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)=4x−x3f(x) = 4x - $$x^3$$

Accepted Answer

Identify the degree and leading coefficient of the polynomial function $$f(x) = 4x - x^3$. The highest power of x$ is 3, so the degree is 3, and the leading coefficient is the coefficient of x^3$, which is -1$. Apply the Leading Coefficient Test to determine the end behavior: Since the degree is odd (3) and the leading coefficient is negative (-1$), as x$$ 	o \infty$$, f(x$$) 	o -\infty$$, and as x$$ 	o -\infty$$, f(x$$) 	o \infty$$. Check for symmetry by testing f(-x$$)$$ and comparing it to f(x$$)$$ and -f(x$$)$$. Calculate f(-x) = 4(-x$$) - $$(-x)^3 = -4x$$ + $$x^3. $$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, start by plotting key points such as the y-intercept at $$x=0$$, where $$f(0) = 0$$, and other points by substituting values of $$x (e.g$$., $$x=1, -1, 2, -2) to $$understand the shape of the curve. Use the end behavior from step 2 and the points from step 4 to sketch the graph, showing that the graph falls to the right and rises to the left, consistent with the Leading Coefficient Test for an odd degree polynomial with a negative 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. $f (x) = 4 x - x^{3}$

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)=4x−x3f(x) = 4x - x^3