The Function $f(x) = -\frac{2}{7}\left(\frac{5}{3}\right)^x$ Is Reflected Over The $y$-axis To Create \$g(x)$[/tex\]. Which Points Represent Ordered Pairs On $g(x)$? Check All That Apply:- $(-7,

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The Function of Reflection: Understanding the Transformation of g(x)

In mathematics, the concept of reflecting a function over the y-axis is a fundamental idea in understanding transformations. When a function is reflected over the y-axis, the x-values of the original function are negated, resulting in a new function that is a mirror image of the original function across the y-axis. In this article, we will explore the function $f(x) = -\frac{2}{7}\left(\frac{5}{3}\right)^x$ and its reflection over the y-axis to create the function $g(x)$.

Understanding the Original Function f(x)

The original function $f(x) = -\frac{2}{7}\left(\frac{5}{3}\right)^x$ is an exponential function with a base of $\frac{5}{3}$ and a coefficient of $-\frac{2}{7}$. The negative coefficient indicates that the function is decreasing as x increases. To understand the behavior of this function, we can analyze its components. The base $\frac{5}{3}$ is greater than 1, indicating that the function will grow exponentially as x increases. However, the negative coefficient $-\frac{2}{7}$ will cause the function to decrease as x increases.

Reflecting the Function over the y-axis

When reflecting a function over the y-axis, the x-values of the original function are negated. This means that for every point (x, y) on the original function, the corresponding point on the reflected function will be (-x, y). In the case of the function $f(x) = -\frac{2}{7}\left(\frac{5}{3}\right)^x$, the reflected function $g(x)$ will have the form $g(x) = -\frac{2}{7}\left(\frac{5}{3}\right)^{-x}$.

Simplifying the Reflected Function g(x)

To simplify the reflected function $g(x)$, we can use the property of exponents that states $a^{-x} = \frac{1}{a^x}$. Applying this property to the reflected function, we get:

g(x)=−27(53)−x=−27(35)xg(x) = -\frac{2}{7}\left(\frac{5}{3}\right)^{-x} = -\frac{2}{7}\left(\frac{3}{5}\right)^x

This simplified form of the reflected function $g(x)$ shows that the function is still decreasing as x increases, but with a different base and coefficient.

Finding Ordered Pairs on g(x)

To find the ordered pairs on the reflected function $g(x)$, we need to substitute different values of x into the function and calculate the corresponding y-values. Since the function is decreasing as x increases, we can expect the y-values to decrease as x increases.

Let's consider the following points:

  • (−7,_)(-7, \_)
  • (−3,_)(-3, \_)
  • (0,_)(0, \_)
  • (3,_)(3, \_)
  • (7,_)(7, \_)

To find the corresponding y-values, we can substitute these x-values into the reflected function $g(x)$:

  • g(−7)=−27(35)−7g(-7) = -\frac{2}{7}\left(\frac{3}{5}\right)^{-7}
  • g(−3)=−27(35)−3g(-3) = -\frac{2}{7}\left(\frac{3}{5}\right)^{-3}
  • g(0)=−27(35)0g(0) = -\frac{2}{7}\left(\frac{3}{5}\right)^0
  • g(3)=−27(35)3g(3) = -\frac{2}{7}\left(\frac{3}{5}\right)^3
  • g(7)=−27(35)7g(7) = -\frac{2}{7}\left(\frac{3}{5}\right)^7

Evaluating these expressions, we get:

  • g(−7)=−27(53)7g(-7) = -\frac{2}{7}\left(\frac{5}{3}\right)^7
  • g(−3)=−27(53)3g(-3) = -\frac{2}{7}\left(\frac{5}{3}\right)^3
  • g(0)=−27g(0) = -\frac{2}{7}
  • g(3)=−27(35)3g(3) = -\frac{2}{7}\left(\frac{3}{5}\right)^3
  • g(7)=−27(53)7g(7) = -\frac{2}{7}\left(\frac{5}{3}\right)^7

These are the ordered pairs on the reflected function $g(x)$.

