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. $f (x) = x^{3} - x^{2} - 9 x + 9$

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Concetti chiave

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