-intercept: at x equal to zero the value is one, so the curve is a full unit above the axis there.
Explain it
They can graph lines and have never plotted a curve.
Discussion prompt
In no more than four sentences, explain how an exponential function differs from a linear one and what its graph looks like. Then tell them what the two numbers in the equation control.
Hint: Multiplying instead of adding.
Answer:
A usable answer: with a line you add the same amount at every step, and with an exponential you multiply by the same amount instead. That makes the steps get bigger as you go, so the graph curves upwards steeply on one side and flattens out towards the axis on the other without ever touching it.
The number in front tells you where the curve crosses the vertical axis, because anything to the power zero is one. The number being raised to the power tells you what happens at each step — bigger than one and it climbs, between zero and one and it falls.
Exit ticket
Name the weakest spot before you close the deck. That is the one worth ten minutes tonight.
Predict first
Which of these would you least want to be handed cold on a quiz tomorrow?
Correct: Whichever you picked is the right answer — and each one has a specific fix.
Why: Negative inputs are fixed by reciprocating first and multiplying by the coefficient last. Intercept against base is fixed by substituting zero, which makes the base vanish. Direction is fixed by comparing the base with one. Classification is fixed by testing the differences first and the ratios second. Pick yours and do five of that kind tonight rather than twenty mixed ones.
Connect it up
Do this on paper. It is worth more than rereading the slides.
Draw it
At the top of a page write an exponential function with a base greater than one and a coefficient other than one, and tabulate it at six inputs from negative two to three, showing the working for the two negative inputs in full. Plot the points on a coordinate plane and join them with a smooth curve extending past the outermost points in both directions, marking the y-intercept and writing beside the left-hand end that the curve never meets the axis. Repeat the whole thing underneath for a function whose base is between zero and one, and write one sentence comparing the two pictures. In the lower half, write two tables of four numbers, one linear and one exponential, and show the difference test and the ratio test on each, writing the function that produced it. Finally, in the margin, write what the coefficient and the base each control.
Both of your curves should have the coefficient as their y-intercept, whatever the base. If a curve crosses the horizontal axis anywhere, a value was computed as negative — which a positive base and coefficient never produce.
Recap
Five things, and the second is where the two numbers get confused.
| If the question says | Your first move is |
|---|---|
| Find the y-intercept | Substitute zero; the base vanishes |
| Which way does the graph go | Compare the base with 1 |
| Evaluate at a negative input | Reciprocate, then multiply by the coefficient |
| Join the plotted points | Use a smooth curve, extended both ways |
| Is this table linear or exponential | Test the differences, then the ratios |
Lesson 8.4 returns to the algebra and supplies the division properties. Dividing powers subtracts exponents, and a quotient raised to a power distributes — completing the set of rules the rest of the chapter needs.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.3 Graphs of Exponential Functions §8.3, pp. 455-460 — everything on these slides traces back here
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