Which Statements Are True For The Functions G ( X ) = X 2 G(x) = X^2 G ( X ) = X 2 And H ( X ) = − X 2 H(x) = -x^2 H ( X ) = − X 2 ? Check All That Apply.- For Any Value Of X X X , G ( X G(x G ( X ] Will Always Be Greater Than H ( X H(x H ( X ].- For Any Value Of X X X ,

Comparing Quadratic Functions: A Closer Look at $g(x) = x^2$ and $h(x) = -x^2$

When it comes to quadratic functions, understanding their behavior and properties is crucial in mathematics. In this article, we will delve into the world of quadratic functions, specifically focusing on the functions $g(x) = x^2$ and $h(x) = -x^2$. We will examine the given statements and determine which ones are true for these functions.

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 $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. The graph of a quadratic function is a parabola, which is a U-shaped curve.

The Functions $g(x) = x^2$ and $h(x) = -x^2$

The function $g(x) = x^2$ is a quadratic function with a positive leading coefficient, which means the parabola opens upwards. On the other hand, the function $h(x) = -x^2$ is a quadratic function with a negative leading coefficient, which means the parabola opens downwards.

Analyzing the Statements

Let's examine the given statements and determine which ones are true for the functions $g(x) = x^2$ and $h(x) = -x^2$.

Statement 1: For any value of $x$, $g(x)$ will always be greater than $h(x)$.

To determine if this statement is true, we need to compare the values of $g(x)$ and $h(x)$ for any value of $x$. Let's consider a few examples.

For $x = 0$, we have $g(0) = 0^2 = 0$ and $h(0) = -0^2 = 0$. In this case, $g(x)$ is equal to $h(x)$, not greater.

For $x = 1$, we have $g(1) = 1^2 = 1$ and $h(1) = -1^2 = -1$. In this case, $g(x)$ is greater than $h(x)$.

For $x = -1$, we have $g(-1) = (-1)^2 = 1$ and $h(-1) = -(-1)^2 = -1$. In this case, $g(x)$ is greater than $h(x)$.

As we can see, the statement is not true for all values of $x$. In fact, the statement is only true for positive values of $x$.

Statement 2: For any value of $x$, $h(x)$ will always be less than $g(x)$.

This statement is the opposite of the first statement. Let's examine if it's true.

For $x = 0$, we have $g(0) = 0^2 = 0$ and $h(0) = -0^2 = 0$. In this case, $g(x)$ is equal to $h(x)$, not less.

For $x = 1$, we have $g(1) = 1^2 = 1$ and $h(1) = -1^2 = -1$. In this case, $h(x)$ is less than $g(x)$.

For $x = -1$, we have $g(-1) = (-1)^2 = 1$ and $h(-1) = -(-1)^2 = -1$. In this case, $h(x)$ is less than $g(x)$.

As we can see, the statement is true for all values of $x$.

Statement 3: The graph of $g(x)$ is always above the graph of $h(x)$.

This statement is similar to the first statement. Let's examine if it's true.

For $x = 0$, the graphs of $g(x)$ and $h(x)$ intersect at the point $(0, 0)$.

For $x > 0$, the graph of $g(x)$ is above the graph of $h(x)$.

For $x < 0$, the graph of $g(x)$ is below the graph of $h(x)$.

As we can see, the statement is not true for all values of $x$. In fact, the statement is only true for positive values of $x$.

Statement 4: The graph of $h(x)$ is always below the graph of $g(x)$.

This statement is the opposite of the third statement. Let's examine if it's true.

For $x = 0$, the graphs of $g(x)$ and $h(x)$ intersect at the point $(0, 0)$.

For $x > 0$, the graph of $h(x)$ is below the graph of $g(x)$.

For $x < 0$, the graph of $h(x)$ is above the graph of $g(x)$.

As we can see, the statement is true for all values of $x$.

Conclusion

In conclusion, we have examined the given statements and determined which ones are true for the functions $g(x) = x^2$ and $h(x) = -x^2$. We found that:

• Statement 2: For any value of $x$, $h(x)$ will always be less than $g(x)$, is true.
• Statement 4: The graph of $h(x)$ is always below the graph of $g(x)$, is true.

The other statements are not true for all values of $x$. We hope this article has provided a clear understanding of the behavior and properties of the functions $g(x) = x^2$ and $h(x) = -x^2$.
Quadratic Functions Q&A: $g(x) = x^2$ and $h(x) = -x^2$

In our previous article, we explored the properties and behavior of the quadratic functions $g(x) = x^2$ and $h(x) = -x^2$. We examined the given statements and determined which ones are true for these functions. In this article, we will answer some frequently asked questions about these functions.

Q: What is the difference between $g(x) = x^2$ and $h(x) = -x^2$?

A: The main difference between $g(x) = x^2$ and $h(x) = -x^2$ is the sign of the leading coefficient. The function $g(x) = x^2$ has a positive leading coefficient, which means the parabola opens upwards. On the other hand, the function $h(x) = -x^2$ has a negative leading coefficient, which means the parabola opens downwards.

Q: How do the graphs of $g(x)$ and $h(x)$ compare?

A: The graphs of $g(x)$ and $h(x)$ are both parabolas, but they open in opposite directions. The graph of $g(x)$ opens upwards, while the graph of $h(x)$ opens downwards. This means that the graph of $h(x)$ is below the graph of $g(x)$ for all values of $x$.

Q: What is the vertex of the parabola $g(x) = x^2$?

A: The vertex of the parabola $g(x) = x^2$ is at the point $(0, 0)$. This is because the parabola opens upwards and has a minimum value at the origin.

Q: What is the vertex of the parabola $h(x) = -x^2$?

A: The vertex of the parabola $h(x) = -x^2$ is also at the point $(0, 0)$. This is because the parabola opens downwards and has a maximum value at the origin.

Q: How do the functions $g(x)$ and $h(x)$ behave as $x$ approaches infinity?

A: As $x$ approaches infinity, the function $g(x) = x^2$ also approaches infinity. This is because the parabola opens upwards and has no upper bound. On the other hand, the function $h(x) = -x^2$ approaches negative infinity as $x$ approaches infinity. This is because the parabola opens downwards and has no lower bound.

Q: How do the functions $g(x)$ and $h(x)$ behave as $x$ approaches negative infinity?

A: As $x$ approaches negative infinity, the function $g(x) = x^2$ also approaches infinity. This is because the parabola opens upwards and has no upper bound. On the other hand, the function $h(x) = -x^2$ approaches negative infinity as $x$ approaches negative infinity. This is because the parabola opens downwards and has no lower bound.

Q: Can we find the intersection points of the graphs of $g(x)$ and $h(x)$?

A: Yes, we can find the intersection points of the graphs of $g(x)$ and $h(x)$. To do this, we need to set the two functions equal to each other and solve for $x$. This will give us the $x$-coordinates of the intersection points. We can then substitute these values into either function to find the corresponding $y$-coordinates.

Q: How do the functions $g(x)$ and $h(x)$ relate to each other?

A: The functions $g(x)$ and $h(x)$ are related in that they are both quadratic functions. However, they have different signs and therefore behave differently. The function $g(x) = x^2$ has a positive leading coefficient, while the function $h(x) = -x^2$ has a negative leading coefficient. This means that the graph of $h(x)$ is below the graph of $g(x)$ for all values of $x$.

Conclusion

In conclusion, we have answered some frequently asked questions about the quadratic functions $g(x) = x^2$ and $h(x) = -x^2$. We hope this article has provided a clear understanding of the properties and behavior of these functions. If you have any further questions, please don't hesitate to ask.