Graphing the Horizontal Reflection

Oct 13, 2025 by talynt.org 99 views

Understanding the Function

The given function is $$f(x) = (x+1)^2$$. This is a quadratic function in the form of $$(x-h)^2 + k$$, where $$(h,k)$$ is the vertex of the parabola. In this case, the vertex is $$(-1,0)$$.

Graphing the Function

To graph the function, we can start by plotting the vertex at $$(-1,0)$$. Then, we can use the fact that the parabola opens upwards to determine the direction of the curve. We can also use the fact that the axis of symmetry is the vertical line $$x=-1$$ to help us plot the curve.

Graphing the Horizontal Reflection

The horizontal reflection of a function is obtained by reflecting the graph of the function across the x-axis. This means that the y-coordinates of the points on the graph are negated.

Finding the Position of the Point

The point $$(-4,9)$$ is a point on the graph of the function $$f(x) = (x+1)^2$$. To find the position of this point in the reflected function, we need to reflect the point across the x-axis.

Reflection Across the X-Axis

When reflecting a point across the x-axis, the x-coordinate remains the same, but the y-coordinate is negated. Therefore, the point $$(-4,9)$$ reflects to $$(-4,-9)$$ in the reflected function.

Conclusion

In conclusion, the graph of the function $$f(x) = (x+1)^2$$ is a parabola with vertex $$(-1,0)$$. The horizontal reflection of this function is obtained by reflecting the graph across the x-axis. The point $$(-4,9)$$ reflects to $$(-4,-9)$$ in the reflected function.

Step 1: Graph the Function

Step 2: Graph the Horizontal Reflection

The horizontal reflection of a function is obtained by reflecting the graph of the function across the x-axis. This means that the y-coordinates of the points on the graph are negated.

Step 3: Find the Position of the Point

The point $$(-4,9)$$ is a point on the graph of the function $$f(x) = (x+1)^2$$. To find the position of this point in the reflected function, we need to reflect the point across the x-axis.

Step 4: Reflection Across the X-Axis

When reflecting a point across the x-axis, the x-coordinate remains the same, but the y-coordinate is negated. Therefore, the point $$(-4,9)$$ reflects to $$(-4,-9)$$ in the reflected function.

Conclusion

Reflection of Points Across the X-Axis

When reflecting a point across the x-axis, the x-coordinate remains the same, but the y-coordinate is negated. This means that the point $$(x,y)$$ reflects to $$(x,-y)$$.

Example

Suppose we want to reflect the point $$(2,3)$$ across the x-axis. Using the formula above, we can see that the reflected point is $$(2,-3)$$.

Reflection of Points Across the Y-Axis

When reflecting a point across the y-axis, the y-coordinate remains the same, but the x-coordinate is negated. This means that the point $$(x,y)$$ reflects to $$(-x,y)$$.

Example

Suppose we want to reflect the point $$(2,3)$$ across the y-axis. Using the formula above, we can see that the reflected point is $$(-2,3)$$.

Reflection of Points Across the Origin

When reflecting a point across the origin, both the x and y coordinates are negated. This means that the point $$(x,y)$$ reflects to $$(-x,-y)$$.

Example

Suppose we want to reflect the point $$(2,3)$$ across the origin. Using the formula above, we can see that the reflected point is $$(-2,-3)$$.

Reflection of Functions Across the X-Axis

When reflecting a function across the x-axis, the y-coordinates of the points on the graph are negated. This means that the function $$f(x)$$ reflects to $$-f(x)$$.

Example

Suppose we want to reflect the function $$f(x) = x^2$$ across the x-axis. Using the formula above, we can see that the reflected function is $$-f(x) = -x^2$$.

Reflection of Functions Across the Y-Axis

When reflecting a function across the y-axis, the x-coordinates of the points on the graph are negated. This means that the function $$f(x)$$ reflects to $$f(-x)$$.

Example

Suppose we want to reflect the function $$f(x) = x^2$$ across the y-axis. Using the formula above, we can see that the reflected function is $$f(-x) = (-x)^2 = x^2$$.

