# 9. graph

Example 1: f(x) = x4 + 8x3 + 22x2 + 24x + 9

•1. Domf = R

•2. f is continuous and derivable in R because it is a polynomial function.

•3. Symmetry: f is a function which is neither odd nor even.

•4. f is not a periodic function.

•5. intersection points: (-3,0),(-1,0),(0,9)

•6. There are no asymptotes because it is a polynomial function.

•7. f’(x) = 4x3 + 24x2 + 44x + 24                 f’(x) = 0 ↔ x Є {-3, -2, -1}

•8. f’’(x) = 12x2 + 48x + 44      f’’(x) = 0 ↔ x ≈ -2.58, -1.42 •9. Graph Example 2: •1. Domf = R – {-2,2}
•2. f is continuous and derivable in its domain because it is a rational function.
•3. Symmetry: f is an even function.
•4. f is not a periodic function.
•5. intersection points: (-3,0),(3,0),(0,9/4)
•6. Asymptotes: x = 2; x = -2; y = 1
•7. •8. no solution •9. Graph Example 3 (PAEG September 2014): a) Study its continuity in x = -1

b) Find the relative extrema in the interval (1,4)

c) Calculate the intervals of increasing and decreasing in the interval (1,4)

a) f is discontinuous in x = -1, it has a jump discontinuity with jump 1

b) It has a  minimum in x = 2 c) f is decreasing in the interval (1,2) and increasing in the interval (2,4)

Exercise: draw the graphs of the functions:
a) f(x) = x·ex Solutions:
a) b) 