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. F ( X ) = − 1 ( X + 1 ) 2 F(x) = -1(x+1)^2 F ( X ) = − 1 ( X + 1 ) 2

The $y$-intercept of a parabola is 1, and its vertex is at $(1,0)$: What function does the graph represent?

A parabola is a fundamental concept in mathematics, particularly in algebra and geometry. It is a U-shaped curve that can be represented by a quadratic equation in the form of $f(x) = ax^2 + bx + c$. The parabola has several key features, including its vertex, axis of symmetry, and $y$-intercept. In this article, we will focus on the $y$-intercept and the vertex of a parabola to determine the function that represents its graph.

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, it is the point on the $y$-axis that the parabola passes through. The $y$-intercept is denoted by the letter $c$ in the quadratic equation $f(x) = ax^2 + bx + c$. In this case, the $y$-intercept is given as 1, which means that the parabola passes through the point $(0,1)$.

The Vertex of a Parabola

The vertex of a parabola is the highest or lowest point on the curve. It is the point where the parabola changes direction, from opening upwards to opening downwards or vice versa. The vertex is denoted by the letter $(h,k)$ in the quadratic equation $f(x) = a(x-h)^2 + k$. In this case, the vertex is given as $(1,0)$, which means that the parabola opens downwards and has a minimum value of 0 at $x=1$.

Determining the Function

To determine the function that represents the graph of the parabola, we need to use the information given about the $y$-intercept and the vertex. Since the $y$-intercept is 1, we know that the parabola passes through the point $(0,1)$. This means that the value of $c$ in the quadratic equation is 1. Since the vertex is at $(1,0)$, we know that the parabola opens downwards and has a minimum value of 0 at $x=1$. This means that the value of $a$ in the quadratic equation is negative.

Analyzing the Options

Now that we have determined the values of $a$ and $c$, we can analyze the options given to determine which function represents the graph of the parabola.

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

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

Option A represents a parabola that opens upwards and has a vertex at $(1,0)$. However, this is not consistent with the information given, which states that the parabola opens downwards.

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

Option B represents a parabola that opens upwards and has a vertex at $(-1,0)$. This is not consistent with the information given, which states that the vertex is at $(1,0)$.

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

Option C represents a parabola that opens downwards and has a vertex at $(1,0)$. This is consistent with the information given, which states that the parabola opens downwards and has a vertex at $(1,0)$.

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

Option D represents a parabola that opens downwards and has a vertex at $(-1,0)$. This is not consistent with the information given, which states that the vertex is at $(1,0)$.

Conclusion

Based on the analysis of the options, we can conclude that the function that represents the graph of the parabola is:

$f(x) = -1(x-1)^2$

This function represents a parabola that opens downwards and has a vertex at $(1,0)$. It is consistent with the information given about the $y$-intercept and the vertex of the parabola.
The $y$-intercept of a parabola is 1, and its vertex is at $(1,0)$: What function does the graph represent? - Q&A

A parabola is a fundamental concept in mathematics, particularly in algebra and geometry. It is a U-shaped curve that can be represented by a quadratic equation in the form of $f(x) = ax^2 + bx + c$. The parabola has several key features, including its vertex, axis of symmetry, and $y$-intercept. In this article, we will focus on the $y$-intercept and the vertex of a parabola to determine the function that represents its graph.

Q&A: The $y$-Intercept 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.

Q: How do you find the $y$-intercept of a parabola? A: To find the $y$-intercept of a parabola, you need to substitute $x=0$ into the quadratic equation $f(x) = ax^2 + bx + c$. This will give you the value of $y$ when $x$ is equal to zero.

Q: What is the $y$-intercept of the parabola in this problem? A: The $y$-intercept of the parabola in this problem is 1.

Q&A: The Vertex of a Parabola

Q: What is the vertex of a parabola? A: The vertex of a parabola is the highest or lowest point on the curve. It is the point where the parabola changes direction, from opening upwards to opening downwards or vice versa.

Q: How do you find the vertex of a parabola? A: To find the vertex of a parabola, you need to use the formula $f(x) = a(x-h)^2 + k$, where $(h,k)$ is the vertex of the parabola.

Q: What is the vertex of the parabola in this problem? A: The vertex of the parabola in this problem is $(1,0)$.

Q&A: Determining the Function

Q: How do you determine the function that represents the graph of a parabola? A: To determine the function that represents the graph of a parabola, you need to use the information given about the $y$-intercept and the vertex. You can then use this information to write the quadratic equation in the form of $f(x) = ax^2 + bx + c$.

Q: What is the function that represents the graph of the parabola in this problem? A: The function that represents the graph of the parabola in this problem is $f(x) = -1(x-1)^2$.

Q&A: Analyzing the Options

Q: How do you analyze the options to determine the function that represents the graph of a parabola? A: To analyze the options, you need to substitute the values of $x$ into each option and see which one gives you the correct value of $y$.

Q: What are the options for the function that represents the graph of the parabola in this problem? A: The options for the function that represents the graph of the parabola in this problem are:

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

Q: Which option represents the function that represents the graph of the parabola in this problem? A: Option C: $f(x) = -1(x-1)^2$ represents the function that represents the graph of the parabola in this problem.

Conclusion

In this article, we have discussed the $y$-intercept and the vertex of a parabola and how to determine the function that represents the graph of a parabola. We have also analyzed the options to determine the function that represents the graph of the parabola in this problem. The function that represents the graph of the parabola in this problem is $f(x) = -1(x-1)^2$.