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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

how fast can i understand functions without much difficulty
what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this