12.3 The Parabola

What is the equation for the parabola with focus $\left(- \frac{1}{2} , 0\right)$ and directrix $x = \frac{1}{2} ?$

What is the equation for the parabola with focus $\left(0 , \frac{7}{2}\right)$ and directrix $y = - \frac{7}{2} ?$

Balcony-sized solar cookers have been designed for families living in India. The top of a dish has a diameter of 1600 mm. The sun’s rays reflect off the parabolic mirror toward the “cooker,” which is placed 320 mm from the base.

ⓐ Find an equation that models a cross-section of the solar cooker. Assume that the vertex of the parabolic mirror is the origin of the coordinate plane, and that the para…

Define a parabola in terms of its focus and directrix.

If the equation of a parabola is written in standard form and $p$ is negative and the directrix is a horizontal line, then what can we conclude about its graph?

As the graph of a parabola becomes wider, what will happen to the distance between the focus and directrix?

Determine whether the given equation is a parabola. If so, rewrite the equation in standard form.

$y = 4 x^{2}$

Determine whether the given equation is a parabola. If so, rewrite the equation in standard form.

$\left(y - 3\right)^{2} = 8 \left(x - 2\right)$

Rewrite the given equation in standard form, and then determine the vertex $\left(V\right) ,$ focus $\left(F\right) ,$ and directrix $\left(d\right)$ of the parabola.

$x = 8 y^{2}$

Rewrite the given equation in standard form, and then determine the vertex $\left(V\right) ,$ focus $\left(F\right) ,$ and directrix $\left(d\right)$ of the parabola.

$y = −4 x^{2}$

Rewrite the given equation in standard form, and then determine the vertex $\left(V\right) ,$ focus $\left(F\right) ,$ and directrix $\left(d\right)$ of the parabola.

$x = 36 y^{2}$

Rewrite the given equation in standard form, and then determine the vertex $\left(V\right) ,$ focus $\left(F\right) ,$ and directrix $\left(d\right)$ of the parabola.

$\left(x - 1\right)^{2} = 4 \left(y - 1\right)$

Rewrite the given equation in standard form, and then determine the vertex $\left(V\right) ,$ focus $\left(F\right) ,$ and directrix $\left(d\right)$ of the parabola.

$\left(y - 4\right)^{2} = 2 \left(x + 3\right)$

Rewrite the given equation in standard form, and then determine the vertex $\left(V\right) ,$ focus $\left(F\right) ,$ and directrix $\left(d\right)$ of the parabola.

$\left(x + 4\right)^{2} = 24 \left(y + 1\right)$

Rewrite the given equation in standard form, and then determine the vertex $\left(V\right) ,$ focus $\left(F\right) ,$ and directrix $\left(d\right)$ of the parabola.

$y^{2} + 12 x - 6 y + 21 = 0$

Rewrite the given equation in standard form, and then determine the vertex $\left(V\right) ,$ focus $\left(F\right) ,$ and directrix $\left(d\right)$ of the parabola.

$5 x^{2} - 50 x - 4 y + 113 = 0$

Rewrite the given equation in standard form, and then determine the vertex $\left(V\right) ,$ focus $\left(F\right) ,$ and directrix $\left(d\right)$ of the parabola.

$x^{2} - 4 x + 2 y - 6 = 0$

Rewrite the given equation in standard form, and then determine the vertex $\left(V\right) ,$ focus $\left(F\right) ,$ and directrix $\left(d\right)$ of the parabola.

$3 y^{2} - 4 x - 6 y + 23 = 0$

Find the equation of the parabola given information about its graph.

Vertex is $\left(0 , 0\right) ;$ directrix is $y = 4 ,$ focus is $\left(0 , −4\right) .$

Find the equation of the parabola given information about its graph.

Vertex is $\left(2 , 2\right) ;$ directrix is $x = 2 - \sqrt{2} ,$ focus is $\left(2 + \sqrt{2} , 2\right) .$

Find the equation of the parabola given information about its graph.

Vertex is $\left(\sqrt{2} , - \sqrt{3}\right) ;$ directrix is $x = 2 \sqrt{2} ,$ focus is $\left(0 , - \sqrt{3}\right) .$

The vertex and endpoints of the latus rectum of a parabola are given. Find the equation.

$V \left(0 , 0\right)$, Endpoints $\left(−2 , 4\right)$, $\left(−2 , −4\right)$

The vertex and endpoints of the latus rectum of a parabola are given. Find the equation.

$V \left(−3 , −1\right)$, Endpoints $\left(0 , 5\right)$, $\left(0 , −7\right)$

The mirror in an automobile headlight has a parabolic cross-section with the light bulb at the focus. On a schematic, the equation of the parabola is given as $x^{2} = 4 y .$ At what coordinates should you place the light bulb?

A satellite dish is shaped like a paraboloid of revolution. This means that it can be formed by rotating a parabola around its axis of symmetry. The receiver is to be located at the focus. If the dish is 12 feet across at its opening and 4 feet deep at its center, where should the receiver be placed?

The reflector in a searchlight is shaped like a paraboloid of revolution. A light source is located 1 foot from the base along the axis of symmetry. If the opening of the searchlight is 3 feet across, find the depth.

An arch is in the shape of a parabola. It has a span of 100 feet and a maximum height of 20 feet. Find the equation of the parabola, and determine the height of the arch 40 feet from the center.

An object is projected so as to follow a parabolic path given by $y = - x^{2} + 96 x ,$ where $x$ is the horizontal distance traveled in feet and $y$ is the height. Determine the maximum height the object reaches.