Vertical and Horizontal Parabolas: Focal Diameter, Focus, Directrix
What is the vertex/transformation form of an equation for a parabola?
(x-h)²=4p(y-k) or (y-k)²=4p(x-h).
What are the equations for parabolas with a vertex that is not at the origin? *Remember the vertex is now (h,k)*
(x-h)²=4p(y-k) or (y-k)²=4p(x-h). (see image, a=P, first row = horizontal, second=vertical)
How is a parabola created?
A parabola is created with the intersection of a cone and a plane parallel to a straight line on the cone.
How can you find the equation for a parabola if it only tells you how far the F (focus) is from the vertex (0,0) and the direction it opens in?
If its vertical, it is x², and if it is horizontal, it is y². The F for vertical parabolas is (0,P) and for horizontal parabolas (p,0). Use the P and substitute its value in the appropriate equation (4py or 4px, etc.).
What do you need to know to draw a graph for a parabola with a vertex not at the origin?
The horizontal/vertical orientation, the P (negative or positive part shows its direction, the distance from vertex to focus/directrix), and the vertex.
How can you draw the graph of a parabola from the equation, assuming the vertex is (0,0)?
x² is an open-up parabola, -x² is an open-down parabola, y² is an open-right parabola and -y² is an open-left parabola. First draw the vertex, which is (0,0). Then draw the focus point, which should be on the line of symmetry. After that, use the focal diameter to find the distance of the two points from the focus of the parabola along the invisible line that goes through the focus. Then draw the directrix, which should be a straight line that is below the vertex and goes in the opposite direction (vertical/horizontal wise) from the parabola. Finally, connect the dots!
What is the distance formula?
√(x2-x1)²+ (y2-y1)²
How can you find the equation of a parabola if it only tells you the FD and the axis the focus in on?
Fd is I4pI, so find P by dividing the Fd by 4. Remember that absolute value creates a positive and negative version of what is in it, so choose the P according to which axis the F is on. If it is in the negative y-axis, choose -P, and if positive, use P, and cross out the one you don't use, which is the extraneous root. Put this P in the appropriate equation (4px or 4py) and solve.
How can you find the equation of a parabola from a graph, assuming the vertex is (0,0)?
Find the P, which is in the F (how far away it is from the vertex), and plug it into the appropriate equation (4py or 4px, etc.).
How can you find the equation of a parabola?
Find the distance between the focus and a point on the parabola and the distance from that point to the directrix. Equate these two distances and solve for the x^2 or y^2 depending on if the parabola is vertical or horizontal.
What is the rule for where P is placed for each type of equation, assuming the vertex is (0,0)?
P is in the opposite x or y place in the F compared to the equation's squared variable (so if eq. has an x² the F will have the P in the y point). P in the D is aligned with the placement of P in the F (so if F has P in y point, D->y=__). Or, *if F=(0,p) D: y=__ and if F=(p,0) D: x=__*. Related to the equation, the D is aligned with the the non-squared variable on the left side of the equation, so *if x=__y² D: x=__ and if y=__x² D: y=__*.
How can you find the focal diameter, focus, or directrix of an equation that is not in the regular form of an equation of a parabola?
Solve the equation for the squared variable, and substitute that squared variable according to the appropriate equation (x² or y² or -x² or -y²). Solve for P, and apply as necessary to the focus, focal diameter, or the directrix.
How can you find the F, D, FD, and the graph from an equation, assuming the vertex is (0,0)? (Hint: equation looks like y=ax² or x=ay²)
Substitute the squared variable for the appropriate equation (4px or 4py), solve for P, and use it to find the F, FD, and D. Then sketch the graph.
What is a locus and how is a parabola one?
The curve formed by a set of points that all have the same property, such as a distance from a certain point. A parabola is a locus because it is the curve formed by the points equidistant from the focus and from the directrix.
What is the horizontal parabola equation for parabolas not centered at the origin, and what is its directrix? How is the focus changed? *Remember the vertex is now (h,k)*
The focus is changed by: (h+p, k). Directrix: x=-p+h or h-p.
What is the vertical parabola equation for parabolas not centered at the origin, and what is its directrix? How is the focus changed? *Remember the vertex is now (h,k)*
The focus is changed by: (h,k+p). Directrix: y=k-p
What is the focal diameter of a parabola?
The length of a line segment that runs through the focus parallel to the directrix with two endpoints on the parabola. It's equation is I4pI (absolute value of 4p).
What is the geometric definition of a parabola?
The locus of points in the plane that are equidistant from a fixed point F (focus) and a fixed line ℓ (directrix).
What are the equations for vertical and horizontal parabolas, assuming the vertex is (0,0)?
VERTICAL PARABOLAS: *open-up* -> x²=4py or y=x²/4p; *open-down* -> x²=-4py or y=x²/-4p. HORIZONTAL PARABOLAS: *open-right* -> y²=4xp; *open-left* -> y²=-4xp.
What is the form of the focus and directrix for vertical and horizontal parabolas, assuming the vertex is (0,0)?
VERTICAL PARABOLAS: Focus: (0,p); Directrix: y=-p. HORIZONTAL PARABOLAS: Focus: (p,0); Directrix: x=-p.
