Index of Graphs

# Index of Graphs

##### Lines

Lines in the Cartesian plane have the form $y = mx + b$ where $m$ represents the slope of the line and $b$ represents the $y$-coordinate of the $y$-intercept. There is an exception - vertical lines cannot be represented in this form and instead have the form $x = k$ where $k$ is the $x$-coordinate of the $x$-intercept.

Diagonal Line, $m > 0$ Diagonal Line, $m < 0$ Horizontal Line, $y = b$ Vertical Line, $x = k$
##### Parabolas

Upwards and downwards parabolas have the form $y = ax^2 + bx + c$ or $y = a(x - h)^2 + k$ where $a \neq 0$ and where $(h, k)$ represents the coordinates of the vertex of the parabola. Leftwards and rightwards parabolas have the form $x = ay^2 + by + c$ or $x = a(y - h)^2 + k$ where $a \neq 0$ and where $(k, h)$ represents the coordinates of the vertex of the parabola.

Upwards Parabola, $a > 0$ Downwards Parabola, $a < 0$ Rightwards Parabola Leftwards Parabola
##### Circles and Ellipses

Circles have the form $(x - h)^2 + (y - k)^2 = r^2$ where $(h, k)$ represents the coordinates of the center of the circle and $r > 0$ represents the radius of the circle. Ellipses have the form $\displaystyle{\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1}$ where $(h, k)$ represents the coordinates of the center of the ellipse, $a > 0$ represents the horizontal radius of the ellipse and $b > 0$ represents the vertical radius of the ellipse.

Circle Ellipse $a > b$ Ellipse $a < b$
##### Hyperbolas

Left/right hyperbolas have the form $\displaystyle{\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1}$ where $(h, k)$ represents the center of the hyperbola and $a, b > 0$. Up/down hyperbolas have the form $\displaystyle{\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1}$ where $(h, k)$ represents the center of the hyperbola and $a, b > 0$

Left/right Hyperbola Up/down Hyperbola
##### Cubics

Cubics have the form $y = ax^3 + bx^2 + cx + d$ where $a \neq 0$.

Cubic, $a > 0$ Cubic, $a < 0$
##### Absolute Value

Upwards/downwards absolute value graphs have the form $y = a|x - h| + k$ where $(h, k)$ represents the coordinates of the vertex. Rightwards/leftwards absolute value graphs have the form $x = a|y - h| + k$ where $(k, h)$ represents the coordinates of the vertex.

Upwards Absolute Value Downwards Absolute Value Rightwards Absolute Value Leftwards Absolute Value
Squareroot Cuberoot
##### Exponential and Logarithmics

Exponential graphs have the form $y = a^x$ where $a > 0$. Logarithmic graphs have the form $y = \log_a x$ where $a > 0$.

Exponential Logarithmic
##### Sine, Cosine, and Tangent

Sine graphs have the form $y = a \sin (bx + c) + d$ and cosine graphs have the form $y = a \cos (bx + c) + d$ where $a$ represents the amplitude, $b$ is the period, $\displaystyle{\frac{c}{b}}$ is the horizontal shift, and $d$ is the vertical shift. Tangent graphs have the form $y = a \tan (bx + c) + d$.

Sine Cosine Tangent

Note that the graph of a sine curve is similar to a graph of a cosine curve.

##### Secant, Cosecant, and Cotangent
Secant Cosecant Cotangent