Select The Correct Answer.The Y Y Y -intercept Of A Parabola Is 1, And Its Vertex Is At ( 1 , 0 (1,0 ( 1 , 0 ]. What Function Does The Graph Represent?A. F ( X ) = ( X − 1 ) 2 F(x) = (x-1)^2 F ( X ) = ( X − 1 ) 2 B. F ( X ) = ( X + 1 ) 2 F(x) = (x+1)^2 F ( X ) = ( X + 1 ) 2 C. F ( X ) = − 1 ( X − 1 ) 2 F(x) = -1(x-1)^2 F ( X ) = − 1 ( X − 1 ) 2 D.

In mathematics, a parabola is a type of quadratic function that can be represented in various forms. The $y$-intercept and vertex of a parabola are two essential points that help in identifying the function it represents. In this article, we will explore the concept of the $y$-intercept and vertex of a parabola and use this information to select the correct function that represents the given graph.

What is the $y$-Intercept of a Parabola?

The $y$-intercept of a parabola is the point where the parabola intersects the $y$-axis. It is the value of $y$ when $x$ is equal to zero. In other words, the $y$-intercept is the constant term in the equation of the parabola. For example, in the equation $f(x) = ax^2 + bx + c$, the $y$-intercept is given by the value of $c$.

What is the Vertex of a Parabola?

The vertex of a parabola is the highest or lowest point on the graph. It is the point where the parabola changes direction, either from increasing to decreasing or from decreasing to increasing. The vertex is represented by the coordinates $(h, k)$, where $h$ is the $x$-coordinate and $k$ is the $y$-coordinate.

Given Information

We are given that the $y$-intercept of the parabola is 1, and its vertex is at $(1,0)$. This means that the parabola intersects the $y$-axis at the point $(0,1)$, and its vertex is located at the point $(1,0)$.

Selecting the Correct Function

To select the correct function that represents the given graph, we need to consider the properties of the parabola. Since the $y$-intercept is 1, the constant term in the equation of the parabola is 1. This means that the equation of the parabola can be written in the form $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex.

Option A: $f(x) = (x-1)^2$

Option A represents a parabola with its vertex at $(1,0)$. However, the $y$-intercept of this parabola is 0, not 1. Therefore, this option is incorrect.

Option B: $f(x) = (x+1)^2$

Option B represents a parabola with its vertex at $(-1,0)$. However, the $y$-intercept of this parabola is 1, but the vertex is not at $(1,0)$. Therefore, this option is also incorrect.

Option C: $f(x) = -1(x-1)^2$

Option C represents a parabola with its vertex at $(1,0)$. The $y$-intercept of this parabola is 1, which matches the given information. However, the coefficient of the squared term is -1, which means that the parabola opens downwards. This is not consistent with the given information, which states that the vertex is at $(1,0)$. Therefore, this option is incorrect.

Option D: $f(x) = (x-1)^2 + 1$

Option D represents a parabola with its vertex at $(1,0)$. The $y$-intercept of this parabola is 1, which matches the given information. The equation of this parabola can be written in the form $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex. In this case, $h = 1$ and $k = 0$, so the equation becomes $f(x) = (x - 1)^2 + 1$. This option is consistent with the given information and represents the correct function.

Conclusion

In conclusion, the correct function that represents the given graph is $f(x) = (x-1)^2 + 1$. This function has a $y$-intercept of 1 and a vertex at $(1,0)$. The other options do not match the given information and are therefore incorrect.

References

• [1] "Parabolas" by Math Open Reference. Retrieved from https://www.mathopenref.com/parabolas.html
• [2] "Vertex Form of a Parabola" by Purplemath. Retrieved from https://www.purplemath.com/modules/parabolas.htm

Additional Resources

• [1] "Parabolas" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f0f7d7f-parabolas
• [2] "Vertex Form of a Parabola" by Mathway. Retrieved from https://www.mathway.com/subjects/Algebra/Parabolas/Vertex-Form
  Q&A: Understanding the $y$-Intercept and Vertex of a Parabola ===========================================================

In our previous article, we explored the concept of the $y$-intercept and vertex of a parabola and used this information to select the correct function that represents the given graph. In this article, we will answer some frequently asked questions related to the $y$-intercept and vertex of a parabola.

Q: What is the $y$-intercept of a parabola?

A: The $y$-intercept of a parabola is the point where the parabola intersects the $y$-axis. It is the value of $y$ when $x$ is equal to zero. In other words, the $y$-intercept is the constant term in the equation of the parabola.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the highest or lowest point on the graph. It is the point where the parabola changes direction, either from increasing to decreasing or from decreasing to increasing. The vertex is represented by the coordinates $(h, k)$, where $h$ is the $x$-coordinate and $k$ is the $y$-coordinate.

Q: How do I find the $y$-intercept of a parabola?

A: To find the $y$-intercept of a parabola, you need to substitute $x = 0$ into the equation of the parabola. This will give you the value of $y$ when $x$ is equal to zero, which is the $y$-intercept.

Q: How do I find the vertex of a parabola?

A: To find the vertex of a parabola, you need to use the formula $h = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic equation. Once you have the value of $h$, you can substitute it back into the equation to find the value of $k$.

Q: What is the difference between the $y$-intercept and the vertex of a parabola?

A: The $y$-intercept of a parabola is the point where the parabola intersects the $y$-axis, while the vertex is the highest or lowest point on the graph. The $y$-intercept is a single point, while the vertex is a point on the graph.

Q: How do I use the $y$-intercept and vertex to select the correct function?

A: To select the correct function, you need to use the $y$-intercept and vertex to determine the equation of the parabola. The equation of the parabola can be written in the form $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex. You can then substitute the values of $h$ and $k$ into the equation to find the correct function.

Q: What are some common mistakes to avoid when working with the $y$-intercept and vertex of a parabola?

A: Some common mistakes to avoid when working with the $y$-intercept and vertex of a parabola include:

• Confusing the $y$-intercept with the vertex
• Not using the correct formula to find the vertex
• Not substituting the correct values into the equation
• Not checking the units of the answer

Conclusion

In conclusion, the $y$-intercept and vertex of a parabola are two essential points that help in identifying the function it represents. By understanding the concept of the $y$-intercept and vertex, you can select the correct function and avoid common mistakes.

References

• [1] "Parabolas" by Math Open Reference. Retrieved from https://www.mathopenref.com/parabolas.html
• [2] "Vertex Form of a Parabola" by Purplemath. Retrieved from https://www.purplemath.com/modules/parabolas.htm

Additional Resources

• [1] "Parabolas" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f0f7d7f-parabolas
• [2] "Vertex Form of a Parabola" by Mathway. Retrieved from https://www.mathway.com/subjects/Algebra/Parabolas/Vertex-Form