Conclusion

In conclusion, the function $f(x) = -\frac{2}{7}\left(\frac{5}{3}\right)^x$ is reflected over the y-axis to create the function $g(x) = -\frac{2}{7}\left(\frac{3}{5}\right)^x$. The reflected function $g(x)$ has a different base and coefficient than the original function $f(x)$. By substituting different values of x into the reflected function $g(x)$, we can find the corresponding ordered pairs on the function.

Answer Key

Based on the calculations above, the correct ordered pairs on the reflected function $g(x)$ are:

  • (−7,−27(53)7)(-7, -\frac{2}{7}\left(\frac{5}{3}\right)^7)
  • (−3,−27(53)3)(-3, -\frac{2}{7}\left(\frac{5}{3}\right)^3)
  • (0,−27)(0, -\frac{2}{7})
  • (3,−27(35)3)(3, -\frac{2}{7}\left(\frac{3}{5}\right)^3)
  • (7,−27(53)7)(7, -\frac{2}{7}\left(\frac{5}{3}\right)^7)
    Q&A: Reflection of Functions over the y-axis

In the previous article, we explored the concept of reflecting a function over the y-axis and applied it to the function $f(x) = -\frac{2}{7}\left(\frac{5}{3}\right)^x$. In this article, we will answer some frequently asked questions about reflecting functions over the y-axis.

Q: What is the purpose of reflecting a function over the y-axis?

A: Reflecting a function over the y-axis is a way to create a new function that is a mirror image of the original function across the y-axis. This can be useful in various mathematical applications, such as graphing functions, solving equations, and analyzing the behavior of functions.

Q: How do I reflect a function over the y-axis?

A: To reflect a function over the y-axis, you need to negate the x-values of the original function. This means that for every point (x, y) on the original function, the corresponding point on the reflected function will be (-x, y).

Q: What happens to the graph of a function when it is reflected over the y-axis?

A: When a function is reflected over the y-axis, the graph of the function is flipped across the y-axis. This means that the x-values of the original function are negated, resulting in a new graph that is a mirror image of the original graph.

Q: Can I reflect a function over the y-axis using a calculator or computer software?

A: Yes, you can reflect a function over the y-axis using a calculator or computer software. Most graphing calculators and computer software programs have a function that allows you to reflect a graph across the y-axis.

Q: What are some common mistakes to avoid when reflecting a function over the y-axis?

A: Some common mistakes to avoid when reflecting a function over the y-axis include:

  • Negating the y-values instead of the x-values
  • Failing to negate the x-values correctly
  • Not considering the domain and range of the original function when reflecting it over the y-axis

Q: Can I reflect a function over the y-axis multiple times?

A: Yes, you can reflect a function over the y-axis multiple times. Each time you reflect the function, the graph will be flipped across the y-axis, resulting in a new graph that is a mirror image of the previous graph.

Q: What are some real-world applications of reflecting functions over the y-axis?

A: Reflecting functions over the y-axis has many real-world applications, including:

  • Graphing functions in physics and engineering
  • Solving equations in algebra and calculus
  • Analyzing the behavior of functions in economics and finance
  • Creating mirror images in art and design

Q: Can I reflect a function over the y-axis using a mathematical formula?

A: Yes, you can reflect a function over the y-axis using a mathematical formula. The formula for reflecting a function over the y-axis is:

g(x)=f(−x)g(x) = f(-x)

where $g(x)$ is the reflected function and $f(x)$ is the original function.

Q: What are some common functions that can be reflected over the y-axis?

A: Some common functions that can be reflected over the y-axis include:

  • Linear functions
  • Quadratic functions
  • Exponential functions
  • Trigonometric functions

Q: Can I reflect a function over the y-axis using a graphing software?

A: Yes, you can reflect a function over the y-axis using a graphing software. Most graphing software programs have a function that allows you to reflect a graph across the y-axis.

Q: What are some tips for reflecting functions over the y-axis?

A: Some tips for reflecting functions over the y-axis include:

  • Make sure to negate the x-values correctly
  • Consider the domain and range of the original function when reflecting it over the y-axis
  • Use a graphing calculator or computer software to help you reflect the function
  • Practice reflecting functions over the y-axis to become more comfortable with the process.