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.
Determine whether the axis of symmetry is the
x - or
y -axis.
If the given coordinates of the focus have the form
then the axis of symmetry is the
x -axis. Use the standard form
If the given coordinates of the focus have the form
then the axis of symmetry is the
y -axis. Use the standard form
Multiply
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
and
directrix
The focus has the form
so the equation will have the form
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
units horizontally and
units vertically, the vertex will be
This translation results in the standard form of the equation we saw previously with
replaced by
and
replaced by
To graph parabolas with a vertex
other than the origin, we use the standard form
for parabolas that have an axis of symmetry parallel to the
x -axis, and
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
The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include:
1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
asegid
to transport fluids fats proteins and lymphocytes to the blood stream as lymph
Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.
Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems
Kamara
yes
Prince
how does the stomach protect itself from the damaging effects of HCl
the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body
the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature