If the directrix is horizontal, then the parabola opens (text{horizontally}/text{vertically})

Question
Conic sections
If the directrix is horizontal, then the parabola opens $$(\text{horizontally}/\text{vertically})$$

2021-03-07
Step 1
Given:
We must find the missing word in the given sentence.
So, Definition of parabola said:
A parabola is another type of conic section generated by the cross section of a cone intersected by a plane.
An equation of the form:
$$\displaystyle{y}={a}{x}^{2}+{b}{x}+{c}{\left({a}\ne{0}\right)}$$
Step 2
The above equation is parabola opening upward if $$\displaystyle{a}>{0}$$ and opening downward if $$\displaystyle{a}<{0}.$$</span>
So, the geometrically definition of parabola is the set of all points in a plane that the equidistant from a fixed line called directrix and a fixed point called the focus.
Step 3
In the standard form of a parabola, if x and y are replaced by $$\displaystyle{x}-{h}{\quad\text{and}\quad}{y}-{k}$$, then the graph of the parabola is shifted h units horizontally and k units vertically. The vertex is (h, k), and the focus is $$\displaystyle{\left({h},{k}+{p}\right)}.$$
The directrix is the line defined by the equation $$\displaystyle{y}={k}-{p}$$ and the equation of the parabola is:
$$\displaystyle{\left({x}-{h}\right)}^{2}={4}{p}{\left({y}-{k}\right)}$$
Step 4
The equation of directrix $$\displaystyle{y}={k}-{p}$$ is horizontal. So, the parabola opens towards vertically.
Thus, the missing word is Vertically.

