Sketch The Graph Of $y + 3 = 2^x$.a. What Are The Domain And Range Of This Function?b. Does This Function Have A Line Of Symmetry? If So, What Is It?c. What Are The $x$- And $ Y Y Y [/tex]-intercepts?d. Change The

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Introduction

In mathematics, graphing functions is an essential skill that helps us visualize and understand the behavior of mathematical relationships. In this article, we will focus on sketching the graph of the function $y + 3 = 2^x$, analyzing its domain and range, line of symmetry, and intercepts.

Domain and Range

a. Domain and Range of the Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. In the case of the function $y + 3 = 2^x$, the domain is all real numbers, denoted as $(-\infty, \infty)$. This is because the function is defined for any value of x.

The range of a function is the set of all possible output values (y-values) for which the function is defined. To find the range, we need to consider the possible values of y. Since $y + 3 = 2^x$, we can rewrite the equation as $y = 2^x - 3$. The minimum value of $2^x$ is 1, which occurs when x = 0. Therefore, the minimum value of y is $1 - 3 = -2$. As x increases, $2^x$ increases exponentially, and y increases accordingly. There is no maximum value for y, as $2^x$ can become arbitrarily large. Therefore, the range of the function is $(-\infty, \infty)$.

b. Line of Symmetry

A line of symmetry is a vertical line that divides the graph of a function into two congruent halves. To determine if the function has a line of symmetry, we need to examine its equation. The function $y + 3 = 2^x$ is an exponential function, which typically has a horizontal asymptote but no line of symmetry. However, we can rewrite the equation as $y = 2^x - 3$, which is a shifted exponential function. The graph of this function is a horizontal translation of the graph of $y = 2^x$ by 3 units down. Since the graph of $y = 2^x$ is symmetric about the y-axis, the graph of $y = 2^x - 3$ is also symmetric about the y-axis. Therefore, the line of symmetry is the y-axis, which is the vertical line x = 0.

Intercepts

c. x- and y-Intercepts

The x-intercept of a function is the point where the graph of the function crosses the x-axis. To find the x-intercept, we need to set y = 0 and solve for x. In the case of the function $y + 3 = 2^x$, we can rewrite the equation as $0 + 3 = 2^x$, which simplifies to $3 = 2^x$. Taking the logarithm base 2 of both sides, we get $\log_2(3) = x$. Therefore, the x-intercept is $(\log_2(3), 0)$.

The y-intercept of a function is the point where the graph of the function crosses the y-axis. To find the y-intercept, we need to set x = 0 and solve for y. In the case of the function $y + 3 = 2^x$, we can rewrite the equation as $y + 3 = 2^0$, which simplifies to $y + 3 = 1$. Solving for y, we get $y = -2$. Therefore, the y-intercept is $(0, -2)$.

Graphing the Function

To graph the function $y + 3 = 2^x$, we can use the following steps:

  1. Plot the x-intercept $(\log_2(3), 0)$.
  2. Plot the y-intercept $(0, -2)$.
  3. Draw a smooth curve through the points, using the fact that the function is an exponential function.
  4. Label the graph with the function name and any relevant features, such as the line of symmetry and intercepts.

Conclusion

In conclusion, the graph of the function $y + 3 = 2^x$ is an exponential function that has a domain of $(-\infty, \infty)$ and a range of $(-\infty, \infty)$. The function has a line of symmetry at the y-axis and x- and y-intercepts at $(\log_2(3), 0)$ and $(0, -2)$, respectively. By following the steps outlined above, we can graph the function and visualize its behavior.

References

  • [1] Larson, R. (2014). Calculus. Cengage Learning.
  • [2] Rogawski, J. (2011). Calculus: Early Transcendentals. W.H. Freeman and Company.
  • [3] Stewart, J. (2012). Calculus: Early Transcendentals. Cengage Learning.

Glossary

  • Domain: The set of all possible input values (x-values) for which the function is defined.
  • Range: The set of all possible output values (y-values) for which the function is defined.
  • Line of symmetry: A vertical line that divides the graph of a function into two congruent halves.
  • x-intercept: The point where the graph of the function crosses the x-axis.
  • y-intercept: The point where the graph of the function crosses the y-axis.
    Q&A: Sketching the Graph of a Function =============================================

Frequently Asked Questions

Q: What is the domain of the function $y + 3 = 2^x$?

A: The domain of the function $y + 3 = 2^x$ is all real numbers, denoted as $(-\infty, \infty)$.

Q: What is the range of the function $y + 3 = 2^x$?

A: The range of the function $y + 3 = 2^x$ is $(-\infty, \infty)$.

Q: Does the function $y + 3 = 2^x$ have a line of symmetry?

A: Yes, the function $y + 3 = 2^x$ has a line of symmetry at the y-axis, which is the vertical line x = 0.

Q: What are the x- and y-intercepts of the function $y + 3 = 2^x$?

A: The x-intercept of the function $y + 3 = 2^x$ is $(\log_2(3), 0)$, and the y-intercept is $(0, -2)$.

Q: How do I graph the function $y + 3 = 2^x$?

A: To graph the function $y + 3 = 2^x$, follow these steps:

  1. Plot the x-intercept $(\log_2(3), 0)$.
  2. Plot the y-intercept $(0, -2)$.
  3. Draw a smooth curve through the points, using the fact that the function is an exponential function.
  4. Label the graph with the function name and any relevant features, such as the line of symmetry and intercepts.

Q: What are some common mistakes to avoid when graphing the function $y + 3 = 2^x$?

A: Some common mistakes to avoid when graphing the function $y + 3 = 2^x$ include:

  • Failing to plot the x- and y-intercepts.
  • Drawing a curve that is not smooth or continuous.
  • Failing to label the graph with the function name and relevant features.
  • Plotting the graph incorrectly, such as plotting the x-intercept at the wrong location.

Q: How can I use the graph of the function $y + 3 = 2^x$ to solve real-world problems?

A: The graph of the function $y + 3 = 2^x$ can be used to solve real-world problems involving exponential growth and decay. For example, the graph can be used to model population growth, chemical reactions, and financial investments.

Q: What are some real-world applications of the function $y + 3 = 2^x$?

A: Some real-world applications of the function $y + 3 = 2^x$ include:

  • Modeling population growth and decay.
  • Modeling chemical reactions and nuclear decay.
  • Modeling financial investments and compound interest.
  • Modeling the spread of diseases and epidemics.

Conclusion

In conclusion, the graph of the function $y + 3 = 2^x$ is an exponential function that has a domain of $(-\infty, \infty)$ and a range of $(-\infty, \infty)$. The function has a line of symmetry at the y-axis and x- and y-intercepts at $(\log_2(3), 0)$ and $(0, -2)$, respectively. By following the steps outlined above, we can graph the function and visualize its behavior. The graph of the function $y + 3 = 2^x$ can be used to solve real-world problems involving exponential growth and decay.

References

  • [1] Larson, R. (2014). Calculus. Cengage Learning.
  • [2] Rogawski, J. (2011). Calculus: Early Transcendentals. W.H. Freeman and Company.
  • [3] Stewart, J. (2012). Calculus: Early Transcendentals. Cengage Learning.

Glossary

  • Domain: The set of all possible input values (x-values) for which the function is defined.
  • Range: The set of all possible output values (y-values) for which the function is defined.
  • Line of symmetry: A vertical line that divides the graph of a function into two congruent halves.
  • x-intercept: The point where the graph of the function crosses the x-axis.
  • y-intercept: The point where the graph of the function crosses the y-axis.