Determine the features for f(x), g(x), and h(x). f(x)=2x-3 rate of change x-int y-int increasing intervals decreasing intervals posititve intervals negative intervals symmetry end behavior g(x)=-4x+7 rate of change x-int y-int increasing intervals decreasing intervals posititve intervals negative intervals symmetry end behavior h(x)=-5 rate of change x-int y-int increasing intervals decreasing intervals posititve intervals negative intervals symmetry end behavior
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
Determine the features for f(x), g(x), and h(x).
f(x)=2x-3
rate of change
x-int
y-int
increasing intervals
decreasing intervals
posititve intervals
negative intervals
symmetry
end behavior
g(x)=-4x+7
rate of change
x-int
y-int
increasing intervals
decreasing intervals
posititve intervals
negative intervals
symmetry
end behavior
h(x)=-5
rate of change
x-int
y-int
increasing intervals
decreasing intervals
posititve intervals
negative intervals
symmetry
end behavior
1.
x-intercept is a point where y is zero, so by taking
so x-intercept is
y-intercept is the point where x is zero, so by taking
so y-intercept is
For the intervals;
, so there is no critical point.
The domain is . The domain together with the critical point makes monotone interval, here the monotone interval is . The sign of in this interval is positive therefore the function is increasing in .
For symmetry;
we replace x with -x, , but this is neither equal to f(x) nor -f(x). So the function is not symmetrical.
End behavior;
The degree of the highest term is 1 and the leading coefficient is positive so, end behavior will be given as
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