Analyzing Equation Options

Oct 13, 2025 by thesweetdesigns.com 132 views

In mathematics, a parabola is a type of quadratic curve that can be represented by a quadratic equation. The graph of a parabola is a U-shaped curve that can open upwards or downwards. In this article, we will explore the characteristics of parabolas and how to identify their equations.

Vertex Form of a Parabola

The vertex form of a parabola is given by the equation $y=a(x-h)^2+k$ , where $(h,k)$ is the vertex of the parabola. The vertex is the point on the parabola that is the lowest or highest point, depending on whether the parabola opens upwards or downwards.

Given Information

We are given that the vertex of the parabola is at $(-1,-1)$ . This means that the value of $h$ is $-1$ and the value of $k$ is $-1$ .

Equation Options

We are given four equation options:

A. $y=(x-1)^2+1$ B. $y=(x-1)^2-1$ C. $y=(x+1)^2+1$ D. $y=(x+1)^2-1$

Analyzing Equation Options

Let's analyze each equation option to determine which one has a graph that is a parabola with a vertex at $(-1,-1)$ .

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

The equation $y=(x-1)^2+1$ can be rewritten as $y=(x-(-1))^2+1$ . This means that the vertex of the parabola is at $(1,1)$ , not $(-1,-1)$ . Therefore, option A is not the correct answer.

Option B: $y=(x-1)^2-1$

The equation $y=(x-1)^2-1$ can be rewritten as $y=(x-(-1))^2-1$ . This means that the vertex of the parabola is at $(1,-1)$ , not $(-1,-1)$ . Therefore, option B is not the correct answer.

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

The equation $y=(x+1)^2+1$ can be rewritten as $y=(x-(-1))^2+1$ . This means that the vertex of the parabola is at $(-1,1)$ , not $(-1,-1)$ . Therefore, option C is not the correct answer.

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

The equation $y=(x+1)^2-1$ can be rewritten as $y=(x-(-1))^2-1$ . This means that the vertex of the parabola is at $(-1,-1)$ , which matches the given information. Therefore, option D is the correct answer.

Conclusion

In conclusion, the equation that has a graph that is a parabola with a vertex at $(-1,-1)$ is $y=(x+1)^2-1$ . This equation matches the given information and can be verified by analyzing the vertex form of a parabola.

Understanding Quadratic Equations

Quadratic equations are a type of polynomial equation that can be written in the form $ax^2+bx+c=0$ . The graph of a quadratic equation is a parabola that can open upwards or downwards. In this article, we will explore the characteristics of quadratic equations and how to identify their graphs.

Vertex Form of a Quadratic Equation

The vertex form of a quadratic equation is given by the equation $y=a(x-h)^2+k$ , where $(h,k)$ is the vertex of the parabola. The vertex is the point on the parabola that is the lowest or highest point, depending on whether the parabola opens upwards or downwards.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is given by the equation $y=ax^2+bx+c$ . This form is useful for identifying the coefficients of the quadratic equation.

Graphing Quadratic Equations

The graph of a quadratic equation is a parabola that can open upwards or downwards. The vertex of the parabola is the point on the graph that is the lowest or highest point.

Identifying the Vertex of a Parabola

The vertex of a parabola can be identified by analyzing the equation of the parabola. The vertex form of a parabola is given by the equation $y=a(x-h)^2+k$ , where $(h,k)$ is the vertex of the parabola.

Conclusion

In conclusion, quadratic equations are a type of polynomial equation that can be written in the form $ax^2+bx+c=0$ . The graph of a quadratic equation is a parabola that can open upwards or downwards. The vertex form of a quadratic equation is given by the equation $y=a(x-h)^2+k$ , where $(h,k)$ is the vertex of the parabola. The vertex of a parabola can be identified by analyzing the equation of the parabola.

Real-World Applications of Quadratic Equations

Quadratic equations have many real-world applications. Some examples include:

Projectile Motion: Quadratic equations can be used to model the trajectory of a projectile under the influence of gravity.
Optimization: Quadratic equations can be used to optimize functions and find the maximum or minimum value of a function.
Electrical Engineering: Quadratic equations can be used to model the behavior of electrical circuits and find the voltage and current in a circuit.

Conclusion

Frequently Asked Questions About Quadratic Equations

In this article, we will answer some of the most frequently asked questions about quadratic equations.

Q: What is a quadratic equation?

A: A quadratic equation is a type of polynomial equation that can be written in the form $ax^2+bx+c=0$ . The graph of a quadratic equation is a parabola that can open upwards or downwards.

Q: What is the vertex form of a quadratic equation?

A: The vertex form of a quadratic equation is given by the equation $y=a(x-h)^2+k$ , where $(h,k)$ is the vertex of the parabola.

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

A: To identify the vertex of a parabola, you can analyze the equation of the parabola. The vertex form of a parabola is given by the equation $y=a(x-h)^2+k$ , where $(h,k)$ is the vertex of the parabola.

Q: What is the standard form of a quadratic equation?

A: The standard form of a quadratic equation is given by the equation $y=ax^2+bx+c$ . This form is useful for identifying the coefficients of the quadratic equation.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you can use the vertex form of the equation. The vertex form of a quadratic equation is given by the equation $y=a(x-h)^2+k$ , where $(h,k)$ is the vertex of the parabola.

Q: What are some real-world applications of quadratic equations?

A: Quadratic equations have many real-world applications, including:

Projectile Motion: Quadratic equations can be used to model the trajectory of a projectile under the influence of gravity.
Optimization: Quadratic equations can be used to optimize functions and find the maximum or minimum value of a function.
Electrical Engineering: Quadratic equations can be used to model the behavior of electrical circuits and find the voltage and current in a circuit.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ . This formula will give you the solutions to the quadratic equation.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to solve quadratic equations. It is given by the equation $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ .

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of $a$ , $b$ , and $c$ into the formula. The formula will then give you the solutions to the quadratic equation.

Conclusion

In conclusion, quadratic equations are a type of polynomial equation that can be written in the form $ax^2+bx+c=0$ . The graph of a quadratic equation is a parabola that can open upwards or downwards. Quadratic equations have many real-world applications and can be used to model a wide range of phenomena.