Point plotting is plotting points until the shape of the function is apparent.
Intercepts are points that have 0 as either the X or Y coordinates.
Knowing the symmetry of a graph can cut the amount of points you need by half.
In order to test for symmetry, replace your function’s X and/or Y with the corresponding -X and/or -Y for the symmetry you are checking and see if the functions are equal. If they are equal, it is symmetrical around that axis.
| Name | Function | Image | Name | Function | Image |
|---|---|---|---|---|---|
| Line | $y = x$ | |
Squared | $y = x^2$ | |
| Cubed | $y = x^3$ | |
Square root | $y = \sqrt{x}$ | |
| Abs | $y = x$ | |
Rational | $y = \frac{1}{x}$ | |
$slope = m = \frac{rise}{run} = \frac{y_1 - y_2}{x_1 - x_2} = \frac{y_2 - y_1}{x_2 - x_1}$
/, negative slope: \, zero slope: _, undefined slope: || Form Name | Equation |
|---|---|
| Point Slope | $y - y_1 = m(x - x_1)$ |
| Slope Intercept | $y = mx + b$ |
| General | $ax + by + c= 0$ |
| Vertical line | x = a |
| Horizontal line | y = a |
Parallel - two lines have the same slope. $m_1 = m_2$
Perpendicular - two lines have negative reciprocals as slopes. $m_1 = -\frac{1}{m_2}$
$f(x) = a_{n}x^n + a_{n-1}x^{n-1} + … + a_{2}x^2 + a_{1}x^1 + a_0$
Even degree polynomials:
| $a_n > 0$ Positive | $a_n < 0$ Negative |
|---|---|
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Odd degree polynomials:
| $a_n > 0$ Positive | $a_n < 0$ Negative |
|---|---|
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| Name | Function | Example |
|---|---|---|
| Shift left | $f(x + c)$ | $y = (x + 2)^2$ |
| Shift right | $f(x - c)$ | $y = (x - 2)^2$ |
| Shift up | $f(x) + c$ | $y = x^2 + 2$ |
| Shift down | $f(x) - c$ | $y = x^2 - 2$ |
| Reflect y-axis | $f(-x)$ | $y = (-x)^2 = x^2$ |
| Reflect x-axis | $-f(x)$ | $y = -x^2$ |
| Reflect origin | $-f(-x)$ | $y = -(-x)^2 = -x^2$ |
| Squeeze x-axis | $f(ax)$ | $y = (ax)^2$ |
| Stretch x-axis | $f(\frac{x}{a})$ | $y = (\frac{x}{a})^2$ |
Elementary functions are functions that can represent many real world phenomena.
Algebraic functions are functions that can combine multiple algebraic operations on polynomials.
| Operation name | Function Shorthand | Function Extended |
|---|---|---|
| Sum | $(f + g)(x)$ | $f(x) + g(x)$ |
| Difference | $(f - g)(x)$ | $f(x) - g(x)$ |
| Product | $(fg)(x)$ | $f(x) * g(x)$ |
| Quotient | $(f/g)(x)$ | $f(x) / g(x)$ |
| Composition | $(f \circ g)(x)$ | $f(g(x))$ |
Radical functions are the same as square root: $f(x) = \sqrt[n]{g(x)}$