﻿ ﻿Exponential Equations - Equations and Inequalities - High School Algebra I Unlocked (2016)

High School Algebra I Unlocked (2016)

Lesson 4.5. Exponential Equations

Think back to our discussion of exponents. Recall that exponents are used to indicate the number of times a number is multiplied by itself. For example, 33 = 3 × 3 × 3 = 27. However, exponents do not exist in a vacuum; you may encounter questions that require you to interpret or graph exponential equations. Exponential equations are often used to model growth or decay. For example, imagine you started with a single penny, and doubled the amount of money every day. How long would it take you to reach \$100? You may think it would take a long time, but work it out—it would only take you 14 days to have more than \$100. That is exponential growth! Thankfully, the process for graphing exponential equations is identical to that of graphing two-variable linear equations, except that you will generally only be able to plot a few points on an exponential graph.

We covered some
exponent basics in
Lesson 2.1, but now
we turn to a more
exponential equations.
Exponential equations
are often used to
describe the growth
or decay, which occurs
when the rate of
increase or decrease
becomes increasingly
rapid in proportion to
the growing total.

Consider the equation y = 3x. Start by setting x = 0 to identify the y-intercept; y = 3x, y = 30, and y = 1. Thus, the graph includes the point (0, 1). Plot this on the graph, as shown.

Now change the value of x to find other points on the graph. It can be helpful to create a chart such as the following:

 y = 3x x-value y-value −2 1/9 −1 1/3 0 1 1 3 2 9 3 27

Once you’ve identified other points on the graph, plot these points as well.

Start by subtracting 5x from both sides of the equation to isolate y on the left side of the equation.

Since you need the equation to be in the form of y = mx + b, multiply both sides of the equation by −1.

−1(−y) = −1(2x + 6)
y = −2x − 6

Now the equation is in y = mx + b form and you can find that the slope is −2, and the correct answer is (F).

You’ll notice that we didn’t plot the point (3, 27). As stated earlier, when working with exponential equations, you will generally only be able to plot a few points on the graph. Here, the point (3, 27) wouldn’t even show up on the graph! Don’t panic: Once you’ve identified a few points on the graph, you will be able to draw a line that goes through all the points, filling in any gaps. Therefore, the graphical representation of y = 3x is:

Let’s try a question that deals with exponential equations.

EXAMPLE

Graph y = 2x + 1 and identify the y-intercept of the equation.

Start by identifying the y-intercept. Since the y-intercept is the point on the graph where x = 0, the y-intercept of y = 2x + 1 is

y = 2x + 1

= 20 + 1

= 21

= 2

Thus, the y-intercept is the point (0, 2), which is plotted on the following graph:

Next, alter the value of x to find other points on the graph:

 y = 2x + 1 x-value y-value −2 1/2 −1 1 0 2 1 4 2 8 3 16

Plot these points, keeping in mind that you may not be able to fit every point on the graph. Here, for example, we do not plot point (3, 16), as it would not be visible on our graph.

Finally, draw a line that connects all the points, filling in any gaps as you go. Thus, the graphical representation of y = 2x + 1, which has a y-intercept at (0, 2) is the following:

EXAMPLE

Graph y = −2x + 1 and identify the y-intercept.

This equation looks pretty similar to the previous example, but there’s one small, important change—the negative sign—which has a dramatic effect on the graph. Again, start by finding the y-intercept of y = −2x + 1:

y = −2x + 1

= −20 + 1

= −21

= −2

Thus, the y-intercept occurs at (0, −2), as shown on the following graph.

Now change the value of x to identify other points on the graph:

 y = −2x + 1 x-value y-value −2 −1/2 −1 −1 0 −2 1 −4 2 −8 3 −16

Go ahead and plot these points, but keep in mind that you may not be able to fit every point on the graph. Here, for example, we do not plot point (3, −16), as it would not be visible on the graph. You will notice that, unlike the graphs of our previous exponential equations, the points on the graph appear below the x-axis; this is a direct result of the negative sign in front of the equation.

Finally, draw a line that connects all of the points, filling in any gaps as you go. The graphical representation of y = −2x + 1 is the following:

Whenever you are working with exponential equations, make sure that you note the sign associated with each term in the equation. Small changes, such as whether an equation is positive or negative, can have a major impact on the final graph.

Make sure that you have a firm grasp of this content before moving on. In algebra, the concepts build upon one another, so it is essential that you understand this material before tackling the next chapter.

DRILL

CHAPTER 4 PRACTICE QUESTIONS

Directions: Complete the following problems as specified by each question. For extra practice, try using an alternative method to solve the problem or check your work.

