2. A lesson in mathematics and physics. And Santa Claus. Graphing a Curve Background from www.clipartpal.com

3. Graph the following curve: f(x) = 2x 3 +9x 2 -18x-5 Step 1: DO NOT PANIC!! Step 1: DO NOT PANIC!! Step 2: Remember the chart to the left. Step 2: Remember the chart to the left. Points to Solve For WhatHow to Find ItDefinition Zeros (x- intercepts) Set f(x) = 0 to get x-values of coordinates y-values equal zero by definition Where the height of the object being described by the function equals zero Local minima and maxima  intervals of increase and decrease Set f’(x) = 0 to get x-values of coordinates Plug those x-values back into f(x) to get y-value. Do a number line. Since f’(x) describes the velocity of the object, the intervals describe whether the velocity of the object is increasing or decreasing Inflection points  concavity Set f”(x) = 0 to get x-values of coordinates Plug those x-values back into f(x) to get y-values. Do a number line. Since f”(x) describes the acceleration of the object, the intervals describe whether the acceleration of the object is increasing or decreasing

4. REMEMBER… B e f o r e y o u g r a p h a f u n c t i o n, b e s u r e t o i d e n t i f y a n y d i s c o n t i n u i t i e s ! f(x) = 2x 3 +9x 2 -18x-5 is a polynomial, AND LIKE ALL POLYNOMIALS it is continuous on the interval (-∞, ∞).

5. First… Points to Solve For WhatHow to Find ItDefinition Zeros (x-intercepts) Set f(x) = 0 to get x-values of coordinates y-values equal zero by definition Where the height of the object being described by the function equals zero Local minima and maxima  intervals of increase and decrease Set f’(x) = 0 to get x-values of coordinates Plug those x-values back into f(x) to get y-value. Do a number line. Since f’(x) describes the velocity of the object, the intervals describe whether the velocity of the object is increasing or decreasing Inflection points  concavity Set f”(x) = 0 to get x-values of coordinates Plug those x-values back into f(x) to get y-values. Do a number line. Since f”(x) describes the acceleration of the object, the intervals describe whether the acceleration of the object is increasing or decreasing

6. Find the zeros (x-intercepts) f(x) = 2x 3 +9x 2 -18x-5 f(x) = 2x 3 +9x 2 -18x-5 0 = 2x 3 +9x 2 -18x-5 0 = 2x 3 +9x 2 -18x-5 How many roots will there be? How many roots will there be?

8. CORRECT!! There are three roots/zeros/x- intercepts There are three roots/zeros/x- intercepts (-5.9434,0) (-5.9434,0) (-0.2486,0) (-0.2486,0) (1.6921,0) (1.6921,0) All graphs made by GraphFunc Online at http://graph.seriesmathstudy.com/

9. Second… Points to Solve For WhatHow to Find ItDefinition Zeros (x-intercepts) Set f(x) = 0 to get x-values of coordinates y-values equal zero by definition Where the height of the object being described by the function equals zero Local minima and maxima  intervals of increase and decrease Set f’(x) = 0 to get x-values of coordinates Plug those x-values back into f(x) to get y-value. Do a number line. Since f’(x) describes the velocity of the object, the intervals describe whether the velocity of the object is increasing or decreasing Inflection points  concavity Set f”(x) = 0 to get x-values of coordinates Plug those x-values back into f(x) to get y-values. Do a number line. Since f”(x) describes the acceleration of the object, the intervals describe whether the acceleration of the object is increasing or decreasing

10. Find local minima and maxima First, set f’(x) = 0 First, set f’(x) = 0 f’(x) = 6x 2 +18x-18 = 0 f’(x) = 6x 2 +18x-18 = 0 Roots are (-3.7913, 83.6170) and (0.7913, -12.6170) Roots are (-3.7913, 83.6170) and (0.7913, -12.6170) Next, to figure out the intervals… Next, to figure out the intervals…

11. Find intervals of increase and decrease Again, maximum or minimum points are (-3.7913, 83.6170) and (0.7913, - 12.6170) Again, maximum or minimum points are (-3.7913, 83.6170) and (0.7913, - 12.6170) Do a number line: Do a number line: f’(-5) = 42 + f’(0) = -18 - f’(2) = 42 + LOCAL MAXIMUM LOCAL MINIMUM

12. So far we have…

13. Third… Points to Solve For WhatHow to Find ItDefinition Zeros (x-intercepts) Set f(x) = 0 to get x-values of coordinates y-values equal zero by definition Where the height of the object being described by the function equals zero Local minima and maxima  intervals of increase and decrease Set f’(x) = 0 to get x-values of coordinates Plug those x-values back into f(x) to get y-value. Do a number line. Since f’(x) describes the velocity of the object, the intervals describe whether the velocity of the object is increasing or decreasing Inflection points  concavity Set f”(x) = 0 to get x-values of coordinates Plug those x-values back into f(x) to get y-values. Do a number line. Since f”(x) describes the acceleration of the object, the intervals describe whether the acceleration of the object is increasing or decreasing

14. Find inflection point(s) and concavity An inflection point is where f”(x) = 0 An inflection point is where f”(x) = 0 f”(x) = 12x + 18 = 0 f”(x) = 12x + 18 = 0 There is one inflection point at (-1.5, 35.5) There is one inflection point at (-1.5, 35.5) To find concavity, do a number line: To find concavity, do a number line: f”(-2) = -6 - f”(5) = 78 + CONCAVE DOWN CONCAVE UP

15. Now, draw it! Zeros(-5.9434,0) (-0.2486,0) (1.6921,0) Maxima and Minima Max: (-3.7913, 83.6170) Min (0.7913, - 12.6170) Increasing on (-∞, -3.7913). Decreasing on (-3.7913, 0.7913). Increasing on (0.7913, ∞). Inflection point and concavity (-1.5, 35.5)Concave down on (-∞, -1.5). Concave up on (-1.5, ∞).

16. Wait… What about Santa??? “Sleigh Ride” from www.thanksmuch.com

17. We’ll use the function to describe Santa’s sleigh ride! Let the point (0.7913,-12.6170) be your house Let the x-axis represent the time in minutes Let the y-axis represent Santa’s height above the ground in miles Zeros(-5.9434,0) (-0.2486,0) (1.6921,0) Maxima and Minima Max: (-3.7913, 83.6170) Min (0.7913, -12.6170) Increasing on (-∞, -3.7913). Decreasing on (-3.7913, 0.7913). Increasing on (0.7913, ∞). Inflection point and concavity (-1.5, 35.5)Concave down on (-∞, -1.5). Concave up on (-1.5, ∞).

18. We’ll use the function to describe Santa’s sleigh ride! Santa starts his journey in time to get to your house in 6 minutes. His sleigh increases in velocity and climbs to a height of 83.1670 miles, where he turns back towards the ground. His velocity now decreasing, he heads toward your home. At a height of 35.5 miles, he starts increasing his acceleration. But he stops on your front lawn (you live in a valley that is 12.6 miles below the surrounding area) about 1.5 minutes later. He leaves your house and his velocity and acceleration increase without bound as he heads back to the North Pole. Zeros(-5.9434,0) (-0.2486,0) (1.6921,0) Maxima and Minima Max: (-3.7913, 83.6170) Min (0.7913, -12.6170) Increasing on (-∞, -3.7913). Decreasing on (-3.7913, 0.7913). Increasing on (0.7913, ∞). Inflection point and concavity (-1.5, 35.5)Concave down on (-∞, -1.5). Concave up on (-1.5, ∞).