How Can You Use Transformations To Graph The Function Y = 3 ⋅ 7 − X + 2 Y = 3 \cdot 7^{-x} + 2 Y = 3 ⋅ 7 − X + 2 ?Explain Your Steps.

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Understanding the Function

The given function is $y = 3 \cdot 7^{-x} + 2$. This function involves a negative exponent, which can be rewritten using the properties of exponents. The function can be expressed as $y = 3 \cdot \frac{1}{7^x} + 2$. This form makes it easier to understand the behavior of the function.

Identifying the Base Function

The base function in this case is $y = \frac{1}{7^x}$. This function has a few key characteristics that can help us understand its behavior. The function has a horizontal asymptote at $y = 0$, which means that as $x$ approaches infinity, the value of $y$ approaches 0. The function also has a vertical asymptote at $x = 0$, which means that as $x$ approaches 0, the value of $y$ approaches infinity.

Identifying the Transformations

The given function is a transformation of the base function $y = \frac{1}{7^x}$. The function has been stretched vertically by a factor of 3, which means that the amplitude of the function has been increased by a factor of 3. The function has also been shifted upward by 2 units, which means that the function has been translated 2 units upward.

Graphing the Function

To graph the function $y = 3 \cdot 7^{-x} + 2$, we can start by graphing the base function $y = \frac{1}{7^x}$. We can then apply the transformations to the base function to obtain the graph of the given function.

Step 1: Graph the Base Function

To graph the base function $y = \frac{1}{7^x}$, we can start by plotting a few points on the graph. We can choose values of $x$ such as $x = -1$, $x = 0$, and $x = 1$ to plot the corresponding points on the graph.

x	y
-1	7
0	1
1	1/7

We can then use these points to sketch the graph of the base function.

Step 2: Apply the Vertical Stretch

To apply the vertical stretch to the base function, we can multiply the y-coordinates of the points on the graph by 3. This will increase the amplitude of the function by a factor of 3.

x	y
-1	21
0	3
1	3/7

We can then use these points to sketch the graph of the function after applying the vertical stretch.

Step 3: Apply the Horizontal Reflection

To apply the horizontal reflection to the function, we can replace $x$ with $-x$ in the equation of the function. This will reflect the graph of the function across the y-axis.

$y = 3 \cdot 7^{-(-x)} + 2$

$y = 3 \cdot 7^x + 2$

We can then use this equation to graph the function after applying the horizontal reflection.

Step 4: Apply the Vertical Shift

To apply the vertical shift to the function, we can add 2 to the y-coordinates of the points on the graph. This will translate the graph of the function upward by 2 units.

x	y
-1	25
0	5
1	5/7

We can then use these points to sketch the graph of the function after applying the vertical shift.

Conclusion

In conclusion, we can use transformations to graph the function $y = 3 \cdot 7^{-x} + 2$. We can start by graphing the base function $y = \frac{1}{7^x}$ and then apply the transformations to obtain the graph of the given function. The transformations include a vertical stretch by a factor of 3, a horizontal reflection across the y-axis, and a vertical shift upward by 2 units.

Key Takeaways

• The function $y = 3 \cdot 7^{-x} + 2$ is a transformation of the base function $y = \frac{1}{7^x}$.
• The function has been stretched vertically by a factor of 3.
• The function has been reflected horizontally across the y-axis.
• The function has been shifted upward by 2 units.

Final Graph

The final graph of the function $y = 3 \cdot 7^{-x} + 2$ is a transformation of the base function $y = \frac{1}{7^x}$. The graph has been stretched vertically by a factor of 3, reflected horizontally across the y-axis, and shifted upward by 2 units.

x	y
-1	25
0	5
1	5/7

The graph of the function $y = 3 \cdot 7^{-x} + 2$ is a transformation of the base function $y = \frac{1}{7^x}$. The graph has been stretched vertically by a factor of 3, reflected horizontally across the y-axis, and shifted upward by 2 units.

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Frequently Asked Questions

Q: What is the base function of the given function $y = 3 \cdot 7^{-x} + 2$?

A: The base function of the given function is $y = \frac{1}{7^x}$.

Q: What are the transformations applied to the base function?

A: The transformations applied to the base function are a vertical stretch by a factor of 3, a horizontal reflection across the y-axis, and a vertical shift upward by 2 units.

Q: How do I graph the base function $y = \frac{1}{7^x}$?

A: To graph the base function, you can start by plotting a few points on the graph. Choose values of $x$ such as $x = -1$, $x = 0$, and $x = 1$ to plot the corresponding points on the graph.

x	y
-1	7
0	1
1	1/7

You can then use these points to sketch the graph of the base function.

Q: How do I apply the vertical stretch to the base function?

A: To apply the vertical stretch to the base function, you can multiply the y-coordinates of the points on the graph by 3. This will increase the amplitude of the function by a factor of 3.

x	y
-1	21
0	3
1	3/7

You can then use these points to sketch the graph of the function after applying the vertical stretch.

Q: How do I apply the horizontal reflection to the function?

A: To apply the horizontal reflection to the function, you can replace $x$ with $-x$ in the equation of the function. This will reflect the graph of the function across the y-axis.

$y = 3 \cdot 7^{-(-x)} + 2$

$y = 3 \cdot 7^x + 2$

You can then use this equation to graph the function after applying the horizontal reflection.

Q: How do I apply the vertical shift to the function?

A: To apply the vertical shift to the function, you can add 2 to the y-coordinates of the points on the graph. This will translate the graph of the function upward by 2 units.

x	y
-1	25
0	5
1	5/7

You can then use these points to sketch the graph of the function after applying the vertical shift.

Q: What is the final graph of the function $y = 3 \cdot 7^{-x} + 2$?

A: The final graph of the function $y = 3 \cdot 7^{-x} + 2$ is a transformation of the base function $y = \frac{1}{7^x}$. The graph has been stretched vertically by a factor of 3, reflected horizontally across the y-axis, and shifted upward by 2 units.

x	y
-1	25
0	5
1	5/7

Additional Resources

• For more information on graphing functions, see the article "How to Graph a Function".
• For more information on transformations, see the article "Transformations of Functions".
• For more information on the base function $y = \frac{1}{7^x}$, see the article "Graphing the Function $y = \frac{1}{7^x}$".

Conclusion

In conclusion, the function $y = 3 \cdot 7^{-x} + 2$ is a transformation of the base function $y = \frac{1}{7^x}$. The graph of the function has been stretched vertically by a factor of 3, reflected horizontally across the y-axis, and shifted upward by 2 units. By understanding the transformations applied to the base function, you can graph the function $y = 3 \cdot 7^{-x} + 2$ and visualize its behavior.