1. If 4 more than twice the sum of x and 3 is 6, what is the value of x ?

2. After Adam purchases a cherry pie for \$11 and hands the cashier a \$20 bill, he has \$42 left in his wallet. How much change does Adam receive from his cherry pie purchase?

A) 20 − 11 + 42

B) 20 − 11

C) 20 − 42 + 11

D) 42 − 11

3. Which of the following is true if 5x − 16 < 13x + 3 ?

A) x > −

B) x < −

C) x >

D) x <

4. What is the slope of the line y = 4 ?

5. What is the slope of a line that contains the points (−4, −20) and (14, −6) ?

6. Which of the following equations has a slope equal to 12x + 15y − 4 = 0 ?

A) 8y + 10x = 16

B) 5y − 4x + 8 = 0

C) 8x − 10y + 6 = 0

D) 5y = −4x − 12

7. Rewrite the equation 3x + 6y = 9 in slope-intercept form and identify the slope, x-intercept, and y-intercept of the equation.

8. Rewrite the equation −4x − 2y = 13 in slope-intercept form and identify the slope, x-intercept, and y-intercept of the equation.

9. Graph y = 32x + 2 and identify the y-intercept.

10. Identify the graph of y = −4x − 4.

A)

B)

C)

D)

SOLUTIONS TO CHAPTER 4 PRACTICE QUESTIONS

1. −2

Here, you need to translate an English statement into a mathematical equation and then solve for x. You are told that 4 more than twice the sum of x and 3 is 6. Therefore, add 4 to 2(x + 3) and set that quantity equal to 6: 2(x + 3) + 4 = 6. Now you need to solve for x. Start by expanding the equation to find that 2(x + 3) + 4 = 6, 2x + 6 + 4 = 6, and 2x + 10 = 6. Next, subtract 10 from both sides of the equation to find that 2x + 10 = 6, 2x = 6 − 10, and 2x = −4. Finally, divide both sides of the equation by 2 to find the value of x: 2x = −4 and x = −2.

2. B

You’re told that Adam purchases a cherry pie for \$11, hands the cashier a \$20 bill, and then has \$42 left in his wallet. You need to identify the equation that represents how much change Adam receives from his cherry pie purchase. The trick to this question is distinguishing the relevant information from the irrelevant information. Here, the \$42 in Adam’s wallet is irrelevant to the question because you’re looking for the amount of change Adam receives from his purchase. If Adam hands the cashier a \$20 for an \$11 pie, the amount of change Adam receives will be represented by 20 − 11, or (B).

3. A

For this question, you need to find all possible values of the inequality 5x − 16 < 13x + 3. Start by subtracting 13x from both sides of the inequality to find that 5x − 13x − 16 < 3 and −8x − 16 < 3. Next, add 16 to both sides of the inequality to find that −8x < 3 + 16 and −8x < 19. Finally, divide both sides of the equation by −8, remembering to switch the orientation of the inequality sign. Therefore, x > −, and the correct answer is (A).

4. 0

Here you’re looking for the slope of the equation y = 4. If you were to graph the line y = 4, you would notice that this is a horizontal line where y is equal to 4 for all values of x. Horizontal lines always have a slope of 0. Alternatively, rewrite y = 4 in y = mx + b form as y = 0x + 4. Since m = 0, the slope is 0.

5.

Since this question requires you to find the slope of a line that contains the points (−4, −20) and (14, −6), you need to use the point-slope formula, slope = . When you plug the two points into the equation, you find that the slope = . Thus, the slope of a line that contains the points (−4, −20) and (14, −6) is .

6. D

Here, you need to identify the equation that has a slope equal to that of the equation 12x + 15y − 4 = 0. Since the equation is already written in linear form, ax + by + c = 0, the slope is equal to −. Thus, in the equation 12x + 15y − 4 = 0, a = 12, b = 15, and the slope is − = − = −. Now, use the same technique to find the slope of each equation in the answer choices. Choice (A) gives you the equation 8y + 10x = 16, which can be written in linear form as 10x + 8y − 16 = 0. Therefore, a = 10, b = 8, and the slope is − = − = −; eliminate (A). Choice (B) gives you the equation 5y − 4x + 8 = 0, which is written in linear form as −4x + 5y + 8 = 0. Thus, a = −4, b = 5, and the slope is − = −(−) = ; eliminate (B). Next, (C) gives you the equation 8x − 10y + 6 = 0, which is already written in linear form. Thus, a = 8, b = −10, and the slope is − = −(−) = ; eliminate (C). Finally, (D) gives you the equation 5y = −4x − 12, which is written in linear form as 4x + 5y − 12 = 0. Thus, a = 4, b = 5, and the slope is − = −. The correct answer is (D).

