Identify All Transformations Of The Function F ( X ) = 1 4 ⋅ 3 5 − X + 8 F(x) = \frac{1}{4} \cdot 3^{5-x} + 8 F ( X ) = 4 1 ​ ⋅ 3 5 − X + 8 From Its Parent Function. In 3-5 Sentences, Describe The Transformations. - The Parent Function Is G ( X ) = 3 X G(x) = 3^x G ( X ) = 3 X .- The Function F ( X F(x F ( X ]

Understanding Transformations of the Function $f(x) = \frac{1}{4} \cdot 3^{5-x} + 8$

In mathematics, transformations of functions are essential to understand the behavior and characteristics of various functions. The given function $f(x) = \frac{1}{4} \cdot 3^{5-x} + 8$ is a transformation of its parent function $g(x) = 3^x$. In this article, we will identify and describe the transformations of the function $f(x)$ from its parent function $g(x)$.

Parent Function $g(x) = 3^x$

The parent function $g(x) = 3^x$ is an exponential function with base 3. This function has a characteristic "S" shape, where the value of the function increases as the input value $x$ increases. The parent function has a horizontal asymptote at $y = 0$ and a vertical asymptote at $x = -\infty$.

Transformations of the Function $f(x)$

The function $f(x) = \frac{1}{4} \cdot 3^{5-x} + 8$ is a transformation of the parent function $g(x) = 3^x$. To identify the transformations, we need to analyze the given function and compare it with the parent function.

Vertical Stretch

The function $f(x)$ has a vertical stretch factor of $\frac{1}{4}$, which means that the function is stretched vertically by a factor of $\frac{1}{4}$. This transformation affects the amplitude of the function, making it narrower and taller.

Horizontal Shift

The function $f(x)$ has a horizontal shift of $-5$ units, which means that the function is shifted to the right by 5 units. This transformation affects the position of the function on the x-axis, making it move to the right.

Reflection

The function $f(x)$ is reflected about the x-axis, which means that the function is flipped upside down. This transformation affects the sign of the function, making it negative.

Vertical Translation

The function $f(x)$ has a vertical translation of 8 units, which means that the function is shifted up by 8 units. This transformation affects the position of the function on the y-axis, making it move up.

Exponential Function

The function $f(x)$ is an exponential function with base 3, which means that the function has the same base as the parent function. However, the exponent is $5-x$, which is a transformation of the exponent in the parent function.

In conclusion, the function $f(x) = \frac{1}{4} \cdot 3^{5-x} + 8$ is a transformation of its parent function $g(x) = 3^x$. The transformations include a vertical stretch, horizontal shift, reflection, vertical translation, and an exponential function with base 3. Understanding these transformations is essential to analyze and graph the function $f(x)$.

The graph of the function $f(x)$ is a transformation of the graph of the parent function $g(x)$. The graph of $f(x)$ has a vertical stretch, horizontal shift, reflection, and vertical translation compared to the graph of $g(x)$.

• The function $f(x) = \frac{1}{4} \cdot 3^{5-x} + 8$ is a transformation of its parent function $g(x) = 3^x$.
• The transformations include a vertical stretch, horizontal shift, reflection, vertical translation, and an exponential function with base 3.
• Understanding these transformations is essential to analyze and graph the function $f(x)$.

In this article, we identified and described the transformations of the function $f(x) = \frac{1}{4} \cdot 3^{5-x} + 8$ from its parent function $g(x) = 3^x$. The transformations include a vertical stretch, horizontal shift, reflection, vertical translation, and an exponential function with base 3. Understanding these transformations is essential to analyze and graph the function $f(x)$.
Q&A: Understanding Transformations of the Function $f(x) = \frac{1}{4} \cdot 3^{5-x} + 8$

In our previous article, we discussed the transformations of the function $f(x) = \frac{1}{4} \cdot 3^{5-x} + 8$ from its parent function $g(x) = 3^x$. In this article, we will answer some frequently asked questions about the transformations of the function $f(x)$.

Q: What is the parent function $g(x) = 3^x$?

A: The parent function $g(x) = 3^x$ is an exponential function with base 3. This function has a characteristic "S" shape, where the value of the function increases as the input value $x$ increases.

Q: What are the transformations of the function $f(x)$?

A: The transformations of the function $f(x)$ include a vertical stretch, horizontal shift, reflection, vertical translation, and an exponential function with base 3.

Q: What is the vertical stretch factor of the function $f(x)$?

A: The vertical stretch factor of the function $f(x)$ is $\frac{1}{4}$, which means that the function is stretched vertically by a factor of $\frac{1}{4}$.

Q: What is the horizontal shift of the function $f(x)$?

A: The horizontal shift of the function $f(x)$ is $-5$ units, which means that the function is shifted to the right by 5 units.

Q: Is the function $f(x)$ reflected about the x-axis?

A: Yes, the function $f(x)$ is reflected about the x-axis, which means that the function is flipped upside down.

Q: What is the vertical translation of the function $f(x)$?

A: The vertical translation of the function $f(x)$ is 8 units, which means that the function is shifted up by 8 units.

Q: What is the base of the exponential function $f(x)$?

A: The base of the exponential function $f(x)$ is 3, which is the same as the base of the parent function $g(x)$.

Q: How do the transformations affect the graph of the function $f(x)$?

A: The transformations affect the graph of the function $f(x)$ by stretching it vertically, shifting it horizontally, reflecting it about the x-axis, translating it vertically, and changing the exponent.

Q: Why is it essential to understand the transformations of the function $f(x)$?

A: Understanding the transformations of the function $f(x)$ is essential to analyze and graph the function. It helps to identify the characteristics of the function and its behavior.

In conclusion, the function $f(x) = \frac{1}{4} \cdot 3^{5-x} + 8$ is a transformation of its parent function $g(x) = 3^x$. The transformations include a vertical stretch, horizontal shift, reflection, vertical translation, and an exponential function with base 3. Understanding these transformations is essential to analyze and graph the function $f(x)$.

• The function $f(x) = \frac{1}{4} \cdot 3^{5-x} + 8$ is a transformation of its parent function $g(x) = 3^x$.
• The transformations include a vertical stretch, horizontal shift, reflection, vertical translation, and an exponential function with base 3.
• Understanding these transformations is essential to analyze and graph the function $f(x)$.

In this article, we answered some frequently asked questions about the transformations of the function $f(x) = \frac{1}{4} \cdot 3^{5-x} + 8$. We hope that this article has provided a better understanding of the transformations of the function and its behavior.