Cartesian graphing

Transformations

With the graph for $y = f(x)$ already known.

Translating/shifting:

• $y = f(x - c)$
  Replacing $x$ with $x - c$ will translate/shift the graph along the $x$-axis, moving it rightward by $c$.
• $y = f(x) + c$
  Replacing $y$ with $y - c$ will translate/shift the graph along the $y$-axis, moving it upward by $c$.

Stretching/shrinking:

• $y = f(a\cdot x)$
  Replacing $x$ with $a\cdot x$ will shrink the graph on the $x$-axis by a factor of $a$.
• $y = \frac{f(x)}{a}$
  Replacing $y$ with $a\cdot y$ will shrink the graph on the $y$-axis by a factor of $a$.
• $y = f(\frac{x}{a})$
  Replacing $x$ with $\frac{x}{a}$ will stretch the graph on the $x$-axis by a factor of $a$.
• $y = a\cdot f(x)$
  Replacing $y$ with $\frac{y}{a}$ will stretch the graph on the $y$-axis by a factor of $a$.

Flipping/mirroring:

• $y = f(-x)$
  Replacing $x$ with $-x$ will create flipped mirror image with $y = 0$ as the symmetry axis.
• $y = -f(x)$
  Replacing $y$ with $-y$ will create flipped mirror image with $x = 0$ as the symmetry axis.
• $y = f^{-1}(x)$ or $x = f(y)$
  Swapping $x$ and $y$ is sometimes no longer a function, but will create the inverse graph which is a flipped mirror image with $x = y$ as the symmetry axis.

Strategy

Strategy for drawing/graphing arbitrary functions (some steps only apply to polinomials):

1. Find intersections with $x$-axis and $y$-axis
   • Find $f(0)$ and $f(x) = 0$
   • Use Newton’s method to find approximate roots
2. Find critical points
   • Where $f’(x)$ equals 0 or doesn’t exist
3. Find local minima and maxima and where function rises or falls
   • Use the sign of $f’(x)$
   • Use the sign of $f^{\prime\prime}(x)$ where $f’(x) = 0$ ($f^{\prime\prime}(x) = 0$ won’t help)
4. Find inflection points / concavity
   • Where $f^{\prime\prime}(x) = 0$ and sign changes
   • If $f^{\prime\prime}(x) > 0$, function is concave upwards
   • If $f^{\prime\prime}(x) < 0$, function is concave downwards
5. Find asymptotes
   • Vertical asymptote
     • Look for where function might divide by zero
   • Horizontal asymptote
     • Not applicable if function is only defined on a limited interval
     • Possible if function contains division
     • Attempt limit of $f(x)$ as $x$ approaches $- \infty$ and $\infty$ (left and right)
     • Divide fraction by $x^n$ (above and below) with $n$ the highest power in the numerator
   • Diagonal asymptote
     • Not applicable if function is only defined on a limited interval
     • Same as for horizontal asymptote
     • Use long division te remove division
     • Put formula in the form $f(x) - (mx + c)$