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}

Example 2: f(x) = x³ + 3x² - 4

•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: (-2,0),(1,0),(0,-4)

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

•7. f’(x) = 3x² + 6x

f’(x) = 0 ↔ x Є {-2, 0}

•8. f’’(x) = 6x + 6

  f’’(x) = 0 ↔ x =-1

•9. Graph

 

Example 3:

•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

 

 

Exercise: draw the graphs of the functions:

 

a) f(x) = x⁴ - 2x² - 8

 

b) f(x) = x·e^x

 

 

 

 

 

 

 

 

Solutions:

a)

b)