Which Equation Has A Graph That Is A Parabola With A Vertex At $(-1,-1$\]?A. $y=(x-1)^2+1$B. $y=(x-1)^2-1$C. $y=(x+1)^2+1$D. $y=(x+1)^2-1$

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Understanding Parabolas and Their Vertices

A parabola is a type of quadratic equation that can be represented in various forms, including the standard form y=ax2+bx+cy = ax^2 + bx + c. The vertex of a parabola is the point at which the parabola changes direction, and it is represented by the coordinates (h,k)(h, k). In this case, we are looking for an equation that has a graph that is a parabola with a vertex at (−1,−1)(-1,-1).

Analyzing the Options

Let's analyze each of the given options to determine which one has a graph that is a parabola with a vertex at (−1,−1)(-1,-1).

Option A: y=(x−1)2+1y=(x-1)^2+1

The equation y=(x−1)2+1y=(x-1)^2+1 can be rewritten as y=(x−1)2+1y=(x-1)^2+1. This equation represents a parabola with a vertex at (1,1)(1,1). However, we are looking for a parabola with a vertex at (−1,−1)(-1,-1), so this option does not meet our requirements.

Option B: y=(x−1)2−1y=(x-1)^2-1

The equation y=(x−1)2−1y=(x-1)^2-1 can be rewritten as y=(x−1)2−1y=(x-1)^2-1. This equation represents a parabola with a vertex at (1,−1)(1,-1). Although the x-coordinate of the vertex is correct, the y-coordinate is not, so this option does not meet our requirements.

Option C: y=(x+1)2+1y=(x+1)^2+1

The equation y=(x+1)2+1y=(x+1)^2+1 can be rewritten as y=(x+1)2+1y=(x+1)^2+1. This equation represents a parabola with a vertex at (−1,1)(-1,1). Although the x-coordinate of the vertex is correct, the y-coordinate is not, so this option does not meet our requirements.

Option D: y=(x+1)2−1y=(x+1)^2-1

The equation y=(x+1)2−1y=(x+1)^2-1 can be rewritten as y=(x+1)2−1y=(x+1)^2-1. This equation represents a parabola with a vertex at (−1,−1)(-1,-1). The x-coordinate of the vertex is −1-1, and the y-coordinate is also −1-1, which meets our requirements.

Conclusion

Based on our analysis, the equation that has a graph that is a parabola with a vertex at (−1,−1)(-1,-1) is y=(x+1)2−1y=(x+1)^2-1. This equation represents a parabola with a vertex at (−1,−1)(-1,-1), which meets our requirements.

Why is this Equation Correct?

This equation is correct because it represents a parabola with a vertex at (−1,−1)(-1,-1). The x-coordinate of the vertex is −1-1, and the y-coordinate is also −1-1, which meets our requirements. Additionally, the equation is in the form y=(x−h)2+ky=(x-h)^2+k, where (h,k)(h,k) is the vertex of the parabola. In this case, h=−1h=-1 and k=−1k=-1, which confirms that the equation represents a parabola with a vertex at (−1,−1)(-1,-1).

What is the Significance of the Vertex?

The vertex of a parabola is the point at which the parabola changes direction. In this case, the vertex is at (−1,−1)(-1,-1), which means that the parabola changes direction at this point. The vertex is also the minimum or maximum point of the parabola, depending on the direction of the parabola. In this case, the parabola opens upwards, so the vertex is the minimum point.

How to Identify the Vertex of a Parabola?

To identify the vertex of a parabola, we need to rewrite the equation in the form y=(x−h)2+ky=(x-h)^2+k. The vertex is then represented by the coordinates (h,k)(h,k). In this case, we can rewrite the equation y=(x+1)2−1y=(x+1)^2-1 as y=(x−(−1))2−1y=(x-(-1))^2-1, which confirms that the vertex is at (−1,−1)(-1,-1).

What are the Applications of Parabolas?

Parabolas have many applications in mathematics, science, and engineering. Some of the applications of parabolas include:

  • Projectile Motion: Parabolas are used to model the trajectory of projectiles, such as balls and rockets.
  • Optics: Parabolas are used to design lenses and mirrors that can focus light and images.
  • Electrical Engineering: Parabolas are used to design antennas and other electrical components.
  • Computer Graphics: Parabolas are used to create smooth and realistic curves in computer graphics.

