analyze the graph f(x)=(x-2)^3(x-3)^2(x-4)
a)end behavior: find the power function that the graph of f resembles for large values of absolute value of x
triple root at x=2
double root at x=3
root at x=4
the graph crosses the x-axis at x=2 and x=4, but is tangent at x=3
for huge values of x, the graph resembles that of the highest power: x^6
f(x) -> +infinity for large x, plus or minus.
enter the function at wolframalpha.com to see the graph
To determine the end behavior of the graph, we need to observe the powers of the function as x approaches positive and negative infinity.
The given function is f(x) = (x - 2)^3(x - 3)^2(x - 4).
As x approaches positive infinity, each term with a factor of (x - a) tends to positive infinity, where 'a' is a constant. In this case, the terms are (x - 2)^3, (x - 3)^2, and (x - 4). All of these terms have even powers, so they will not change sign as x becomes large. Therefore, the graph of f(x) resembles a power function with positive leading coefficient for large values of x.
Hence, the end behavior of the graph of f(x) resembles a positive power function for large values of |x|.
To determine the end behavior of the graph of the function f(x) = (x - 2)^3(x - 3)^2(x - 4), we need to examine the degrees of the factors in the function.
The degree of a polynomial term is the highest power of the variable in that term. In this case, we have a product of three polynomial terms: (x - 2)^3, (x - 3)^2, and (x - 4).
The degree of the first term, (x - 2)^3, is 3 since it represents three factors of (x - 2). Similarly, the degree of the second term, (x - 3)^2, is 2, and the degree of the third term, (x - 4), is 1.
To find the degree of the entire function, we add up the degrees of each term. Therefore, the degree of f(x) = (x - 2)^3(x - 3)^2(x - 4) is 3 + 2 + 1 = 6.
Now, let's consider the sign of the leading coefficient. The leading coefficient is the coefficient of the term with the highest power of the variable, which in this case is the coefficient of the term with x raised to the power of 6. Since there is no numerical coefficient explicitly given before the highest power term, we assume the coefficient is 1.
Based on the degree of the function and the sign of the leading coefficient, we can infer the end behavior of the graph of f(x).
For large positive values of x, as x approaches positive infinity, the term with the highest power, (x - 2)^3(x - 3)^2(x - 4), dominates the function. Since the degree of the function is 6 and the leading coefficient is positive, the graph of f(x) will resemble the end behavior of a power function with an even degree and a positive leading coefficient.
Therefore, for large values of absolute value of x, the graph of f(x) will resemble that of a positive even-powered function, rising on both sides as x moves away from the origin.