Graphing the Reflected Function

Oct 13, 2025 by windailys.com 70 views

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Understanding Reflections in the y-axis

In mathematics, reflections are an essential concept in graphing and geometry. When a function is reflected in the y-axis, it means that the x-coordinates of the function are negated, while the y-coordinates remain the same. This process creates a mirror image of the original function on the opposite side of the y-axis.

Graphing the Function ${$ h $}$

To graph the function ${$ h $}$, we need to understand that it is the reflection of the function ${$ f $}$ in the y-axis. This means that for every point (x, y) on the graph of ${$ f $}$, there is a corresponding point (-x, y) on the graph of ${$ h $}$.

Let's consider an example to illustrate this concept. Suppose we have a function ${$ f $}$ with the equation ${$ y = x^2 + 3x - 2 $}$. To graph the reflection of this function in the y-axis, we need to negate the x-coordinates of the function.

The equation of the reflected function ${$ h $}$ is ${$ y = (-x)^2 + 3(-x) - 2 $}$. Simplifying this equation, we get ${$ y = x^2 - 3x - 2 $}$.

Graphing the Reflected Function

To graph the reflected function ${$ h $}$, we can use the same techniques as graphing the original function ${$ f $}$. However, we need to keep in mind that the x-coordinates are negated.

The graph of the reflected function ${$ h $}$ is a mirror image of the graph of the original function ${$ f $}$ on the opposite side of the y-axis.

Finding the Coordinates of the Vertex of ${$ h $}$

The vertex of a quadratic function is the point on the graph where the function changes from decreasing to increasing or vice versa. To find the coordinates of the vertex of the reflected function ${$ h $}$, we need to use the formula for the x-coordinate of the vertex, which is ${$ x = -\frac{b}{2a} $}$.

In the equation ${$ y = x^2 - 3x - 2 $}$, the coefficients are a = 1 and b = -3. Plugging these values into the formula, we get ${$ x = -\frac{-3}{2(1)} = \frac{3}{2} $}$.

To find the y-coordinate of the vertex, we need to plug the x-coordinate into the equation of the function. Substituting x = \frac{3}{2} into the equation, we get ${$ y = \left(\frac{3}{2}\right)^2 - 3\left(\frac{3}{2}\right) - 2 $}$.

Simplifying this expression, we get ${$ y = \frac{9}{4} - \frac{9}{2} - 2 = \frac{9}{4} - \frac{18}{4} - \frac{8}{4} = -\frac{17}{4} $}$.

Therefore, the coordinates of the vertex of the reflected function ${$ h $}$ are ${$ \left(\frac{3}{2}, -\frac{17}{4}\right) $}$.

Equation of the Reflected Function

The equation of the reflected function ${$ h $}$ is ${$ y = x^2 - 3x - 2 $}$. This equation represents the mirror image of the original function ${$ f $}$ on the opposite side of the y-axis.

Graphing the Reflected Function

To graph the reflected function ${$ h $}$, we can use the same techniques as graphing the original function ${$ f $}$. However, we need to keep in mind that the x-coordinates are negated.

The graph of the reflected function ${$ h $}$ is a mirror image of the graph of the original function ${$ f $}$ on the opposite side of the y-axis.

Conclusion

In conclusion, reflecting a function in the y-axis involves negating the x-coordinates of the function. This process creates a mirror image of the original function on the opposite side of the y-axis. The equation of the reflected function can be found by negating the x-coordinates of the original function. The coordinates of the vertex of the reflected function can be found using the formula for the x-coordinate of the vertex.

Key Takeaways

Reflections in the y-axis involve negating the x-coordinates of the function.
The equation of the reflected function can be found by negating the x-coordinates of the original function.
The coordinates of the vertex of the reflected function can be found using the formula for the x-coordinate of the vertex.

Final Thoughts

Reflections in the y-axis are an essential concept in graphing and geometry. Understanding how to reflect a function in the y-axis can help us visualize and analyze the behavior of functions in different coordinate systems. By applying the concepts learned in this article, we can gain a deeper understanding of the properties and behavior of functions in mathematics.

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Understanding Reflections in the y-axis

Q: What is the equation of the reflected function?

A: The equation of the reflected function can be found by negating the x-coordinates of the original function.

Q: How do I graph the reflected function?

A: To graph the reflected function, we can use the same techniques as graphing the original function. However, we need to keep in mind that the x-coordinates are negated.

Q: What are the coordinates of the vertex of the reflected function?

A: The coordinates of the vertex of the reflected function can be found using the formula for the x-coordinate of the vertex, which is ${$ x = -\frac{b}{2a} $}$.

Q: How do I find the y-coordinate of the vertex of the reflected function?

A: To find the y-coordinate of the vertex, we need to plug the x-coordinate into the equation of the function.

Q: What is the relationship between the original function and the reflected function?

A: The reflected function is a mirror image of the original function on the opposite side of the y-axis.

Q: Can I reflect a function in the x-axis?

A: Yes, you can reflect a function in the x-axis by negating the y-coordinates of the function.

Q: How do I graph a function that is reflected in the x-axis?

A: To graph a function that is reflected in the x-axis, we can use the same techniques as graphing the original function. However, we need to keep in mind that the y-coordinates are negated.

Q: What are the coordinates of the vertex of a function that is reflected in the x-axis?

A: The coordinates of the vertex of a function that is reflected in the x-axis can be found using the formula for the y-coordinate of the vertex, which is ${$ y = -\frac{c}{2a} $}$.

Q: How do I find the x-coordinate of the vertex of a function that is reflected in the x-axis?

A: To find the x-coordinate of the vertex, we need to plug the y-coordinate into the equation of the function.

Q: What is the relationship between the original function and the function that is reflected in the x-axis?

A: The function that is reflected in the x-axis is a mirror image of the original function on the opposite side of the x-axis.

Conclusion

In conclusion, reflections in the y-axis and x-axis are essential concepts in graphing and geometry. Understanding how to reflect a function in the y-axis and x-axis can help us visualize and analyze the behavior of functions in different coordinate systems. By applying the concepts learned in this article, we can gain a deeper understanding of the properties and behavior of functions in mathematics.

Key Takeaways

Reflections in the y-axis involve negating the x-coordinates of the function.
Reflections in the x-axis involve negating the y-coordinates of the function.
The equation of the reflected function can be found by negating the x-coordinates or y-coordinates of the original function.
The coordinates of the vertex of the reflected function can be found using the formula for the x-coordinate or y-coordinate of the vertex.

Final Thoughts

Reflections in the y-axis and x-axis are fundamental concepts in mathematics that can help us understand the behavior of functions in different coordinate systems. By applying the concepts learned in this article, we can gain a deeper understanding of the properties and behavior of functions in mathematics.