The Graph Of A Quadratic Function, $y = X^2$, Is Reflected Over The $x$-axis. Which Of The Following Is The Equation Of The Transformed Graph?A. $y = -x^2$ B. $y = (-x)^2$ C. $y = \sqrt{-x}$ D. $y =

Understanding Quadratic Functions

A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $a$ cannot be zero. The graph of a quadratic function is a parabola, which is a U-shaped curve that opens upwards or downwards.

Reflection Over the x-Axis

When a graph is reflected over the x-axis, the y-coordinates of the points on the graph are negated. This means that if a point $(x, y)$ is on the original graph, the corresponding point on the reflected graph is $(x, -y)$. In other words, the reflection of a point over the x-axis is obtained by changing the sign of its y-coordinate.

Transforming the Graph of y = x^2

The graph of $y = x^2$ is a parabola that opens upwards, with its vertex at the origin $(0, 0)$. To reflect this graph over the x-axis, we need to negate the y-coordinate of each point on the graph. This can be achieved by multiplying the entire function by $-1$, which gives us $y = -x^2$.

Analyzing the Options

Now, let's analyze the options given:

A. $y = -x^2$ - This is the correct equation of the transformed graph, as we have negated the y-coordinate of each point on the original graph.

B. $y = (-x)^2$ - This option is incorrect, as the expression $(-x)^2$ is equal to $x^2$, not $-x^2$. The negation of the x-coordinate is not the same as negating the y-coordinate.

C. $y = \sqrt{-x}$ - This option is also incorrect, as the square root of a negative number is not a real number. The graph of $y = \sqrt{-x}$ is not a valid transformation of the graph of $y = x^2$.

D. $y = -\sqrt{x}$ - This option is incorrect, as the square root of a negative number is not a real number. The graph of $y = -\sqrt{x}$ is not a valid transformation of the graph of $y = x^2$.

Conclusion

In conclusion, the equation of the transformed graph of $y = x^2$ after reflection over the x-axis is $y = -x^2$. This is because we have negated the y-coordinate of each point on the original graph, resulting in a parabola that opens downwards.

Key Takeaways

• A quadratic function is a polynomial function of degree two, with the general form $y = ax^2 + bx + c$.
• The graph of a quadratic function is a parabola that opens upwards or downwards.
• Reflection over the x-axis involves negating the y-coordinate of each point on the graph.
• The equation of the transformed graph of $y = x^2$ after reflection over the x-axis is $y = -x^2$.

Further Reading

For more information on quadratic functions and their graphs, we recommend the following resources:

• Khan Academy: Quadratic Functions
• Math Is Fun: Quadratic Functions
• Wolfram MathWorld: Quadratic Function

Practice Problems

Try the following practice problems to test your understanding of quadratic functions and their graphs:

• Find the equation of the graph of $y = x^2 + 3x - 4$ after reflection over the x-axis.
• Find the equation of the graph of $y = 2x^2 - 5x + 1$ after reflection over the x-axis.
• Find the equation of the graph of $y = x^2 - 2x + 1$ after reflection over the x-axis.

Solutions

• The equation of the graph of $y = x^2 + 3x - 4$ after reflection over the x-axis is $y = -(x^2 + 3x - 4)$.
• The equation of the graph of $y = 2x^2 - 5x + 1$ after reflection over the x-axis is $y = -2(x^2 - \frac{5}{2}x + \frac{1}{2})$.
• The equation of the graph of $y = x^2 - 2x + 1$ after reflection over the x-axis is $y = -(x^2 - 2x + 1)$.

Conclusion

In conclusion, the equation of the transformed graph of $y = x^2$ after reflection over the x-axis is $y = -x^2$. This is because we have negated the y-coordinate of each point on the original graph, resulting in a parabola that opens downwards.

Understanding Quadratic Functions

A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $a$ cannot be zero. The graph of a quadratic function is a parabola, which is a U-shaped curve that opens upwards or downwards.

