Whether the graph of a polynomial rises or falls can be determined by the Leading Coefficient Tests.
P(x) = a_{n}x^{n} + a_{n-1}x^{n-1} +............. a_{1}x + a_{0}
In the above polynomial, n is the degree and a_{n} is the leading coefficient.
Case |
End Behavior of Graph |
When n is odd and a_{n} is positive
Graph falls to the left and rises to the right
When n is odd and a_{n} is negative
Graph rises to the left and falls to the right
When n is even and a_{n} is positive
Graph rises to the left and right
When n is even and a_{n} is negative
Graph falls to the left and right
Example 1 :
Find the right-hand and left-hand behaviors of the graph of
f(x) = x^{5} + 2x^{3} - 3x + 5
Solution :
Because the degree is odd and the leading coefficient is positive, the graph falls to the left and rises to the right as shown in the figure.
Example 2 :
Determine the end behavior of the graph of the polynomial function below using Leading Coefficient Test.
P(x) = -x^{3} + 5x
Solution :
Because the degree is odd and the leading coefficient is negative, the graph rises to the left and falls to the right as shown in the figure.
Example 3 :
Determine the end behavior of the graph of the polynomial function below using Leading Coefficient Test.
P(x) = 2x^{2} - 2
Solution :
Because the degree is even and the leading coefficient is positive, the graph rises to the left and right as shown in the figure.
Example 4 :
Determine the end behavior of the graph of the polynomial function below using Leading Coefficient Test.
P(x) = -x^{2} + 1
Solution :
Because the degree is even and the leading coefficient is negative, the graph falls to the left and right as shown in the figure.
Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.
You can also visit our following web pages on different stuff in math.
WORD PROBLEMS
Word problems on simple equations
Word problems on linear equations
Word problems on quadratic equations
Area and perimeter word problems
Word problems on direct variation and inverse variation
Word problems on comparing rates
Converting customary units word problems
Converting metric units word problems
Word problems on simple interest
Word problems on compound interest
Word problems on types of angles
Complementary and supplementary angles word problems
Trigonometry word problems
Markup and markdown word problems
Word problems on mixed fractrions
One step equation word problems
Linear inequalities word problems
Ratio and proportion word problems
Word problems on sets and venn diagrams
Pythagorean theorem word problems
Percent of a number word problems
Word problems on constant speed
Word problems on average speed
Word problems on sum of the angles of a triangle is 180 degree
OTHER TOPICS
Time, speed and distance shortcuts
Ratio and proportion shortcuts
Domain and range of rational functions
Domain and range of rational functions with holes
Graphing rational functions with holes
Converting repeating decimals in to fractions
Decimal representation of rational numbers
Finding square root using long division
L.C.M method to solve time and work problems
Translating the word problems in to algebraic expressions
Remainder when 2 power 256 is divided by 17
Remainder when 17 power 23 is divided by 16
Sum of all three digit numbers divisible by 6
Sum of all three digit numbers divisible by 7
Sum of all three digit numbers divisible by 8
Sum of all three digit numbers formed using 1, 3, 4
Sum of all three four digit numbers formed with non zero digits