Conics- Parabola Question

In summary, to find the vertex, equation of the axis of symmetry, and direction of opening for the equation (y-2)^2= 4(x-3), you need to use the formula PF=PD. Make sure to note that the right hand side of the equation is positive and find the minimum value of x and corresponding value of y to determine the vertex. Remember that p=1 represents the distance from the vertex to the focus and that the parabola is horizontal.
  • #1
Alain12345
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If I am given (y-2)2= 4 (x-3), how would I find the vertex, equation of the axis of symmetry, and the direction of the opening? I'm guessing that I have to use PF=PD, but it's confusing because it looks different from other things that I have done involving PF=PD.

Thanks.
 
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  • #2
the right hand side of your equation is positive right?, what is the minimum value of x for which this is posible? and the corresponding value of y? this is the vertex. Now rebember that p=1 is the distance from the vertex to the focus and that you have and horizontal parabola.
 
  • #3


I can provide a clear and concise explanation for finding the vertex, equation of the axis of symmetry, and the direction of the opening in this parabola equation. To find the vertex, we can use the standard form of a parabola equation: y = a(x-h)^2 + k, where (h,k) represents the coordinates of the vertex. In your given equation, we can see that the vertex is located at (3,2) since it is in the form (x-h)^2.

To find the equation of the axis of symmetry, we can use the formula x = h, which is the vertical line passing through the vertex. In this case, the equation of the axis of symmetry is x = 3.

As for the direction of the opening, we can determine it by looking at the coefficient of the x-term in the standard form equation. If the coefficient is positive, the parabola opens upwards, and if it is negative, the parabola opens downwards. In this case, the coefficient is 1, which means the parabola opens upwards.

You mentioned using the formula PF=PD, which stands for the distance from the focus (F) to any point (P) on the parabola is equal to the distance from the directrix (D) to that same point. This formula is used to find the focus and directrix of a parabola, but it is not needed in this case since we are only asked to find the vertex, equation of the axis of symmetry, and direction of opening.

I hope this explanation helps clarify any confusion you had about using PF=PD in this problem. As a scientist, it is important to understand and use formulas correctly, but it is also important to know when they are applicable and when they are not necessary. Keep practicing and you will become more comfortable with solving conic equations.
 

1. What is a parabola?

A parabola is a type of conic section, which is a curve formed by slicing a cone at different angles. It is defined by the equation y = ax^2 + bx + c, where a, b, and c are constants. A parabola is characterized by its symmetry and its focus and directrix relationship.

2. How is a parabola graphed?

A parabola can be graphed by plotting points using the equation y = ax^2 + bx + c, or by using the focus and directrix relationship. The focus is a point on the parabola that is equidistant from all points on the curve, while the directrix is a line that is perpendicular to the axis of symmetry and passes through the focus. The parabola will always be symmetrical around the axis of symmetry.

3. What is the significance of the vertex of a parabola?

The vertex of a parabola is the point where the parabola changes direction and is located at the minimum or maximum point of the curve. It is also the point where the axis of symmetry intersects the parabola. The coordinates of the vertex can be found using the formula (-b/2a, c - b^2/4a).

4. What are some real-life applications of parabolas?

Parabolas have many real-life applications, including in engineering, architecture, and physics. Some examples include the design of satellite dishes, the shape of water fountains, and the trajectory of a thrown ball. They are also used in reflecting telescopes and solar cookers.

5. How are parabolas related to other conic sections?

A parabola is one of the four types of conic sections, along with circles, ellipses, and hyperbolas. It can be thought of as a special case of an ellipse or a hyperbola, where the eccentricity is equal to 1. This means that the parabola has only one focus, while ellipses and hyperbolas have two. The circle is a special case of a parabola where the focus and directrix coincide.

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