Functions and Curves
Interactive Curve Sketcher & Transformer
Select a base function, define its parameters, apply transformations, and see the original and transformed curves plotted.
1. Select Base Function & Parameters
2. Apply Transformations (Optional)
Relevant Tools
To further enhance your learning and problem-solving skills, explore these additional resources
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Standard Curves & Their Equations
This section provides an overview of common mathematical curves and their standard equations, with example plots.
Graphical Transformations
Transformations change the position, shape, or size of a graph. Understanding these allows you to predict the appearance of a function based on its equation.
- $y = f(x) + k$: Vertical Shift. If $k>0$, shifts up. If $k<0$, shifts down.
- $y = f(x-h)$: Horizontal Shift. If $h>0$, shifts right. If $h<0$ (i.e., $f(x+c)$ where $c>0$), shifts left.
- $y = a \cdot f(x)$: Vertical Stretch/Compression & Reflection.
- If $|a|>1$, stretches vertically.
- If $0<|a|<1$, compresses vertically.
- If $a<0$, reflects across the x-axis (in addition to any scaling).
- $y = f(bx)$: Horizontal Stretch/Compression & Reflection.
- If $|b|>1$, compresses horizontally by a factor of $1/|b|$.
- If $0<|b|<1$, stretches horizontally by a factor of $1/|b|$.
- If $b<0$, reflects across the y-axis (in addition to any scaling).
- $y = -f(x)$: Reflection in the x-axis. (This is a special case of $a \cdot f(x)$ where $a=-1$).
- $y = f(-x)$: Reflection in the y-axis. (This is a special case of $f(bx)$ where $b=-1$).
Function Properties
Periodic Functions
A function $f(x)$ is periodic if there exists a positive number $P$ (the period) such that $f(x+P) = f(x)$ for all $x$. Example: $\sin(x)$ has period $2\pi$.
Continuous & Discontinuous Functions
A continuous function can be drawn without lifting the pen. Discontinuous functions have breaks, jumps, or holes. Example of discontinuity: $y = 1/x$ at $x=0$.
Odd & Even Functions
Even function: $f(-x) = f(x)$ (symmetric about y-axis). Example: $y=x^2, y=\cos(x)$.
Odd function: $f(-x) = -f(x)$ (symmetric about origin). Example: $y=x^3, y=\sin(x)$.
Inverse Functions
If $f(x)$ maps $x$ to $y$, its inverse $f^{-1}(y)$ maps $y$ back to $x$. The graph of $f^{-1}(x)$ is a reflection of $f(x)$ in the line $y=x$. A function must be one-to-one (pass the horizontal line test) to have a well-defined inverse function over its entire domain. Example: If $f(x) = 2x+1$, then $f^{-1}(x) = (x-1)/2$.
Asymptotes
Asymptotes are lines that a curve approaches as it heads towards infinity or a point of discontinuity.
- Vertical Asymptote (VA): A vertical line $x=a$ is a VA if $f(x) \to \pm\infty$ as $x \to a^+$ or $x \to a^-$. Often found where the denominator of a rational function is zero (and the numerator is non-zero). Example: $y=\frac{1}{x-2}$ has a VA at $x=2$.
- Horizontal Asymptote (HA): A horizontal line $y=L$ is an HA if $f(x) \to L$ as $x \to \pm\infty$. For rational functions, compare degrees of numerator and denominator. Example: $y=\frac{2x}{x-1}$ has an HA at $y=2$.
- Oblique (Slant) Asymptote: A line $y=mx+c$ (where $m \neq 0$) that the curve approaches as $x \to \pm\infty$. Occurs in rational functions if the degree of the numerator is exactly one greater than the degree of the denominator. Example: $y=\frac{x^2+1}{x} = x + \frac{1}{x}$ has an oblique asymptote $y=x$.