Question

In Exercises 19–24, (a) Use the Leading Coefficient Test to determine the graph's end behavior. (b) Determine whether the graph has y-axis symmetry, origin symmetry, or neither. (c) Graph the function. f(x)=x3−x2−9x+9f(x) = $$x^3$$ - $$x^2 - 9x + 9$$

Accepted Answer

Identify the degree and leading coefficient of the polynomial function $$f(x) = x^3$ - x^2$ - 9x + 9. $$The degree is the highest power of $$x$$, which is 3, and the leading coefficient is the coefficient of $$x^3$$, which is 1. Apply the Leading Coefficient Test to determine the end behavior: Since the degree is odd (3) and the leading coefficient is positive (1), as $$x 	o \infty$$, $$f(x) 	o \infty$$, and as $$x 	o -\infty$$, $$f(x) 	o -\infty. $$Check for symmetry by evaluating $$f(-x)$$ and comparing it to $$f(x)$$ and $$-f(x). $$Calculate $$f(-x) = (-x)^3$ - (-x)^2$ - 9(-x) + 9 = -x^3$ - x^2$ + 9x + 9. $$Since $$f(-x) 
eq f(x)$$ and $$f(-x) 
eq -f(x)$$, the function has neither y-axis symmetry nor origin symmetry. To graph the function, first find the x-intercepts by solving $$f(x) = 0$$, which means solving $$x^3$$ - $$x^2 - 9x + 9 = 0$. Use factoring techniques or the Rational Root Theorem to find possible roots. Next, find the y-intercept by evaluating f(0) = 9$. Plot the intercepts and use the end behavior and symmetry information to sketch the graph, noting the general shape of a cubic function with the given leading coefficient.

In Exercises 19–24, (a) Use the Leading Coefficient Test to determine the graph's end behavior. (b) Determine whether the graph has y-axis symmetry, origin symmetry, or neither. (c) Graph the function. $f (x) = x^{3} - x^{2} - 9 x + 9$

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Leading Coefficient Test

Symmetry of Functions

Graphing Polynomial Functions

In Exercises 19–24, (a) Use the Leading Coefficient Test to determine the graph's end behavior. (b) Determine whether the graph has y-axis symmetry, origin symmetry, or neither. (c) Graph the function. f(x)=x3−x2−9x+9f(x) = x^3 - x^2 - 9x + 9