EXAMPLE 5 Graph polynomial functions Graph (a) f (x) = –x3 + x2 + 3x – 3 and (b) f (x) = x4 – x3 – 4x2 + 4. SOLUTION To graph the function, make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior. a. The degree is odd and leading coefficient is negative. So, f (x) → +∞ as x → –∞ and f (x) → –∞ as x → +∞.

EXAMPLE 5 Graph polynomial functions To graph the function, make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior. b. The degree is even and leading coefficient is positive. So, f (x) → +∞ as x → –∞ and f (x) → +∞ as x → +∞.

EXAMPLE 6 Solve a multi-step problem The energy E (in foot-pounds) in each square foot of a wave is given by the model E = 0.0029s4 where s is the wind speed (in knots). Graph the model. Use the graph to estimate the wind speed needed to generate a wave with 1000 foot-pounds of energy per square foot. Physical Science

EXAMPLE 6 Solve a multi-step problem SOLUTION Make a table of values. The model only deals with positive values of s. STEP 1

 EXAMPLE 6 Solve a multi-step problem STEP 2 Plot the points and connect them with a smooth curve. Because the leading coefficient is positive and the degree is even, the graph rises to the right. STEP 3 Examine the graph to see that s 24 when E = 1000.  The wind speed needed to generate the wave is about 24 knots. ANSWER

GUIDED PRACTICE for Examples 5 and 6 Graph the polynomial function. 9. f(x) = x4 + 6x2 – 3 10. f(x) = –x3 + x2 + x – 1

GUIDED PRACTICE for Examples 5 and 6 11. f(x) = 4 – 2x3

GUIDED PRACTICE for Examples 5 and 6 WHAT IF? If wind speed is measured in miles per hour, the model in Example 6 becomes E = 0.0051s4. Graph this model. What wind speed is needed to generate a wave with 2000 foot-pounds of energy per square foot? 12. ANSWER about 25 mi/h.