Question

Use an end behavior diagram, as shown below, to describe the end behavior of the graph of each polynomial function. ƒ(x)=5x5+2x3-3x+4

Accepted Answer

Identify the leading term of the polynomial function. For the given function $$f(x) = 5x^5$ + 2x^3$ - 3x + 4$$, the leading term is $$5x^5$$ because it has the highest power of $$x. $$Determine the degree and the leading coefficient of the polynomial. Here, the degree is 5 (an odd number), and the leading coefficient is 5 (a positive number). Recall the general end behavior rules for polynomials based on degree and leading coefficient:  
- If the degree is odd and the leading coefficient is positive, then as $$x 	o \infty$$, $$f(x) 	o \infty$$, and as $$x 	o -\infty$$, $$f(x) 	o -\infty.
- If $$the degree is odd and the leading coefficient is negative, the end behaviors are reversed.
- If the degree is even and the leading coefficient is positive, both ends go to $$\infty.
- If $$the degree is even and the leading coefficient is negative, both ends go to $$-\infty. $$Apply the rule to the given polynomial: since the degree is odd (5) and the leading coefficient is positive (5), the graph falls to negative infinity on the left and rises to positive infinity on the right. Use the end behavior diagram notation to describe this: as $$x 	o -\infty$$, $$f(x) 	o -\infty ($$downward arrow), and as $$x 	o \infty$$, $$f(x) 	o \infty ($$upward arrow). This can be represented as $$\downarrow \quad \uparrow.$$

Use an end behavior diagram, as shown below, to describe the end behavior of the graph of each polynomial function. ƒ(x)=5x⁵+2x³-3x+4

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Key Concepts

Polynomial Degree and Leading Term

End Behavior of Polynomial Functions

Using End Behavior Diagrams