The Function F ( X ) = 2 X F(x) = 2^x F ( X ) = 2 X Is Translated Left 3 Units And Down 2 Units. Which Is The Equation Of The Translated Function?A. G ( X ) = 2 X − 3 − 2 G(x) = 2^{x-3} - 2 G ( X ) = 2 X − 3 − 2 B. G ( X ) = 2 X − 2 − 3 G(x) = 2^{x-2} - 3 G ( X ) = 2 X − 2 − 3 C. G ( X ) = 2 X − 2 + 3 G(x) = 2^{x-2} + 3 G ( X ) = 2 X − 2 + 3 D. $g(x) =

Understanding Function Translation

In mathematics, function translation is a process of shifting a function's graph to a new position on the coordinate plane. This can be achieved by changing the input values (x-values) or the output values (y-values) of the function. In this article, we will explore how to translate the function $f(x) = 2^x$ left 3 units and down 2 units.

What is Function Translation?

Function translation is a fundamental concept in mathematics, particularly in algebra and calculus. It involves shifting the graph of a function to a new position on the coordinate plane. There are two types of function translations:

• Horizontal translation: This involves shifting the graph of a function to the left or right by changing the input values (x-values).
• Vertical translation: This involves shifting the graph of a function up or down by changing the output values (y-values).

Translating the Function $f(x) = 2^x$

To translate the function $f(x) = 2^x$ left 3 units and down 2 units, we need to understand how to perform horizontal and vertical translations.

Horizontal Translation

To translate the function $f(x) = 2^x$ left 3 units, we need to change the input values (x-values) by adding 3 to the original function. This can be represented as:

$f(x + 3) = 2^{x+3}$

Vertical Translation

To translate the function $f(x) = 2^x$ down 2 units, we need to change the output values (y-values) by subtracting 2 from the original function. This can be represented as:

$f(x) - 2 = 2^x - 2$

Combining Horizontal and Vertical Translations

To translate the function $f(x) = 2^x$ left 3 units and down 2 units, we need to combine the horizontal and vertical translations. This can be represented as:

$f(x + 3) - 2 = 2^{x+3} - 2$

Simplifying the equation, we get:

$g(x) = 2^{x+3} - 2$

Simplifying the Equation

To simplify the equation, we can use the properties of exponents. Specifically, we can use the property that $a^{b+c} = a^b \cdot a^c$. Applying this property to the equation, we get:

$g(x) = 2^x \cdot 2^3 - 2$

Simplifying further, we get:

$g(x) = 8 \cdot 2^x - 2$

Comparing with the Options

Comparing the simplified equation with the options, we can see that the correct equation is:

$g(x) = 2^{x-3} - 2$

This is option A.

Conclusion

In conclusion, to translate the function $f(x) = 2^x$ left 3 units and down 2 units, we need to combine the horizontal and vertical translations. The correct equation is:

$g(x) = 2^{x-3} - 2$

This is option A.

Final Answer

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Understanding Function Translation

In mathematics, function translation is a process of shifting a function's graph to a new position on the coordinate plane. This can be achieved by changing the input values (x-values) or the output values (y-values) of the function. In this article, we will explore the concept of function translation and answer some frequently asked questions.

Q: What is function translation?

A: Function translation is a process of shifting a function's graph to a new position on the coordinate plane. This can be achieved by changing the input values (x-values) or the output values (y-values) of the function.

Q: What are the two types of function translations?

A: The two types of function translations are:

• Horizontal translation: This involves shifting the graph of a function to the left or right by changing the input values (x-values).
• Vertical translation: This involves shifting the graph of a function up or down by changing the output values (y-values).

Q: How do I translate a function left 3 units?

A: To translate a function left 3 units, you need to change the input values (x-values) by adding 3 to the original function. This can be represented as:

$f(x + 3) = 2^{x+3}$

Q: How do I translate a function down 2 units?

A: To translate a function down 2 units, you need to change the output values (y-values) by subtracting 2 from the original function. This can be represented as:

$f(x) - 2 = 2^x - 2$

Q: How do I combine horizontal and vertical translations?

A: To combine horizontal and vertical translations, you need to add the horizontal translation to the original function and then subtract the vertical translation. This can be represented as:

$f(x + 3) - 2 = 2^{x+3} - 2$

Q: What is the equation of the translated function?

A: The equation of the translated function is:

$g(x) = 2^{x-3} - 2$

Q: How do I simplify the equation of the translated function?

A: To simplify the equation of the translated function, you can use the properties of exponents. Specifically, you can use the property that $a^{b+c} = a^b \cdot a^c$. Applying this property to the equation, you get:

$g(x) = 2^x \cdot 2^3 - 2$

Simplifying further, you get:

$g(x) = 8 \cdot 2^x - 2$

Q: What is the final answer?

A: The final answer is option A: $g(x) = 2^{x-3} - 2$.

Common Mistakes to Avoid

• Not understanding the concept of function translation: Function translation is a fundamental concept in mathematics, and it's essential to understand it before attempting to translate a function.
• Not using the correct notation: Using the correct notation is crucial when translating a function. Make sure to use the correct symbols and equations.
• Not simplifying the equation: Simplifying the equation of the translated function is essential to get the correct answer.

Conclusion

In conclusion, function translation is a process of shifting a function's graph to a new position on the coordinate plane. By understanding the concept of function translation and using the correct notation, you can translate a function left 3 units and down 2 units. The equation of the translated function is:

$g(x) = 2^{x-3} - 2$

This is option A.

Final Answer

The final answer is option A: $g(x) = 2^{x-3} - 2$.