Definition of Curve Sketching  Curve Sketching is the process of using the first and second derivative and information gathered from the original equation to graph a function.

What does f’(x) tell us about f(x)? Increasing 0r Decreasing Intervals  Find f’(x)  If f’(x) > o  f(x) is increasing (positive slope)  If f’(x) < 0  f(x) is decreasing (negative slope)  If f’(x) = 0  f(x) has a turning point

Local maximums and minimums  Take the first derivative of function and equal it to zero.  Find the critical points:  If f’(x) changes from positive to negative at c, there is a local maximum at c.  If f’(x) changes from negative to positive at c, there is a local minimum at c.  If f’(x) does not change sign at c, there is no maximum or minimum at c.  C is a critical number

What does f”(x) tell us about f(x)?  Solve for the second derivative. If:  F”(x) = 0 or f(x) is undefined  f(x) changes concavity (inflection point)  F”(x) > 0  the graph of f(x) is concave up  F”(x) < 0  the graph of f(x) is concave down

Asymptotes  Find the horizonal and vertical astymptotes of f(x), if any.  Horizontal- look at the degree in both numerator and denominator. If they are the same, divide the coefficients. If numerator is larger, then there is no asy. If denominator is larger, then the asy. is zero.  Vertical- factor the denominator and solve for zero.

Sketch the Curve  Draw graph and label X and Y lines  Make dotted lines for asymptotes, if any  Plot the local max and mins  Sketch the line using positive and negative direction and the up/down concavity info that was determined.

Example 1 f(x)=2x 4 -4x 2 +2  No asymptotes  Find f’(x) and find the zeros  Put the zeros on a number line. Choose a number that is higher or lower than the zeroes and plug it back into the f’(x). If it comes out with a positive number it’s increasing if it is negative its decreasing. Keep note of the results  F’(x)= 8x 3 -8x =8x(x 2 -1) =8x(x-1)(x+1) X= 0,±1  f’(-2) 0, f’(1/2) 0  Dec(-∞.-1)Inc(-1,0)Dec(0,1)Inc(1, ∞)

Example 1 cont.  Find f”(x) to determine the concavity of f(x) at specific intervals  Put zeroes on a number line, then plug in numbers on either side of the zeroes back into f”(x). If the result is positive, the graph of f(x) is concave upward at this interval. If the result is negative, the graph of f(x) is concave downward at this interval.  F’(x)=8x 3 -8x F”(x)=24x 2 -8 =8(3x 2 -1) X=√3/3, -√3/3  f”(-1)=16, f”(0)=-8, f”(1)=16  f”(-1)>0, f”(0) 0  Concave up (-∞, √3/3) concave down(-√3/3, √3/3) concave up (√3/3, ∞)

Example 1 Cont. Sketch  With the information provided, sketch the asymptotes, if any, plot the max(s) and min(s), and sketch the line based on its slope and concavity. y x

Example 2 f(x)=5x 3 +4x 2 +2  No asymptotes  Find f’(x) and find the zeros  Put the zeros on a number line. Choose a number that is higher or lower than the zeroes and plug it back into the f’(x). If it comes out with a positive number it’s increasing if it is negative its decreasing. Keep note of the results  F’(x)=15x 2 +8x = x(15x+8) x= -8/15,0  f’(-1)>0, f’(-1/2) 0  Inc (-∞,-15/8)dec(-15/8,0)inc(0, ∞)

Example 2 cont.  Find f”(x) to determine the concavity of f(x) at specific intervals  Put zeroes on a number line, then plug in numbers on either side of the zeroes back into f”(x). If the result is positive, the graph of f(x) is concave upward at this interval. If the result is negative, the graph of f(x) is concave downward at this interval.  F’(x)=15x 2 +8x F”(x)=30x+8 = 2 (15x+4) X=-4/15  F”(-1)=-22. f”(1)=38 F”(-1) 0  concave down(-∞,-4/15) concave up (-4/15, ∞)

Example 2 Cont. Sketch  With the information provided, sketch the asymptotes, if any, plot the max(s) and min(s), and sketch the line based on its slope and concavity. Y x

Examples  Sketch the curve and show the first and second derivatives.  f(x)=2x 2 +16x+5  f(x)=√x+2x+2

Answers  1. f’(x)=4x+16, f”(x)=4  f(x)=1/2√x+2,  f”(x)=-1/(4x 3/2 ) x y y x

Resources  calculus-intro/precalculus-algebra/18-rational- functions-finding-horizontal-slant-asymptotes-01 calculus-intro/precalculus-algebra/18-rational- functions-finding-horizontal-slant-asymptotes-01 