Reflection of Functions Across the Origin

When reflecting a function across the origin, both the x and y coordinates are negated. This means that the function $$f(x)$$ reflects to $$-f(-x)$$.

Example

Suppose we want to reflect the function $$f(x) = x^2$$ across the origin. Using the formula above, we can see that the reflected function is $$-f(-x) = -(-x)^2 = -x^2$$.

Reflection of Functions in the Coordinate Plane

When reflecting a function in the coordinate plane, we can use the following formulas:

Reflection across the x-axis: $$f(x) \rightarrow -f(x)$$
Reflection across the y-axis: $$f(x) \rightarrow f(-x)$$
Reflection across the origin: $$f(x) \rightarrow -f(-x)$$

Example

Suppose we want to reflect the function $$f(x) = x^2$$ in the coordinate plane. Using the formulas above, we can see that the reflected function is $$-f(-x) = -(-x)^2 = -x^2$$.

Conclusion

Q: What is the reflection of a function across the x-axis?

A: The reflection of a function across the x-axis is obtained by negating the y-coordinates of the points on the graph. This means that the function $$f(x)$$ reflects to $$-f(x)$$.

Q: What is the reflection of a function across the y-axis?

A: The reflection of a function across the y-axis is obtained by negating the x-coordinates of the points on the graph. This means that the function $$f(x)$$ reflects to $$f(-x)$$.

Q: What is the reflection of a function across the origin?

A: The reflection of a function across the origin is obtained by negating both the x and y coordinates of the points on the graph. This means that the function $$f(x)$$ reflects to $$-f(-x)$$.

Q: How do I graph the reflection of a function across the x-axis?

A: To graph the reflection of a function across the x-axis, you can use the following steps:

Graph the original function.
Negate the y-coordinates of the points on the graph.
Plot the new points on the graph.

Q: How do I graph the reflection of a function across the y-axis?

A: To graph the reflection of a function across the y-axis, you can use the following steps:

Graph the original function.
Negate the x-coordinates of the points on the graph.
Plot the new points on the graph.

Q: How do I graph the reflection of a function across the origin?

A: To graph the reflection of a function across the origin, you can use the following steps:

Graph the original function.
Negate both the x and y coordinates of the points on the graph.
Plot the new points on the graph.

Q: What is the difference between reflecting a function across the x-axis and reflecting a function across the y-axis?

A: The main difference between reflecting a function across the x-axis and reflecting a function across the y-axis is the direction of the reflection. When reflecting a function across the x-axis, the y-coordinates of the points on the graph are negated. When reflecting a function across the y-axis, the x-coordinates of the points on the graph are negated.

Q: Can I reflect a function across the origin using a combination of reflections across the x-axis and y-axis?

A: Yes, you can reflect a function across the origin using a combination of reflections across the x-axis and y-axis. To do this, you can first reflect the function across the x-axis, and then reflect the result across the y-axis.

Q: How do I find the reflection of a point across the x-axis?

A: To find the reflection of a point across the x-axis, you can use the following formula:

(x,y) \rightarrow (x,-y)

Q: How do I find the reflection of a point across the y-axis?

A: To find the reflection of a point across the y-axis, you can use the following formula:

(x,y) \rightarrow (-x,y)

Q: How do I find the reflection of a point across the origin?

A: To find the reflection of a point across the origin, you can use the following formula:

(x,y) \rightarrow (-x,-y)

Q: Can I reflect a function using a combination of reflections across the x-axis, y-axis, and origin?

A: Yes, you can reflect a function using a combination of reflections across the x-axis, y-axis, and origin. To do this, you can first reflect the function across the x-axis, then reflect the result across the y-axis, and finally reflect the result across the origin.

Conclusion

In conclusion, reflecting a function in the coordinate plane is an important concept in mathematics. By understanding how to reflect a function across the x-axis, y-axis, and origin, you can graph and analyze functions in a more efficient and effective way.