The Minimum Of A Parabola Is Located At ( − 1 , − 3 (-1,-3 ( − 1 , − 3 ]. The Point ( 0 , 1 (0,1 ( 0 , 1 ] Is Also On The Graph. Which Equation Can Be Solved To Determine The Value Of A A A In The Function Representing The Parabola?A. $1 = A(0 + 1)^2 -

$$

Introduction

In mathematics, a parabola is a quadratic function that can be represented in the form $y = ax^2 + bx + c$. The minimum value of a parabola occurs at its vertex, which can be found using the formula $x = -\frac{b}{2a}$. In this article, we will explore how to find the value of $a$ in a parabola given the minimum point and another point on the graph.

The Minimum Point

The minimum point of a parabola is given as $(-1,-3)$. This means that the vertex of the parabola is located at $x = -1$ and $y = -3$. Since the vertex is the minimum point, we know that the parabola opens upwards.

The Other Point

The other point on the graph is given as $(0,1)$. This means that when $x = 0$, the value of $y$ is $1$.

The Equation of the Parabola

The equation of a parabola can be represented in the form $y = ax^2 + bx + c$. Since the parabola opens upwards, we know that $a > 0$. We can use the two points to create a system of equations to solve for $a$.

Using the Minimum Point

We know that the minimum point is $(-1,-3)$. Plugging this point into the equation of the parabola, we get:

$$-3 = a(-1)^2 + b(-1) + c$$

Simplifying the equation, we get:

$$-3 = a - b + c$$

Using the Other Point

We know that the other point is $(0,1)$. Plugging this point into the equation of the parabola, we get:

$$1 = a(0)^2 + b(0) + c$$

Simplifying the equation, we get:

$$1 = c$$

Solving for $a$

Now that we have two equations, we can solve for $a$. We can substitute the value of $c$ into the first equation:

$$-3 = a - b + 1$$

Simplifying the equation, we get:

$$-4 = a - b$$

We can also use the fact that the vertex is the minimum point to find the value of $b$. Since the vertex is $x = -1$, we know that $b = 2a$.

Substituting $b$

Substituting $b = 2a$ into the equation $-4 = a - b$, we get:

$$-4 = a - 2a$$

Simplifying the equation, we get:

$$-4 = -a$$

Solving for $a$, we get:

$$a = 4$$

Conclusion

In this article, we have shown how to find the value of $a$ in a parabola given the minimum point and another point on the graph. We used the equation of the parabola and the two points to create a system of equations, which we solved to find the value of $a$. The final answer is:

The final answer is: $\boxed{4}$

Discussion

This problem is a classic example of how to use the equation of a parabola to solve for the value of $a$. The key is to use the two points to create a system of equations, which can be solved to find the value of $a$. This problem is a great exercise in algebra and can be used to help students understand the concept of quadratic functions.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "College Algebra" by James Stewart

Related Topics

• Quadratic functions
• Parabolas
• Algebra
• Trigonometry

Keywords

• Parabola
• Quadratic function
• Algebra
• Trigonometry
• Minimum point
• Vertex
• Equation of a parabola
  The Minimum of a Parabola: Q&A =====================================

Introduction

In our previous article, we explored how to find the value of $a$ in a parabola given the minimum point and another point on the graph. In this article, we will answer some common questions related to the minimum of a parabola.

Q: What is the minimum point of a parabola?

A: The minimum point of a parabola is the point where the parabola changes direction from opening upwards to opening downwards. It is also the vertex of the parabola.

Q: How do I find the minimum point of a parabola?

A: To find the minimum point of a parabola, you can use the formula $x = -\frac{b}{2a}$. This will give you the x-coordinate of the vertex. To find the y-coordinate, you can plug the x-coordinate into the equation of the parabola.

Q: What is the equation of a parabola?

A: The equation of a parabola is given by $y = ax^2 + bx + c$. This equation represents a parabola that opens upwards or downwards.

Q: How do I find the value of $a$ in a parabola?

A: To find the value of $a$ in a parabola, you can use the equation of the parabola and the two points on the graph. You can create a system of equations and solve for $a$.

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

A: The vertex of a parabola is the minimum or maximum point of the parabola. It is also the point where the parabola changes direction.

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

A: To find the x-coordinate of the vertex of a parabola, you can use the formula $x = -\frac{b}{2a}$. This will give you the x-coordinate of the vertex.

Q: What is the y-coordinate of the vertex of a parabola?

A: To find the y-coordinate of the vertex of a parabola, you can plug the x-coordinate into the equation of the parabola.

Q: Can a parabola have more than one minimum point?

A: No, a parabola can only have one minimum point. However, a parabola can have more than one maximum point.

Q: Can a parabola have a minimum point at the origin?

A: Yes, a parabola can have a minimum point at the origin. This occurs when the equation of the parabola is of the form $y = ax^2$.

Conclusion

In this article, we have answered some common questions related to the minimum of a parabola. We have discussed the equation of a parabola, the minimum point, and the vertex of a parabola. We have also provided formulas and examples to help you understand the concepts.

Discussion

This article is a great resource for students who are learning about parabolas and quadratic functions. It provides a comprehensive overview of the concepts and formulas related to the minimum of a parabola.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "College Algebra" by James Stewart

Related Topics

• Quadratic functions
• Parabolas
• Algebra
• Trigonometry

Keywords

• Parabola
• Quadratic function
• Algebra
• Trigonometry
• Minimum point
• Vertex
• Equation of a parabola