Drag Each Sign Or Value To The Correct Location In The Equation. Each Sign Or Value Can Be Used More Than Once, But Not All Signs And Values Will Be Used.The Focus Of A Parabola Is $(-4,-5)$, And Its Directrix Is $y=-1$. Fill In
Introduction
In mathematics, a parabola is a type of quadratic equation that has a specific focus and directrix. The focus of a parabola is a fixed point that is equidistant from the directrix and the parabola itself. In this article, we will explore how to solve a parabola equation using the given focus and directrix.
Understanding the Focus and Directrix
The focus of a parabola is a point that is equidistant from the directrix and the parabola itself. The directrix is a line that is perpendicular to the axis of symmetry of the parabola. In this case, the focus is given as $(-4,-5)$, and the directrix is given as $y=-1$.
The Parabola Equation
A parabola equation can be written in the form $y = a(x-h)^2 + k$, where $(h,k)$ is the vertex of the parabola. The focus of the parabola is given by the equation $y = k + \frac{1}{4a}$.
Step 1: Find the Value of a
To find the value of $a$, we can use the fact that the focus is equidistant from the directrix and the parabola itself. We can write an equation using the given focus and directrix:
Solving for $a$, we get:
Step 2: Find the Value of h
The value of $h$ is the x-coordinate of the vertex of the parabola. Since the focus is given as $(-4,-5)$, we can write:
Step 3: Find the Value of k
The value of $k$ is the y-coordinate of the vertex of the parabola. Since the directrix is given as $y=-1$, we can write:
Step 4: Write the Parabola Equation
Now that we have found the values of $a$, $h$, and $k$, we can write the parabola equation:
Conclusion
In this article, we have solved a parabola equation using the given focus and directrix. We have found the values of $a$, $h$, and $k$, and written the parabola equation in the form $y = a(x-h)^2 + k$. This equation can be used to graph the parabola and find its vertex, focus, and directrix.
Drag Each Sign or Value to the Correct Location in the Equation
| Sign or Value | Correct Location |
|---|---|
| $\frac{1}{4}$ | $a$ |
| $-4$ | $h$ |
| $-1$ | $k$ |
| $x+4$ | $x-h$ |
| $-1$ | $y$ |
Drag Each Sign or Value to the Correct Location in the Equation: A Step-by-Step Guide
Step 1: Identify the Correct Location for Each Sign or Value
-
\frac{1}{4}$ is the value of $a$.
-
-4$ is the value of $h$.
-
-1$ is the value of $k$.
-
x+4$ is the expression for $x-h$.
-
-1$ is the value of $y$.
Step 2: Drag Each Sign or Value to the Correct Location in the Equation
- Drag $\frac{1}{4}$ to the location of $a$.
- Drag $-4$ to the location of $h$.
- Drag $-1$ to the location of $k$.
- Drag $x+4$ to the location of $x-h$.
- Drag $-1$ to the location of $y$.
Step 3: Write the Parabola Equation
- The parabola equation is $y = \frac{1}{4}(x+4)^2 - 1$.
The Final Answer
Introduction
In our previous article, we explored how to solve a parabola equation using the given focus and directrix. In this article, we will answer some frequently asked questions about parabola equations.
Q: What is the focus of a parabola?
A: The focus of a parabola is a fixed point that is equidistant from the directrix and the parabola itself.
Q: What is the directrix of a parabola?
A: The directrix of a parabola is a line that is perpendicular to the axis of symmetry of the parabola.
Q: How do I find the value of a in a parabola equation?
A: To find the value of $a$, you can use the fact that the focus is equidistant from the directrix and the parabola itself. You can write an equation using the given focus and directrix, and then solve for $a$.
Q: How do I find the value of h in a parabola equation?
A: The value of $h$ is the x-coordinate of the vertex of the parabola. You can find the value of $h$ by using the given focus and directrix.
Q: How do I find the value of k in a parabola equation?
A: The value of $k$ is the y-coordinate of the vertex of the parabola. You can find the value of $k$ by using the given directrix.
Q: What is the vertex of a parabola?
A: The vertex of a parabola is the point where the parabola changes direction. It is the minimum or maximum point of the parabola.
Q: How do I write a parabola equation in the form y = a(x-h)^2 + k?
A: To write a parabola equation in the form $y = a(x-h)^2 + k$, you need to find the values of $a$, $h$, and $k$. You can use the given focus and directrix to find these values.
Q: What is the axis of symmetry of a parabola?
A: The axis of symmetry of a parabola is a line that passes through the vertex of the parabola and is perpendicular to the directrix.
Q: How do I graph a parabola?
A: To graph a parabola, you can use the given focus and directrix to find the vertex and axis of symmetry. You can then use these values to graph the parabola.
Q: What is the equation of a parabola in standard form?
A: The equation of a parabola in standard form is $y = a(x-h)^2 + k$, where $(h,k)$ is the vertex of the parabola.
Q: How do I find the x-intercepts of a parabola?
A: To find the x-intercepts of a parabola, you can set $y = 0$ and solve for $x$.
Q: How do I find the y-intercept of a parabola?
A: To find the y-intercept of a parabola, you can set $x = 0$ and solve for $y$.
Conclusion
In this article, we have answered some frequently asked questions about parabola equations. We hope that this article has been helpful in understanding parabola equations and how to solve them.
Parabola Equation Q&A: A Step-by-Step Guide
Step 1: Identify the Correct Location for Each Sign or Value
-
\frac{1}{4}$ is the value of $a$.
-
-4$ is the value of $h$.
-
-1$ is the value of $k$.
-
x+4$ is the expression for $x-h$.
-
-1$ is the value of $y$.
Step 2: Drag Each Sign or Value to the Correct Location in the Equation
- Drag $\frac{1}{4}$ to the location of $a$.
- Drag $-4$ to the location of $h$.
- Drag $-1$ to the location of $k$.
- Drag $x+4$ to the location of $x-h$.
- Drag $-1$ to the location of $y$.
Step 3: Write the Parabola Equation
- The parabola equation is $y = \frac{1}{4}(x+4)^2 - 1$.
The Final Answer
The final answer is $y = \frac{1}{4}(x+4)^2 - 1$.