# 10.3 The parabola  (Page 6/11)

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## Key equations

 Parabola, vertex at origin, axis of symmetry on x -axis ${y}^{2}=4px$ Parabola, vertex at origin, axis of symmetry on y -axis ${x}^{2}=4py$ Parabola, vertex at $\text{\hspace{0.17em}}\left(h,k\right),$ axis of symmetry on x -axis ${\left(y-k\right)}^{2}=4p\left(x-h\right)$ Parabola, vertex at $\text{\hspace{0.17em}}\left(h,k\right),$ axis of symmetry on y -axis ${\left(x-h\right)}^{2}=4p\left(y-k\right)$

## Key concepts

• A parabola is the set of all points $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ in a plane that are the same distance from a fixed line, called the directrix, and a fixed point (the focus) not on the directrix.
• The standard form of a parabola with vertex $\text{\hspace{0.17em}}\left(0,0\right)\text{\hspace{0.17em}}$ and the x -axis as its axis of symmetry can be used to graph the parabola. If $\text{\hspace{0.17em}}p>0,$ the parabola opens right. If $\text{\hspace{0.17em}}p<0,$ the parabola opens left. See [link] .
• The standard form of a parabola with vertex $\text{\hspace{0.17em}}\left(0,0\right)\text{\hspace{0.17em}}$ and the y -axis as its axis of symmetry can be used to graph the parabola. If $\text{\hspace{0.17em}}p>0,$ the parabola opens up. If $\text{\hspace{0.17em}}p<0,$ the parabola opens down. See [link] .
• When given the focus and directrix of a parabola, we can write its equation in standard form. See [link] .
• The standard form of a parabola with vertex $\text{\hspace{0.17em}}\left(h,k\right)\text{\hspace{0.17em}}$ and axis of symmetry parallel to the x -axis can be used to graph the parabola. If $\text{\hspace{0.17em}}p>0,$ the parabola opens right. If $\text{\hspace{0.17em}}p<0,$ the parabola opens left. See [link] .
• The standard form of a parabola with vertex $\text{\hspace{0.17em}}\left(h,k\right)\text{\hspace{0.17em}}$ and axis of symmetry parallel to the y -axis can be used to graph the parabola. If $\text{\hspace{0.17em}}p>0,$ the parabola opens up. If $\text{\hspace{0.17em}}p<0,$ the parabola opens down. See [link] .
• Real-world situations can be modeled using the standard equations of parabolas. For instance, given the diameter and focus of a cross-section of a parabolic reflector, we can find an equation that models its sides. See [link] .

## Verbal

Define a parabola in terms of its focus and directrix.

A parabola is the set of points in the plane that lie equidistant from a fixed point, the focus, and a fixed line, the directrix.

If the equation of a parabola is written in standard form and $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ is positive and the directrix is a vertical line, then what can we conclude about its graph?

If the equation of a parabola is written in standard form and $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ is negative and the directrix is a horizontal line, then what can we conclude about its graph?

The graph will open down.

What is the effect on the graph of a parabola if its equation in standard form has increasing values of $\text{\hspace{0.17em}}p\text{?}$

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

The distance between the focus and directrix will increase.

## Algebraic

For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form.

${y}^{2}=4-{x}^{2}$

$y=4{x}^{2}$

yes $\text{\hspace{0.17em}}y=4\left(1\right){x}^{2}$

$3{x}^{2}-6{y}^{2}=12$

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

yes $\text{\hspace{0.17em}}{\left(y-3\right)}^{2}=4\left(2\right)\left(x-2\right)$

${y}^{2}+12x-6y-51=0$

For the following exercises, rewrite the given equation in standard form, and then determine the vertex $\text{\hspace{0.17em}}\left(V\right),$ focus $\text{\hspace{0.17em}}\left(F\right),$ and directrix of the parabola.

$x=8{y}^{2}$

${y}^{2}=\frac{1}{8}x,V:\left(0,0\right);F:\left(\frac{1}{32},0\right);d:x=-\frac{1}{32}$

$y=\frac{1}{4}{x}^{2}$

$y=-4{x}^{2}$

${x}^{2}=-\frac{1}{4}y,V:\left(0,0\right);F:\left(0,-\frac{1}{16}\right);d:y=\frac{1}{16}$

$x=\frac{1}{8}{y}^{2}$

$x=36{y}^{2}$

${y}^{2}=\frac{1}{36}x,V:\left(0,0\right);F:\left(\frac{1}{144},0\right);d:x=-\frac{1}{144}$

$x=\frac{1}{36}{y}^{2}$

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

${\left(x-1\right)}^{2}=4\left(y-1\right),V:\left(1,1\right);F:\left(1,2\right);d:y=0$

${\left(y-2\right)}^{2}=\frac{4}{5}\left(x+4\right)$

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

${\left(y-4\right)}^{2}=2\left(x+3\right),V:\left(-3,4\right);F:\left(-\frac{5}{2},4\right);d:x=-\frac{7}{2}$

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

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

${\left(x+4\right)}^{2}=24\left(y+1\right),V:\left(-4,-1\right);F:\left(-4,5\right);d:y=-7$

The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations