A Parabola Opening Up Or Down Has A Vertex At \[$(0, 3)\$\] And Passes Through \[$(-12, -9)\$\]. Write Its Equation In Vertex Form.

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Introduction

In mathematics, a parabola is a quadratic equation that can be represented in various forms, including standard form, vertex form, and factored form. The vertex form of a parabola is a convenient way to express its equation, as it provides a clear representation of the parabola's vertex and direction. In this article, we will explore how to write the equation of a parabola in vertex form, given its vertex and a point it passes through.

Understanding Vertex Form

The vertex form of a parabola is given by the equation:

y=a(x−h)2+ky = a(x - h)^2 + k

where (h,k)(h, k) is the vertex of the parabola, and aa is a constant that determines the direction and width of the parabola. If aa is positive, the parabola opens upward, and if aa is negative, the parabola opens downward.

Given Information

We are given that the parabola has a vertex at (0,3)(0, 3) and passes through the point (−12,−9)(-12, -9). We can use this information to find the value of aa in the vertex form of the parabola's equation.

Finding the Value of aa

To find the value of aa, we can substitute the given point (−12,−9)(-12, -9) into the vertex form of the parabola's equation:

−9=a(−12−0)2+3-9 = a(-12 - 0)^2 + 3

Simplifying the equation, we get:

−9=144a+3-9 = 144a + 3

Subtracting 3 from both sides, we get:

−12=144a-12 = 144a

Dividing both sides by 144, we get:

a=−12144=−112a = -\frac{12}{144} = -\frac{1}{12}

Writing the Equation in Vertex Form

Now that we have found the value of aa, we can write the equation of the parabola in vertex form:

y=−112(x−0)2+3y = -\frac{1}{12}(x - 0)^2 + 3

Simplifying the equation, we get:

y=−112x2+3y = -\frac{1}{12}x^2 + 3

Conclusion

In this article, we have explored how to write the equation of a parabola in vertex form, given its vertex and a point it passes through. We have used the given information to find the value of aa in the vertex form of the parabola's equation and have written the equation in vertex form. The vertex form of a parabola is a convenient way to express its equation, as it provides a clear representation of the parabola's vertex and direction.

Example Use Cases

The vertex form of a parabola has many practical applications in mathematics and science. For example, it can be used to model the trajectory of a projectile, the path of a thrown ball, or the shape of a satellite dish. It can also be used to find the maximum or minimum value of a quadratic function.

Tips and Tricks

When writing the equation of a parabola in vertex form, it is essential to remember that the vertex is the point (h,k)(h, k) in the equation. The value of aa determines the direction and width of the parabola, and the point (x,y)(x, y) can be used to find the value of aa.

Common Mistakes

One common mistake when writing the equation of a parabola in vertex form is to forget to include the value of aa. Another mistake is to substitute the wrong point into the equation. It is essential to double-check the given information and the equation to ensure that it is correct.

Final Thoughts

In conclusion, the vertex form of a parabola is a powerful tool for expressing its equation. By understanding the vertex form and how to write its equation, we can model real-world phenomena and solve problems in mathematics and science.

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Introduction

In our previous article, we explored how to write the equation of a parabola in vertex form, given its vertex and a point it passes through. In this article, we will answer some frequently asked questions about writing the equation of a parabola in vertex form.

Q&A

Q: What is the vertex form of a parabola?

A: The vertex form of a parabola is given by the equation:

y=a(x−h)2+ky = a(x - h)^2 + k

where (h,k)(h, k) is the vertex of the parabola, and aa is a constant that determines the direction and width of the parabola.

Q: How do I find the value of aa in the vertex form of the parabola's equation?

A: To find the value of aa, you can substitute the given point into the vertex form of the parabola's equation. For example, if the parabola passes through the point (x,y)(x, y), you can substitute xx and yy into the equation:

y=a(x−h)2+ky = a(x - h)^2 + k

Solving for aa, you can find its value.

Q: What if the parabola opens downward?

A: If the parabola opens downward, the value of aa will be negative. For example, if the parabola has a vertex at (h,k)(h, k) and passes through the point (x,y)(x, y), you can substitute xx and yy into the equation:

y=−a(x−h)2+ky = -a(x - h)^2 + k

Solving for aa, you can find its value.

Q: Can I use the vertex form of a parabola to find the maximum or minimum value of a quadratic function?

A: Yes, you can use the vertex form of a parabola to find the maximum or minimum value of a quadratic function. The vertex of the parabola is the point where the function reaches its maximum or minimum value.

Q: How do I write the equation of a parabola in vertex form if I only know its vertex and a point it passes through?

A: To write the equation of a parabola in vertex form, you can use the given point and the vertex to find the value of aa. Then, you can substitute the vertex and the value of aa into the vertex form of the parabola's equation.

Q: Can I use the vertex form of a parabola to model real-world phenomena?

A: Yes, you can use the vertex form of a parabola to model real-world phenomena. For example, you can use it to model the trajectory of a projectile, the path of a thrown ball, or the shape of a satellite dish.

Example Use Cases

The vertex form of a parabola has many practical applications in mathematics and science. For example, it can be used to:

  • Model the trajectory of a projectile
  • Find the maximum or minimum value of a quadratic function
  • Model the path of a thrown ball
  • Model the shape of a satellite dish

Tips and Tricks

When writing the equation of a parabola in vertex form, it is essential to remember that the vertex is the point (h,k)(h, k) in the equation. The value of aa determines the direction and width of the parabola, and the point (x,y)(x, y) can be used to find the value of aa.

Common Mistakes

One common mistake when writing the equation of a parabola in vertex form is to forget to include the value of aa. Another mistake is to substitute the wrong point into the equation. It is essential to double-check the given information and the equation to ensure that it is correct.

Final Thoughts

In conclusion, the vertex form of a parabola is a powerful tool for expressing its equation. By understanding the vertex form and how to write its equation, we can model real-world phenomena and solve problems in mathematics and science.