# 10.3 The parabola  (Page 3/11)

 Page 3 / 11

Graph $\text{\hspace{0.17em}}{y}^{2}=-16x.\text{\hspace{0.17em}}$ Identify and label the focus, directrix, and endpoints of the latus rectum.

Focus: $\text{\hspace{0.17em}}\left(-4,0\right);\text{\hspace{0.17em}}$ Directrix: $\text{\hspace{0.17em}}x=4;\text{\hspace{0.17em}}$ Endpoints of the latus rectum: $\text{\hspace{0.17em}}\left(-4,±8\right)$

## Graphing a parabola with vertex (0, 0) and the y -axis as the axis of symmetry

Graph $\text{\hspace{0.17em}}{x}^{2}=-6y.\text{\hspace{0.17em}}$ Identify and label the focus , directrix    , and endpoints of the latus rectum    .

The standard form that applies to the given equation is $\text{\hspace{0.17em}}{x}^{2}=4py.\text{\hspace{0.17em}}$ Thus, the axis of symmetry is the y -axis. It follows that:

• $-6=4p,$ so $\text{\hspace{0.17em}}p=-\frac{3}{2}.\text{\hspace{0.17em}}$ Since $\text{\hspace{0.17em}}p<0,$ the parabola opens down.
• the coordinates of the focus are $\text{\hspace{0.17em}}\left(0,p\right)=\left(0,-\frac{3}{2}\right)$
• the equation of the directrix is $\text{\hspace{0.17em}}y=-p=\frac{3}{2}$
• the endpoints of the latus rectum can be found by substituting into the original equation, $\text{\hspace{0.17em}}\left(±3,-\frac{3}{2}\right)$

Next we plot the focus, directrix, and latus rectum, and draw a smooth curve to form the parabola    .

Graph $\text{\hspace{0.17em}}{x}^{2}=8y.\text{\hspace{0.17em}}$ Identify and label the focus, directrix, and endpoints of the latus rectum.

Focus: $\text{\hspace{0.17em}}\left(0,2\right);\text{\hspace{0.17em}}$ Directrix: $\text{\hspace{0.17em}}y=-2;\text{\hspace{0.17em}}$ Endpoints of the latus rectum: $\text{\hspace{0.17em}}\left(±4,2\right).$

## Writing equations of parabolas in standard form

In the previous examples, we used the standard form equation of a parabola to calculate the locations of its key features. We can also use the calculations in reverse to write an equation for a parabola when given its key features.

Given its focus and directrix, write the equation for a parabola in standard form.

1. Determine whether the axis of symmetry is the x - or y -axis.
1. If the given coordinates of the focus have the form $\text{\hspace{0.17em}}\left(p,0\right),$ then the axis of symmetry is the x -axis. Use the standard form $\text{\hspace{0.17em}}{y}^{2}=4px.$
2. If the given coordinates of the focus have the form $\text{\hspace{0.17em}}\left(0,p\right),$ then the axis of symmetry is the y -axis. Use the standard form $\text{\hspace{0.17em}}{x}^{2}=4py.$
2. Multiply $\text{\hspace{0.17em}}4p.$
3. Substitute the value from Step 2 into the equation determined in Step 1.

## Writing the equation of a parabola in standard form given its focus and directrix

What is the equation for the parabola    with focus $\text{\hspace{0.17em}}\left(-\frac{1}{2},0\right)\text{\hspace{0.17em}}$ and directrix     $\text{\hspace{0.17em}}x=\frac{1}{2}?$

The focus has the form $\text{\hspace{0.17em}}\left(p,0\right),$ so the equation will have the form $\text{\hspace{0.17em}}{y}^{2}=4px.$

• Multiplying $\text{\hspace{0.17em}}4p,$ we have $\text{\hspace{0.17em}}4p=4\left(-\frac{1}{2}\right)=-2.$
• Substituting for $\text{\hspace{0.17em}}4p,$ we have $\text{\hspace{0.17em}}{y}^{2}=4px=-2x.$

Therefore, the equation for the parabola is $\text{\hspace{0.17em}}{y}^{2}=-2x.$

What is the equation for the parabola with focus $\text{\hspace{0.17em}}\left(0,\frac{7}{2}\right)\text{\hspace{0.17em}}$ and directrix $\text{\hspace{0.17em}}y=-\frac{7}{2}?$

${x}^{2}=14y.$

## Graphing parabolas with vertices not at the origin

Like other graphs we’ve worked with, the graph of a parabola can be translated. If a parabola is translated $\text{\hspace{0.17em}}h\text{\hspace{0.17em}}$ units horizontally and $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ units vertically, the vertex will be $\text{\hspace{0.17em}}\left(h,k\right).\text{\hspace{0.17em}}$ This translation results in the standard form of the equation we saw previously with $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ replaced by $\text{\hspace{0.17em}}\left(x-h\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ replaced by $\text{\hspace{0.17em}}\left(y-k\right).$

To graph parabolas with a vertex $\text{\hspace{0.17em}}\left(h,k\right)\text{\hspace{0.17em}}$ other than the origin, we use the standard form $\text{\hspace{0.17em}}{\left(y-k\right)}^{2}=4p\left(x-h\right)\text{\hspace{0.17em}}$ for parabolas that have an axis of symmetry parallel to the x -axis, and $\text{\hspace{0.17em}}{\left(x-h\right)}^{2}=4p\left(y-k\right)\text{\hspace{0.17em}}$ for parabolas that have an axis of symmetry parallel to the y -axis. These standard forms are given below, along with their general graphs and key features.

## Standard forms of parabolas with vertex ( h , k )

[link] and [link] summarize the standard features of parabolas with a vertex at a point $\text{\hspace{0.17em}}\left(h,k\right).$

 Axis of Symmetry Equation Focus Directrix Endpoints of Latus Rectum $y=k$ ${\left(y-k\right)}^{2}=4p\left(x-h\right)$ $x=h-p$ $x=h$ ${\left(x-h\right)}^{2}=4p\left(y-k\right)$ $y=k-p$

what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
Â
Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
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
the range is twice of the natural number which is the domain
Morolake
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
For Plan A to reach $27/month to surpass Plan B's$26.50 monthly payment, you'll need 3,000 texts which will cost an additional \$10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert
can I see the picture
How would you find if a radical function is one to one?