Unit 3 - Curve Sketching

a function is concave down when

f’’(x) < 0 (second derivative is negative)

a function is concave up when

f’’(x) > 0 (second derivative is positive)

a function is increasing & concave up when

f’(x) > 0 and f’’(x) > 0 (first and second derivative are positive)

a function is decreasing and concave up when

f’(x) < 0 and f’’(x) > 0 (first derivative is negative and second derivative is positive)

a function is increasing and concave down when

f’(x) > 0 and f’’(x) < 0 (first derivative is positive and second derivative is negative)

a function is decreasing and concave down when

f’(x) < 0 and f’’(x) < 0 (first and second derivative are negative)

how to sketch a curve

step 1 factor and find any asymptotes and or holes

step 2 find first derivative and solve for zero to get critical numbers for y

step 3 find second derivative and solve for zero to get critical numbers for y’

step 4 create number line chart & find the value of the first and second derivatives between the critical numbers and or asymptotes

step 5 use values of first & second derivatives to determine shape, local maximums, local minimums, and inflection points

step 6 use all info to graph