Graph plane curves described by parametric equations by plotting points.
Graph parametric equations.
It is the bottom of the ninth inning, with two outs and two men on base. The home team is losing by two runs. The batter swings and hits the baseball at 140 feet per second and at an angle of approximately
to the horizontal. How far will the ball travel? Will it clear the fence for a game-winning home run? The outcome may depend partly on other factors (for example, the wind), but mathematicians can model the path of a projectile and predict approximately how far it will travel using
parametric equations . In this section, we’ll discuss parametric equations and some common applications, such as projectile motion problems.
Graphing parametric equations by plotting points
In lieu of a graphing calculator or a computer graphing program, plotting points to represent the graph of an equation is the standard method. As long as we are careful in calculating the values, point-plotting is highly dependable.
Given a pair of parametric equations, sketch a graph by plotting points.
Construct a table with three columns:
Evaluate
and
for values of
over the interval for which the functions are defined.
Plot the resulting pairs
Sketching the graph of a pair of parametric equations by plotting points
Sketch the graph of the
parametric equations
Construct a table of values for
and
as in
[link] , and plot the points in a plane.
The graph is a
parabola with vertex at the point
opening to the right. See
[link] .
Sketching the graph of trigonometric parametric equations
Construct a table of values for the given parametric equations and sketch the graph:
Construct a table like that in
[link] using angle measure in radians as inputs for
and evaluating
and
Using angles with known sine and cosine values for
makes calculations easier.
By the symmetry shown in the values of
and
we see that the parametric equations represent an
ellipse . The
ellipse is mapped in a counterclockwise direction as shown by the arrows indicating increasing
values.
Graphing parametric equations and rectangular form together
Graph the parametric equations
and
First, construct the graph using data points generated from the
parametric form . Then graph the
rectangular form of the equation. Compare the two graphs.
Next, translate the parametric equations to rectangular form. To do this, we solve for
in either
or
and then substitute the expression for
in the other equation. The result will be a function
if solving for
as a function of
or
if solving for
as a function of
for the "hiking" mix, there are 1,000 pieces in the mix, containing 390.8 g of fat, and 165 g of protein. if there is the same amount of almonds as cashews, how many of each item is in the trail mix?
an object is traveling around a circle with a radius of 13 meters .if in 20 seconds a central angle of 1/7 Radian is swept out what are the linear and angular speed of the object
like this: (2)/(2-x)
the aim is to see what will not be compatible with this rational expression. If x= 0 then the fraction is undefined since we cannot divide by zero. Therefore, the domain consist of all real numbers except 2.
functions can be understood without a lot of difficulty.
Observe the following:
f(2) 2x - x
2(2)-2= 2
now observe this:
(2,f(2)) ( 2, -2)
2(-x)+2 = -2
-4+2=-2
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
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
158.5
This number can be developed by using algebra and logarithms.
Begin by moving log(2) to the right hand side of the equation like this:
t/100 log(2)= log(3)
step 1: divide each side by log(2)
t/100=1.58496250072
step 2: multiply each side by 100 to isolate t.
t=158.49
Dan
what is the importance knowing the graph of circular functions?
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
how to find x:
12x = 144
notice how 12 is being multiplied by x. Therefore division is needed to isolate x
and whatever we do to one side of the equation we must do to the other.
That develops this:
x= 144/12
divide 144 by 12 to get x.
addition:
12+x= 14
subtract 12 by each side. x =2
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.