Reflection Over the x-Axis

When a graph is reflected over the x-axis, the y-coordinates of the points on the graph are negated. This means that if a point $(x, y)$ is on the original graph, the corresponding point on the reflected graph is $(x, -y)$. In other words, the reflection of a point over the x-axis is obtained by changing the sign of its y-coordinate.

Q&A

Q: What is the equation of the transformed graph of $y = x^2$ after reflection over the x-axis?

A: The equation of the transformed graph of $y = x^2$ after reflection over the x-axis is $y = -x^2$.

Q: How do you reflect a graph over the x-axis?

A: To reflect a graph over the x-axis, you need to negate the y-coordinate of each point on the graph. This can be achieved by multiplying the entire function by $-1$.

Q: What is the difference between reflecting a graph over the x-axis and reflecting a graph over the y-axis?

A: When a graph is reflected over the x-axis, the y-coordinates of the points on the graph are negated. When a graph is reflected over the y-axis, the x-coordinates of the points on the graph are negated.

Q: Can you give an example of a quadratic function that is reflected over the x-axis?

A: Yes, the quadratic function $y = -x^2$ is an example of a quadratic function that is reflected over the x-axis.

Q: How do you find the equation of the transformed graph of a quadratic function after reflection over the x-axis?

A: To find the equation of the transformed graph of a quadratic function after reflection over the x-axis, you need to negate the y-coordinate of each point on the original graph. This can be achieved by multiplying the entire function by $-1$.

Q: Can you give an example of a quadratic function that is not reflected over the x-axis?

A: Yes, the quadratic function $y = x^2 + 3x - 4$ is an example of a quadratic function that is not reflected over the x-axis.

Q: How do you determine if a quadratic function is reflected over the x-axis?

A: To determine if a quadratic function is reflected over the x-axis, you need to check if the y-coordinate of each point on the graph is negated. If the y-coordinate is negated, then the graph is reflected over the x-axis.

Conclusion

In conclusion, the equation of the transformed graph of $y = x^2$ after reflection over the x-axis is $y = -x^2$. This is because we have negated the y-coordinate of each point on the original graph, resulting in a parabola that opens downwards. We hope that this Q&A article has helped you to understand quadratic functions and their graphs, as well as the concept of reflection over the x-axis.

Key Takeaways

• A quadratic function is a polynomial function of degree two, with the general form $y = ax^2 + bx + c$.
• The graph of a quadratic function is a parabola, which is a U-shaped curve that opens upwards or downwards.
• Reflection over the x-axis involves negating the y-coordinate of each point on the graph.
• The equation of the transformed graph of $y = x^2$ after reflection over the x-axis is $y = -x^2$.

Further Reading

For more information on quadratic functions and their graphs, we recommend the following resources:

• Khan Academy: Quadratic Functions
• Math Is Fun: Quadratic Functions
• Wolfram MathWorld: Quadratic Function

Practice Problems

Try the following practice problems to test your understanding of quadratic functions and their graphs:

• Find the equation of the graph of $y = x^2 + 3x - 4$ after reflection over the x-axis.
• Find the equation of the graph of $y = 2x^2 - 5x + 1$ after reflection over the x-axis.
• Find the equation of the graph of $y = x^2 - 2x + 1$ after reflection over the x-axis.

Solutions

• The equation of the graph of $y = x^2 + 3x - 4$ after reflection over the x-axis is $y = -(x^2 + 3x - 4)$.
• The equation of the graph of $y = 2x^2 - 5x + 1$ after reflection over the x-axis is $y = -2(x^2 - \frac{5}{2}x + \frac{1}{2})$.
• The equation of the graph of $y = x^2 - 2x + 1$ after reflection over the x-axis is $y = -(x^2 - 2x + 1)$.

Conclusion

In conclusion, the equation of the transformed graph of $y = x^2$ after reflection over the x-axis is $y = -x^2$. This is because we have negated the y-coordinate of each point on the original graph, resulting in a parabola that opens downwards.