Conclusion

In conclusion, the equation that has a graph that is a parabola with a vertex at (−1,−1)(-1,-1) is y=(x+1)2−1y=(x+1)^2-1. This equation represents a parabola with a vertex at (−1,−1)(-1,-1), which meets our requirements. The vertex of a parabola is the point at which the parabola changes direction, and it is represented by the coordinates (h,k)(h,k). Parabolas have many applications in mathematics, science, and engineering, and they are used to model the trajectory of projectiles, design lenses and mirrors, and create smooth and realistic curves in computer graphics.

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Q: What is a parabola?

A parabola is a type of quadratic equation that can be represented in various forms, including the standard form y=ax2+bx+cy = ax^2 + bx + c. The vertex of a parabola is the point at which the parabola changes direction.

Q: What is the vertex of a parabola?

The vertex of a parabola is the point at which the parabola changes direction. It is represented by the coordinates (h,k)(h,k).

Q: How to identify the vertex of a parabola?

To identify the vertex of a parabola, we need to rewrite the equation in the form y=(x−h)2+ky=(x-h)^2+k. The vertex is then represented by the coordinates (h,k)(h,k).

Q: What is the significance of the vertex of a parabola?

The vertex of a parabola is the point at which the parabola changes direction. It is also the minimum or maximum point of the parabola, depending on the direction of the parabola.

Q: How to determine the direction of a parabola?

To determine the direction of a parabola, we need to look at the coefficient of the x2x^2 term. If the coefficient is positive, the parabola opens upwards. If the coefficient is negative, the parabola opens downwards.

Q: What are the applications of parabolas?

Parabolas have many applications in mathematics, science, and engineering. Some of the applications of parabolas include:

  • Projectile Motion: Parabolas are used to model the trajectory of projectiles, such as balls and rockets.
  • Optics: Parabolas are used to design lenses and mirrors that can focus light and images.
  • Electrical Engineering: Parabolas are used to design antennas and other electrical components.
  • Computer Graphics: Parabolas are used to create smooth and realistic curves in computer graphics.

Q: How to graph a parabola?

To graph a parabola, we need to identify the vertex and the direction of the parabola. We can then use this information to plot the parabola on a coordinate plane.

Q: What is the equation of a parabola with a vertex at (h,k)?

The equation of a parabola with a vertex at (h,k)(h,k) is y=(x−h)2+ky=(x-h)^2+k.

Q: How to find the equation of a parabola given its vertex?

To find the equation of a parabola given its vertex, we need to use the vertex form of the equation, which is y=(x−h)2+ky=(x-h)^2+k. We can then substitute the values of hh and kk into the equation to get the final equation of the parabola.

Q: What is the difference between a parabola and a circle?

A parabola is a type of quadratic equation that can be represented in various forms, including the standard form y=ax2+bx+cy = ax^2 + bx + c. A circle, on the other hand, is a type of quadratic equation that can be represented in the form (x−h)2+(y−k)2=r2(x-h)^2 + (y-k)^2 = r^2, where (h,k)(h,k) is the center of the circle and rr is the radius.

Q: How to determine if a graph is a parabola or a circle?

To determine if a graph is a parabola or a circle, we need to look at the equation of the graph. If the equation is in the form y=ax2+bx+cy = ax^2 + bx + c, it is a parabola. If the equation is in the form (x−h)2+(y−k)2=r2(x-h)^2 + (y-k)^2 = r^2, it is a circle.

Q: What is the significance of the axis of symmetry of a parabola?

The axis of symmetry of a parabola is the vertical line that passes through the vertex of the parabola. It is a line of symmetry, meaning that the parabola is symmetric about this line.

Q: How to find the axis of symmetry of a parabola?

To find the axis of symmetry of a parabola, we need to identify the vertex of the parabola. The axis of symmetry is then the vertical line that passes through the vertex.

Q: What is the relationship between the axis of symmetry and the vertex of a parabola?

The axis of symmetry of a parabola is the vertical line that passes through the vertex of the parabola. The vertex is the point at which the parabola changes direction, and it is represented by the coordinates (h,k)(h,k). The axis of symmetry is then the vertical line that passes through the point (h,k)(h,k).