7. y = −x + and m = −; y-intercept: (0, ); x-intercept: (3, 0)

In order to find the slope of the equation 3x + 6y = 9, rewrite the equation in slope-intercept form, or y = mx + b form. Accordingly, 3x + 6y = 9, 6y = −3x + 9, and y = −x + . Now that the equation is in slope-intercept form, you can find the slope, which is equal to m, and the y-intercept, which is equal to b. Thus, the slope of the line 3x + 6y = 9 is −1/2 and the y-intercept occurs at (0, 3/2). Now, find the x-intercept by setting y = 0 and solving for x: 3x + 6y = 9, 3x + 6(0) = 9, 3x = 9, and x = 3. Thus, the x-intercept of the equation occurs at (3, 0).

8. y = −2x and m = −2; y-intercept: (0, −); x-intercept: (−, 0)

This question requires you to identify the slope, x-intercept, and y-intercept of the equation −4x − 2y = 13. Start by rewriting the equation in slope-intercept form. Therefore, −4x − 2y = 13, 2y = −4x − 13, and y = − 2x − 13/2. So, the slope of the line is −2, and they-intercept of the equation is (0, −13/2). Now find the x-intercept by setting y = 0 and solving for x. Thus, y = −2x − 13/2, 0 = −2x − 13/2, 2x = −13/2, and x = − 13/4. The x-intercept of the equation is (−13/4, 0).

9. y-intercept: (0, 9)

This question requires you to graph y = 32x + 2 and identify the y-intercept of the equation. Start by setting x = 0 to find the y-intercept of the equation: y = 32x + 2, y = 32(0) + 2, y = 32, and y = 9; therefore, the y-intercept occurs at (0, 9). Now, change the value of xto identify other points on the graph:

 y = 32x + 2 x-value y-value −3 1/81 −2 1/9 −1 1 −1/2 3 1/2 27

Plot the points and draw a line that connects all the points, filling in any gaps as you go. The representation of y = 32x + 2 is as follows:

10. D

In order to identify the graph of y = −4x − 4, start by plugging in values for x to find the associated value of y to identify points of the graph.

 y = −4x − 4 x-value y-value 0 −1/256 3 −1 /4 4 −1 5 −4 6 −16

Once you’ve identified the appropriate x- and y-values, compare these points to the provided graphs. Since the y-values are negative for all the x-values, you can immediately eliminate (A) and (B). Now take a look at the graphs in (C) and (D). The only graph that depicts the points found in the table above is (D).

REFLECT

Congratulations on completing Chapter 4!

Here’s what we just covered.

•Explain the properties of linear equations

1 2 3 4 5

•Solve single-variable linear equations and inequalities

1 2 3 4 5

•Construct linear equations from word problems

1 2 3 4 5

•Solve two-variable linear equations

1 2 3 4 5

•Rewrite equations in slope-intercept form

1 2 3 4 5

•Determine the slope of a line using multiple methods:

1 2 3 4 5

•Find the x- and y-intercepts of linear equations

1 2 3 4 5

•Identify equivalent equations

1 2 3 4 5

•Graph exponential equations and graphs

1 2 3 4 5

If you rated any of these topics lower than you’d like, consider reviewing the corresponding lesson before moving on, especially if you found yourself unable to correctly answer one of the related end-of-chapter questions.

CHAPTER 4 KEY POINTS

In order to solve a single-variable linear equation, isolate the variable on one side of the equals sign and the numbers on the other side of the equals sign.

Equations are equivalent when they have the same solution.

Inequalities work like normal equations, unless you multiply or divide by a negative number. If you perform either of these operations, you need to flip the inequality sign.

Some problems can be “translated” from English to math:

is, are, were, did, does means =

of means ×

out of means ÷

what means a variable, such as a, b, or c

To graph an x- or y-coordinate on a coordinate plane, count over to the right (for positive) or left (for negative) to plot the x-coordinate, and then up (for positive) or down (for negative) to plot the y-coordinate.

Many two-variable linear equation questions can be solved by putting them into slope-intercept form, y + mx = b, where m is the slope of the line, and b is the y-intercept.

The slope of a line is the change in y-values (rise) divided by the change in x-values (run). It is generally expressed as a fraction.

Slope can also be calculated from two points on a coordinate plane using the following formula:

If you have an equation in ax + by + c = 0 form, the slope of the line is equal to −.

Graph exponential equations using the same method used to graph linear equations; determine the y-intercept, plot points on the graph, and connect the points.

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