Which Function Has A Vertex On The Y Y Y -axis?A. F ( X ) = ( X − 2 ) 2 F(x)=(x-2)^2 F ( X ) = ( X − 2 ) 2 B. F ( X ) = X ( X + 2 F(x)=x(x+2 F ( X ) = X ( X + 2 ]C. F ( X ) = ( X − 2 ) ( X + 2 F(x)=(x-2)(x+2 F ( X ) = ( X − 2 ) ( X + 2 ]D. F ( X ) = ( X + 1 ) ( X − 2 F(x)=(x+1)(x-2 F ( X ) = ( X + 1 ) ( X − 2 ]

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Introduction

In mathematics, a vertex is a point where a curve changes direction. In the context of functions, a vertex is a point where the function changes from increasing to decreasing or vice versa. The $y$-axis is a vertical line that passes through the origin, and a vertex on the $y$-axis means that the function has a minimum or maximum point at $x=0$. In this article, we will explore which function among the given options has a vertex on the $y$-axis.

Understanding the Options

Before we dive into the analysis, let's understand the given options:

• Option A: $f(x)=(x-2)^2$
• Option B: $f(x)=x(x+2)$
• Option C: $f(x)=(x-2)(x+2)$
• Option D: $f(x)=(x+1)(x-2)$

We need to analyze each option to determine which one has a vertex on the $y$-axis.

Analyzing Option A

Option A is a quadratic function in the form of $f(x)=(x-h)^2$, where $h$ is the $x$-coordinate of the vertex. In this case, $h=2$, which means the vertex is at $(2,0)$. Since the vertex is not on the $y$-axis, Option A is not the correct answer.

Analyzing Option B

Option B is a quadratic function in the form of $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are constants. To find the vertex, we can use the formula $x=-\frac{b}{2a}$. In this case, $a=1$ and $b=2$, so $x=-\frac{2}{2(1)}=-1$. This means the vertex is at $(-1,0)$, which is on the $y$-axis. Therefore, Option B is a possible answer.

Analyzing Option C

Option C is a quadratic function in the form of $f(x)=(x-h)(x-k)$, where $h$ and $k$ are constants. To find the vertex, we can use the formula $x=\frac{h+k}{2}$. In this case, $h=2$ and $k=-2$, so $x=\frac{2-2}{2}=0$. This means the vertex is at $(0,0)$, which is on the $y$-axis. Therefore, Option C is a possible answer.

Analyzing Option D

Option D is a quadratic function in the form of $f(x)=(x-h)(x-k)$, where $h$ and $k$ are constants. To find the vertex, we can use the formula $x=\frac{h+k}{2}$. In this case, $h=-1$ and $k=2$, so $x=\frac{-1+2}{2}=\frac{1}{2}$. This means the vertex is not on the $y$-axis, so Option D is not the correct answer.

Conclusion

In conclusion, both Option B and Option C have a vertex on the $y$-axis. However, since the question asks for a single function, we need to choose one of them as the correct answer. Based on the analysis, we can see that both functions have a vertex at $x=0$, but Option C has a more straightforward form that makes it easier to identify the vertex.

Final Answer

Therefore, the final answer is:

• Option C: $f(x)=(x-2)(x+2)$

This function has a vertex on the $y$-axis, and it is the correct answer to the given question.

Additional Information

In addition to the analysis above, we can also use the concept of symmetry to determine which function has a vertex on the $y$-axis. A function is symmetric about the $y$-axis if it is unchanged when $x$ is replaced by $-x$. In this case, we can see that Option B and Option C are symmetric about the $y$-axis, while Option A and Option D are not. This means that Option B and Option C are more likely to have a vertex on the $y$-axis.

Real-World Applications

The concept of a vertex on the $y$-axis has many real-world applications in fields such as physics, engineering, and economics. For example, in physics, the vertex of a parabola can represent the maximum or minimum point of a physical system, such as the trajectory of a projectile or the motion of a pendulum. In engineering, the vertex of a parabola can represent the maximum or minimum point of a structural system, such as the deflection of a beam or the stress on a material. In economics, the vertex of a parabola can represent the maximum or minimum point of a economic system, such as the supply and demand curve of a market.

Conclusion

In conclusion, the concept of a vertex on the $y$-axis is an important concept in mathematics that has many real-world applications. By analyzing the given options, we can see that both Option B and Option C have a vertex on the $y$-axis, but Option C has a more straightforward form that makes it easier to identify the vertex. Therefore, the final answer is:

• Option C: $f(x)=(x-2)(x+2)$

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Introduction

In our previous article, we explored which function among the given options has a vertex on the $y$-axis. We analyzed each option and determined that both Option B and Option C have a vertex on the $y$-axis. However, since the question asks for a single function, we need to choose one of them as the correct answer. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information.

Q: What is a vertex on the $y$-axis?

A: A vertex on the $y$-axis is a point where a curve changes direction and is located on the $y$-axis. In other words, it is a point where the function has a minimum or maximum value at $x=0$.

Q: How do I determine if a function has a vertex on the $y$-axis?

A: To determine if a function has a vertex on the $y$-axis, you can use the following methods:

• Method 1: Analyze the function and determine if it has a minimum or maximum point at $x=0$.
• Method 2: Use the concept of symmetry to determine if the function is symmetric about the $y$-axis.
• Method 3: Use the formula $x=-\frac{b}{2a}$ to find the $x$-coordinate of the vertex.

Q: What is the difference between Option B and Option C?

A: Option B is a quadratic function in the form of $f(x)=ax^2+bx+c$, while Option C is a quadratic function in the form of $f(x)=(x-h)(x-k)$. Although both functions have a vertex on the $y$-axis, Option C has a more straightforward form that makes it easier to identify the vertex.

Q: Can you provide more examples of functions with a vertex on the $y$-axis?

A: Yes, here are a few more examples of functions with a vertex on the $y$-axis:

• Example 1: $f(x)=x^2-4x+4$
• Example 2: $f(x)=(x-1)(x+1)$
• Example 3: $f(x)=x^2+2x+1$

Q: How do I apply the concept of a vertex on the $y$-axis in real-world scenarios?

A: The concept of a vertex on the $y$-axis has many real-world applications in fields such as physics, engineering, and economics. For example, in physics, the vertex of a parabola can represent the maximum or minimum point of a physical system, such as the trajectory of a projectile or the motion of a pendulum. In engineering, the vertex of a parabola can represent the maximum or minimum point of a structural system, such as the deflection of a beam or the stress on a material. In economics, the vertex of a parabola can represent the maximum or minimum point of a economic system, such as the supply and demand curve of a market.

Q: Can you provide more information on the concept of symmetry in relation to the $y$-axis?

A: Yes, the concept of symmetry is an important concept in mathematics that can be used to determine if a function is symmetric about the $y$-axis. A function is symmetric about the $y$-axis if it is unchanged when $x$ is replaced by $-x$. In other words, if $f(x)=f(-x)$, then the function is symmetric about the $y$-axis.

Conclusion

In conclusion, the concept of a vertex on the $y$-axis is an important concept in mathematics that has many real-world applications. By analyzing the given options and using the concept of symmetry, we can determine which function has a vertex on the $y$-axis. We hope that this Q&A section has provided additional information and clarified any doubts. If you have any further questions, please don't hesitate to ask.

Final Answer

Therefore, the final answer is:

• Option C: $f(x)=(x-2)(x+2)$

This function has a vertex on the $y$-axis, and it is the correct answer to the given question.