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Answer:
(A) y= -1.026x^2 + 1016.402x - 162075
Step-by-step explanation:
Here is the options to the question
Which equation best represents the data?
A) y= -1.026x^2 + 1016.402x - 162075
B) y= -1.036x^2 + 1024.771x - 163710
C) y= 298.214x - 66317.667
D) y= 196.2x – 18710
(A) y= -1.026x^2 + 1016.402x - 162075
The data above represents the company's profit based on number of items produced.
To get the equation which best represents the data, we have to substitute the value of x into the given equations
By substituting x=100 in first equation, we have:
y = -1.02x² + 1016.402x – 162075
y = -1.026(100)² + 1016.402(100) – 162075
y = -70500
y = -70500
Option A satisfies.
Using the second equation
When x = 100
y = -1.036x^2 + 1024.771x - 163710
y = -1.036(100)² + 1024.771(100) - 163710
y = -10360 + 102477.1 – 163710
y = -71592.9
Option B doesn’t satisfies
We can also replace the various values of x in the equations to check whether or not they satisfy.
Let's consider the functions one by one.
i)
,
x --> -infinity means that x is a very very small negative number. To model it in our mind let's think of
. This number to an even power becomes 
... etc.
Indeed, we can see that the smaller the x, the greater the value
. In fact, as x --> -infinity, f(x)-->+infinity.
ii)
y=x,
this clearly means that the behaviors of x and y are identical.
As x --> -infinity, y --> -infinity as well.
iii) y=|x|,
We can think of x --> -infinity, again, as a very very small number, like. For this value of x, y is 
, that is 
.
Indeed, the smaller the x, the greater the y. As in the function in part (i), as x --> -infinity, f(x)-->+infinity.
iv)
y=1/x
consider the values of y for the following values of x:
for x=-10, -100, -1000, y is respectively -0.1, -0.01, -0.001.
We can see that the smaller the x, the closer y gets to 0.
Thus, <span>f(x)-->0 as x --> -infinity
</span>
v) n'th root of x in not defined for negative x, when n is even.
vi) y=b^x, b>0
Here note that b cannot be equal to 1, otherwise the function is not exponential.
Let b=5, consider the values of y for the following values of x:
for x=-10, -100, -1000, y is respectively
.
That is, the smaller the value of x, the closer y gets to 0.
Thus, <span>f(x)-->0 as x --> -infinity</span>
i)
x --> -infinity means that x is a very very small negative number. To model it in our mind let's think of
Indeed, we can see that the smaller the x, the greater the value
ii)
y=x,
this clearly means that the behaviors of x and y are identical.
As x --> -infinity, y --> -infinity as well.
iii) y=|x|,
We can think of x --> -infinity, again, as a very very small number, like
Indeed, the smaller the x, the greater the y. As in the function in part (i), as x --> -infinity, f(x)-->+infinity.
iv)
y=1/x
consider the values of y for the following values of x:
for x=-10, -100, -1000, y is respectively -0.1, -0.01, -0.001.
We can see that the smaller the x, the closer y gets to 0.
Thus, <span>f(x)-->0 as x --> -infinity
</span>
v) n'th root of x in not defined for negative x, when n is even.
vi) y=b^x, b>0
Here note that b cannot be equal to 1, otherwise the function is not exponential.
Let b=5, consider the values of y for the following values of x:
for x=-10, -100, -1000, y is respectively
That is, the smaller the value of x, the closer y gets to 0.
Thus, <span>f(x)-->0 as x --> -infinity</span>