Graph of the Translated Function

Oct 13, 2025 by windailys.com 59 views

Model Function: $y=x^2$ Translated Function: $f(x)=-4(x-3)^2+2$

Understanding the Translated Function

The given translated function is $f(x)=-4(x-3)^2+2$ . This function is a transformation of the basic quadratic function $y=x^2$ . To understand the translated function, we need to analyze the different components of the given function.

Horizontal Shift

The given function is in the form $f(x)=a(x-h)^2+k$ , where $(h,k)$ is the vertex of the parabola. In this case, the vertex is $(3,2)$ . The value of $h$ represents the horizontal shift of the function. A positive value of $h$ indicates a shift to the right, while a negative value indicates a shift to the left.

Horizontal Shift: $\boxed{3}$ [Select]

Vertical Shift

The value of $k$ in the function $f(x)=a(x-h)^2+k$ represents the vertical shift of the function. A positive value of $k$ indicates a shift upwards, while a negative value indicates a shift downwards.

Vertical Shift: $\boxed{2}$ [Select]

Reflection

The coefficient $a$ in the function $f(x)=a(x-h)^2+k$ represents the vertical stretch or compression of the function. A negative value of $a$ indicates a reflection across the x-axis.

Reflection: $\boxed{Yes}$ [Select]

Discussion

The translated function $f(x)=-4(x-3)^2+2$ is a transformation of the basic quadratic function $y=x^2$ . The function has been shifted horizontally by 3 units to the right and vertically by 2 units upwards. Additionally, the function has been reflected across the x-axis due to the negative coefficient $a$ .

Understanding the Graph

To visualize the graph of the translated function, we need to consider the individual transformations. The horizontal shift of 3 units to the right means that the graph of the function will be shifted 3 units to the right. The vertical shift of 2 units upwards means that the graph of the function will be shifted 2 units upwards. Finally, the reflection across the x-axis means that the graph of the function will be reflected across the x-axis.

Graph of the Translated Function

The graph of the translated function $f(x)=-4(x-3)^2+2$ is a parabola that has been shifted horizontally by 3 units to the right and vertically by 2 units upwards. The graph has also been reflected across the x-axis due to the negative coefficient $a$ .

Key Takeaways

The translated function $f(x)=-4(x-3)^2+2$ is a transformation of the basic quadratic function $y=x^2$ .
The function has been shifted horizontally by 3 units to the right and vertically by 2 units upwards.
The function has been reflected across the x-axis due to the negative coefficient $a$ .
The graph of the translated function is a parabola that has been shifted horizontally and vertically, and reflected across the x-axis.

Conclusion

In conclusion, the translated function $f(x)=-4(x-3)^2+2$ is a transformation of the basic quadratic function $y=x^2$ . The function has been shifted horizontally by 3 units to the right and vertically by 2 units upwards, and reflected across the x-axis. Understanding the individual transformations and how they affect the graph of the function is essential in visualizing the graph of the translated function.
Model Function: $y=x^2$ Translated Function: $f(x)=-4(x-3)^2+2$ Q&A

Understanding the Translated Function

Q&A

Q: What is the horizontal shift of the function? A: The horizontal shift of the function is 3 units to the right.

Q: What is the vertical shift of the function? A: The vertical shift of the function is 2 units upwards.

Q: Is the function reflected across the x-axis? A: Yes, the function is reflected across the x-axis due to the negative coefficient $a$ .

Q: What is the vertex of the parabola? A: The vertex of the parabola is $(3,2)$ .

Q: What is the effect of the negative coefficient $a$ on the graph of the function? A: The negative coefficient $a$ causes the graph of the function to be reflected across the x-axis.

Q: How does the value of $a$ affect the graph of the function? A: The value of $a$ affects the vertical stretch or compression of the graph of the function. A negative value of $a$ causes the graph to be compressed vertically.

Q: What is the effect of the value of $h$ on the graph of the function? A: The value of $h$ affects the horizontal shift of the graph of the function. A positive value of $h$ causes the graph to be shifted to the right, while a negative value causes the graph to be shifted to the left.

Q: What is the effect of the value of $k$ on the graph of the function? A: The value of $k$ affects the vertical shift of the graph of the function. A positive value of $k$ causes the graph to be shifted upwards, while a negative value causes the graph to be shifted downwards.

Graph of the Translated Function

Key Takeaways

The translated function $f(x)=-4(x-3)^2+2$ is a transformation of the basic quadratic function $y=x^2$ .
The function has been shifted horizontally by 3 units to the right and vertically by 2 units upwards.
The function has been reflected across the x-axis due to the negative coefficient $a$ .
The graph of the translated function is a parabola that has been shifted horizontally and vertically, and reflected across the x-axis.

Conclusion

Frequently Asked Questions

Q: What is the vertex of the parabola? A: The vertex of the parabola is $(3,2)$ .

Q: What is the effect of the negative coefficient $a$ on the graph of the function? A: The negative coefficient $a$ causes the graph of the function to be reflected across the x-axis.

Additional Resources

For more information on the translated function $f(x)=-4(x-3)^2+2$ , please refer to the original article.
For more information on the basic quadratic function $y=x^2$ , please refer to the original article.
For more information on the transformations of the function, please refer to the original article.