SOLUTION: what is the x-intercept and y-intercept, domain, range, interval of increase, interval of decrease, maximum, minimum, and the end behavior of the graph f(x)=x^2-4?
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Question 639145: what is the x-intercept and y-intercept, domain, range, interval of increase, interval of decrease, maximum, minimum, and the end behavior of the graph f(x)=x^2-4?
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
\ =\ x^2\ -\ 4)
Solve
for
The 
-intercepts are then
)
and
)
Evaluate )
, then )
is the 
-intercept.
NOTE: These are NOT the answers. You have to calculate the actual values of 
and 
This is a polynomial function with real coefficients, therefore the domain is:
To define the range and intervals of increase and decrease, we first need to calculate the function maximum.
Take the first derivative using the power rule:
\right)}{dx}\ =\ 2x)
Set the first derivative equal to zero and solve:
Hence a local extremum exists at 
Take the second derivative
\right)}{dx^2}\ =\ 2)
Which is positive 
, therefore the extremum at is a local minimum, and the minimum value is \ =\ -4)
Since this is a polynomial function it is continuous over the entire domain, hence the range is:
The first derivative has a single root, namely , hence there are two intervals to consider: )
and )
.
Choose a value in the first interval, say -1. \ =\ -2)
, hence the interval is an interval of decrease.
Choose a value in the second interval, say 1. \ =\ 2)
, hence the interval is an interval of increase.
This is an even-degree polynomial with a positive lead coefficient, hence both ends take off to positive infinity.
John
My calculator said it, I believe it, that settles it
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