Relevant Questions

A latus rectum of a conic section is a chord through a focus parallel to the directrix. Find the area bounded by the parabola $$\displaystyle{y}={x}^{2}\text{/}{\left({4}{c}\right)}$$ and its latus rectum.
Write the equation of each conic section, given the following characteristics:
a) Write the equation of an ellipse with center at (3, 2) and horizontal major axis with length 8. The minor axis is 6 units long.
b) Write the equation of a hyperbola with vertices at (3, 3) and (-3,3). The foci are located at (4, 3) and (-4, 3).
c) Write the equation of a parabola with vertex at (-2,4) and focus at (-4, 4)
Write the equation of each conic section, given the following characteristics:
a) Write the equation of an ellipse with center at (3, 2) and horizontal major axis with length 8. The minor axis is 6 units long.
b) Write the equation of a hyperbola with vertices at (3, 3) and (-3, 3). The foci are located at (4, 3) and (-4, 3).
c) Write the equation of a parabola with vertex at (-2, 4) and focus at (-4, 4)
For Exercise, a. Identify the equation as representing a circle, an ellipse, a hyperbola, or a parabola. b. Graph the curve. c. Identify key features of the graph. That is. If the equation represents a circle, identify the center and radius. • If the equation represents an ellipse, identify the center, vertices, endpoints of the minor axis, foci, and eccentricity. • If the equation represents a hyperbola, identify the center, vertices, foci, equations of the asymptotes, and eccentricity. • If the equation represents a parabola, identify the vertex, focus, endpoints of the latus rectum, equation of the directrix, and equation of the axis of symmetry. $$x2\+\ y2\ −\ 4x\ −\ 6y\ +\ 1 = 0$$
If a conic section is written as a polar equation, and the denominator involves $$\displaystyle{\sin{\ }}\theta$$, what conclusion can be drawn about the directrix?
We need to verify theorem 5 Focus-Directrix definition: $$\displaystyle{0}\ {<}\ {e}\ {<}\ {1}\ \text{and}\ {c}={d}{\left({e}\ -\ {2}\ -\ {1}\right)}$$
Decide if the equation defines an ellipse, a hyperbola, a parabola, or no conic section at all.
$$\displaystyle{\left({a}\right)}{4}{x}{2}-{9}{y}{2}={12}{\left({b}\right)}-{4}{x}+{9}{y}{2}={0}$$
$$\displaystyle{\left({c}\right)}{4}{y}{2}+{9}{x}{2}={12}{\left({d}\right)}{4}{x}{3}+{9}{y}{3}={12}$$
An automobile tire manufacturer collected the data in the table relating tire pressure x​ (in pounds per square​ inch) and mileage​ (in thousands of​ miles). A mathematical model for the data is given by
$$\displaystyle​ f{{\left({x}\right)}}=-{0.554}{x}^{2}+{35.5}{x}-{514}.$$
$$\begin{array}{|c|c|} \hline x & Mileage \\ \hline 28 & 45 \\ \hline 30 & 51\\ \hline 32 & 56\\ \hline 34 & 50\\ \hline 36 & 46\\ \hline \end{array}$$
​(A) Complete the table below.
$$\begin{array}{|c|c|} \hline x & Mileage & f(x) \\ \hline 28 & 45 \\ \hline 30 & 51\\ \hline 32 & 56\\ \hline 34 & 50\\ \hline 36 & 46\\ \hline \end{array}$$
​(Round to one decimal place as​ needed.)
$$A. 20602060xf(x)$$
A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (28,45), (30,51), (32,56), (34,50), and (36,46). A parabola opens downward and passes through the points (28,45.7), (30,52.4), (32,54.7), (34,52.6), and (36,46.0). All points are approximate.
$$B. 20602060xf(x)$$
Acoordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2.
Data points are plotted at (43,30), (45,36), (47,41), (49,35), and (51,31). A parabola opens downward and passes through the points (43,30.7), (45,37.4), (47,39.7), (49,37.6), and (51,31). All points are approximate.
$$C. 20602060xf(x)$$
A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (43,45), (45,51), (47,56), (49,50), and (51,46). A parabola opens downward and passes through the points (43,45.7), (45,52.4), (47,54.7), (49,52.6), and (51,46.0). All points are approximate.
$$D.20602060xf(x)$$
A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (28,30), (30,36), (32,41), (34,35), and (36,31). A parabola opens downward and passes through the points (28,30.7), (30,37.4), (32,39.7), (34,37.6), and (36,31). All points are approximate.
​(C) Use the modeling function​ f(x) to estimate the mileage for a tire pressure of 29
$$\displaystyle​\frac{{{l}{b}{s}}}{{{s}{q}}}\in.$$ and for 35
$$\displaystyle​\frac{{{l}{b}{s}}}{{{s}{q}}}\in.$$
The mileage for the tire pressure $$\displaystyle{29}\frac{{{l}{b}{s}}}{{{s}{q}}}\in.$$ is
The mileage for the tire pressure $$\displaystyle{35}\frac{{{l}{b}{s}}}{{{s}{q}}}$$ in. is
(Round to two decimal places as​ needed.)
(D) Write a brief description of the relationship between tire pressure and mileage.
A. As tire pressure​ increases, mileage decreases to a minimum at a certain tire​ pressure, then begins to increase.
B. As tire pressure​ increases, mileage decreases.
C. As tire pressure​ increases, mileage increases to a maximum at a certain tire​ pressure, then begins to decrease.
D. As tire pressure​ increases, mileage increases.
$$\displaystyle{2}\sqrt{{3}}{x}^{2}-{6}{x}{y}+\sqrt{{3}}{x}+{3}{y}={0}$$
b) Use a rotation of axes to eliminate the xy-term. (Write an equation in XY-coordinates. Use a rotation angle that satisfies $$\displaystyle{0}\le\varphi\le\pi\text{/}{2}$$)
Determine whether the statement If $$\displaystyle{D}\ne{0}{\quad\text{or}\quad}{E}\ne{0}$$,
then the graph of $$\displaystyle{y}^{2}-{x}^{2}+{D}{x}+{E}{y}={0}$$ is a hyperbolais true or false. If it is false, explain why or give an example that shows it is false.