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Introduction

Exponential functions are a fundamental concept in mathematics, and graphing them is an essential skill for understanding their behavior and properties. In this article, we will explore how to graph the function $y = 3^x$ and determine its $y$-intercept. We will then use the graph to approximate the value of the expression $3^{1.2}$.

Graphing the Function $y = 3^x$

The function $y = 3^x$ is an exponential function with base 3. To graph this function, we can start by plotting a few points on the coordinate plane. We can choose values of $x$ and calculate the corresponding values of $y$ using the function.

$x$	$y = 3^x$
-2	$3^{-2} = \frac{1}{9}$
-1	$3^{-1} = \frac{1}{3}$
0	$3^0 = 1$
1	$3^1 = 3$
2	$3^2 = 9$

By plotting these points on the coordinate plane, we can see that the graph of the function $y = 3^x$ is a curve that passes through the points $(x, y)$ listed above. The graph is a continuous curve that increases as $x$ increases.

Determining the $y$-Intercept

The $y$-intercept of a function is the point where the graph of the function intersects the $y$-axis. To find the $y$-intercept of the function $y = 3^x$, we can set $x = 0$ and calculate the corresponding value of $y$.

$y = 3^0 = 1$

Therefore, the $y$-intercept of the function $y = 3^x$ is the point $(0, 1)$.

Approximating the Value of $3^{1.2}$

Now that we have graphed the function $y = 3^x$ and determined its $y$-intercept, we can use the graph to approximate the value of the expression $3^{1.2}$. To do this, we can locate the point on the graph that corresponds to $x = 1.2$ and read off the corresponding value of $y$.

From the graph, we can see that the point $(1.2, y)$ is approximately $(1.2, 3.7)$. Therefore, the approximate value of the expression $3^{1.2}$ is $3.7$.

Conclusion

In this article, we graphed the function $y = 3^x$ and determined its $y$-intercept. We then used the graph to approximate the value of the expression $3^{1.2}$. By graphing exponential functions and using the graph to evaluate expressions, we can gain a deeper understanding of the behavior and properties of these functions.

Discussion

• What are some other ways to evaluate the expression $3^{1.2}$?
• How can we use the graph of the function $y = 3^x$ to determine the value of other expressions?
• What are some real-world applications of exponential functions?

Answer Key

A. $1; 3.7$

B. $0; 1$

Introduction

In our previous article, we explored how to graph the function $y = 3^x$ and determine its $y$-intercept. We then used the graph to approximate the value of the expression $3^{1.2}$. In this article, we will answer some frequently asked questions about graphing exponential functions and evaluating expressions.

Q&A

Q: What is the $y$-intercept of the function $y = 3^x$?

A: The $y$-intercept of the function $y = 3^x$ is the point $(0, 1)$.

Q: How can we graph the function $y = 3^x$?

A: We can graph the function $y = 3^x$ by plotting a few points on the coordinate plane. We can choose values of $x$ and calculate the corresponding values of $y$ using the function.

Q: What is the approximate value of the expression $3^{1.2}$?

A: The approximate value of the expression $3^{1.2}$ is $3.7$.

Q: How can we use the graph of the function $y = 3^x$ to determine the value of other expressions?

A: We can use the graph of the function $y = 3^x$ to determine the value of other expressions by locating the point on the graph that corresponds to the given value of $x$ and reading off the corresponding value of $y$.

Q: What are some real-world applications of exponential functions?

A: Exponential functions have many real-world applications, including:

• Modeling population growth
• Modeling chemical reactions
• Modeling financial investments
• Modeling the spread of diseases

Q: What are some other ways to evaluate the expression $3^{1.2}$?

A: Some other ways to evaluate the expression $3^{1.2}$ include:

• Using a calculator to calculate the value of $3^{1.2}$
• Using a computer program to calculate the value of $3^{1.2}$
• Using a mathematical formula to calculate the value of $3^{1.2}$

Q: How can we determine the value of an exponential expression with a negative exponent?

A: To determine the value of an exponential expression with a negative exponent, we can use the following formula:

$3^{-x} = \frac{1}{3^x}$

For example, to evaluate the expression $3^{-2}$, we can use the formula above to get:

$3^{-2} = \frac{1}{3^2} = \frac{1}{9}$

Q: How can we determine the value of an exponential expression with a fractional exponent?

A: To determine the value of an exponential expression with a fractional exponent, we can use the following formula:

$3^{x/y} = \sqrt[y]{3^x}$

For example, to evaluate the expression $3^{1/2}$, we can use the formula above to get:

$3^{1/2} = \sqrt[2]{3^1} = \sqrt{3}$

Conclusion

In this article, we answered some frequently asked questions about graphing exponential functions and evaluating expressions. We hope that this article has been helpful in clarifying some of the concepts and techniques involved in graphing exponential functions and evaluating expressions.

Discussion

• What are some other ways to evaluate exponential expressions?
• How can we use the graph of the function $y = 3^x$ to determine the value of other expressions?
• What are some real-world applications of exponential functions?

Answer Key

• Q: What is the $y$-intercept of the function $y = 3^x$? A: The $y$-intercept of the function $y = 3^x$ is the point $(0, 1)$.
• Q: How can we graph the function $y = 3^x$? A: We can graph the function $y = 3^x$ by plotting a few points on the coordinate plane.
• Q: What is the approximate value of the expression $3^{1.2}$? A: The approximate value of the expression $3^{1.2}$ is $3.7$.
• Q: How can we use the graph of the function $y = 3^x$ to determine the value of other expressions? A: We can use the graph of the function $y = 3^x$ to determine the value of other expressions by locating the point on the graph that corresponds to the given value of $x$ and reading off the corresponding value of $y$.
• Q: What are some real-world applications of exponential functions? A: Exponential functions have many real-world applications, including modeling population growth, modeling chemical reactions, modeling financial investments, and modeling the spread of diseases.
• Q: What are some other ways to evaluate the expression $3^{1.2}$? A: Some other ways to evaluate the expression $3^{1.2}$ include using a calculator to calculate the value of $3^{1.2}$, using a computer program to calculate the value of $3^{1.2}$, and using a mathematical formula to calculate the value of $3^{1.2}$.
• Q: How can we determine the value of an exponential expression with a negative exponent? A: To determine the value of an exponential expression with a negative exponent, we can use the formula $3^{-x} = \frac{1}{3^x}$.
• Q: How can we determine the value of an exponential expression with a fractional exponent? A: To determine the value of an exponential expression with a fractional exponent, we can use the formula $3^{x/y} = \sqrt[y]{3^x}$.