Exercises - Graphing Functions

  1. Graph the following functions. Label all intercepts, local maxima and minima, and points of inflection, as they exist. Also, identify the intervals over which the function is increasing, decreasing, concave up, and concave down.

    1. $y = 3x^4 - 2x^2$

    2. $y = 3x^4+4x^3-6x^2-12x$

    3. $y = x^3-2x^2+x-2$

    4. $y = x^4-4x^3$

    5. $y = x^3-4x^2+4x$

    6. $y = 2x^3-9x^2+12x$

    7. $y = x^5-5x^3-20x$   (omit $x$-intercepts)

    8. $y = x^4 - 18x^2$

    1. $y = x^5 - 5x + 1$   (omit $x$-intercepts)

    2. $y = x^4 - 8x^3 + 18x^2$

    3. $y = 2x^3 - 3x^2 - 12x + 18$

    4. $y = x^3 - 3x^2 + 1$   (omit $x$-intercepts)

    5. $y = (1-x)^2(1+x)^2$

    6. $y = \frac{1}{16}x^4 - 2x$

    7. $y = x^3+x^2+6x-5$   (omit $x$-intercepts)

    8. $y = 2x^3 + 3x^2 - 12x - 18$


  2. Graph the following functions. Label all intercepts, asymptotes, horizontal tangents, local maxima and minima, and points of inflection, as they exist. Also, identify the intervals over which the function is increasing, decreasing, concave up, and concave down.

    1. $\displaystyle{g(x) = \frac{x^2}{(x+4)^2}}$

    2. $\displaystyle{g(x) = \frac{x^2}{x^2+4}}$

    3. $\displaystyle{g(x) = \frac{x-1}{x^2+3}}$

    4. $\displaystyle{f\,(x) = \frac{x-5}{x^2-9}}$

    5. $\displaystyle{y = \frac{x^2}{(x-4)^2}}$

    6. $\displaystyle{y = \frac{x^3}{2(x^3+1)}}$

    7. $\displaystyle{f\,(x) = \frac{x^2-4}{x^2-x-6}}$

    8. $\displaystyle{f\,(x) = x^2 - \frac{1}{x}}$

    9. $\displaystyle{y = \frac{x^2-9}{x^2-1}}$

    10. $\displaystyle{y = \frac{2-x}{x^2+4x-12}}$

    11. $\displaystyle{y = \frac{x+2}{x^2}}$

    12. $\displaystyle{y = \frac{-2x^2+32}{x^2+16}}$

    1. $\displaystyle{f\,(x) = \frac{x-3}{x^2-9}}$

    2. $\displaystyle{f\,(x) = \frac{x}{(x+1)^2}}$

    3. $\displaystyle{y = \frac{-2x}{x^2+1}}$

    4. $\displaystyle{y = \frac{x+1}{x^2-3x}}$   (omit concavity)

    5. $\displaystyle{y = \frac{4x^2}{2x^2+1}}$

    6. $\displaystyle{y = \frac{2x+10}{9-x^2}}$   (omit concavity)

    7. $\displaystyle{y = \frac{x^2-3}{x^3}}$

    8. $\displaystyle{f(x) = \frac{2x^2+5}{x^2+3}}$

    9. $\displaystyle{y = \frac{3x^2}{2+x-x^2}}$   (omit concavity)

    10. $\displaystyle{y = \frac{2x^3}{x^3+1}}$

    11. $\displaystyle{y = \frac{-x}{(x-1)^2}}$


  3. Graph the following functions. Label all important aspects, including intercepts, asymptotes, horizontal and vertical tangents, cusps, corners, local maxima and minima, and points of inflection, as they exist. Also, identify the intervals over which the function is increasing, decreasing, concave up, and concave down.

    1. $\displaystyle{f\,(x) = x^{2/3}(x-2)^2}$   (omit concavity)

    2. $\displaystyle{y = x^{2/3}(x+5)^{1/3}}$   (omit concavity)

    3. $\displaystyle{y = x(3x+10)^{2/3}}$

    4. $\displaystyle{y = x(1-x)^{1/3}}$

    5. $\displaystyle{y = \frac{x^{2/3}}{x-8}}$   (omit concavity)

    6. $\displaystyle{y = (1+x)^{2/3}(x-4)}$

    7. $\displaystyle{y = \frac{(x-1)^{2/3}}{x}}$   (omit concavity)

    8. $\displaystyle{y = \frac{x^{2/3}}{x-3}}$   (omit concavity)

    9. $\displaystyle{y = x^{2/3}(x-4)^2 - 4}$   (omit $x$-intercepts and concavity)

    10. $\displaystyle{y = x^{1/3}(4-x)^{2/3}+2}$   (omit $x$-intercepts and concavity)

    11. $\displaystyle{y = x^3(9x+11)^{2/3}}$   (omit concavity)

    12. $\displaystyle{y = (x-2)^{2/3}(2x+1)}$   (omit concavity)

    13. $\displaystyle{y = x^{1/3}(x+1)^{2/3}}$   (omit concavity)

    14. $\displaystyle{y = x^2(x+2)^{2/3}}$   (omit concavity)

    15. $\displaystyle{y = x^{1/3}(x+4)}$

    16. $\displaystyle{y = \frac{-x}{(x-1)^2}}$

    17. $\displaystyle{y = x^3-x^2-x+1}$

    18. $\displaystyle{y = x^{2/3}(12-x)^{1/3}}$   (omit concavity)

    19. $\displaystyle{y = x^4-2x^2}$

    20. $\displaystyle{y = \frac{x^2}{x^2+3}}$

    21. $\displaystyle{y = 3x^{2/3} - 2x}$

    22. $\displaystyle{y = x(1-x)^{2/3} }$

    23. $\displaystyle{y = x^4 + 4x^3}$

    24. $\displaystyle{y = \frac{x}{(1-x)^2}}$

    25. $y = x^{2/3} (x-\frac{3}{2})^{1/3}$   (omit concavity)