-`Domain`: List of all valid **x-values** for relation, expressed as $`D = {x, x_2, x_3}`$ or as a relation such as $`D = \{x \in \mathbb{R}, x =\not 0, x <50\}`$
-`Range`: List of all valid **y-values** for relation, expressed as $`R = {y, y_2, y_3}`$ or as a relation such as $`R = \{y \in \mathbb{R}, y =\not 0, y <50\}`$
There are several things you should know about a parabola:
- X-intercepts
- Y-intercepts
- Vertex
- Optimal value
- Axis of Symmetry
- Direction Of Opening
- Step Property
### X-intercepts
- These are the zeroes of the quadratic function, or the solutions you found when solving for a quadratic function in factored form.
- These are the values of $`x`$ where $`y=0`$.
- By using factored form, we can easily see that an equation with factored form of $`y=a(x-r)(x-s)`$ has its x-intercepts at $`(r, 0), (s, 0)`$.
### Y-intercepts
- These are the values of $`y`$ where $`x=0`$.
### Vertex
- This highest/lowest value of $`y`$ that the parabola takes.
- This point tells us alot of things, including the `axis of symmetry` and the `maximum/minimum/optimal` value.
- With vertex form in $`y=a(x-h)^2 + k`$, we know that the vertex is at $`(h, k)`$.
### Optimal value
- This is basically the $`y`$ value of the vertex, and is useful for maximum/minimum word problems
- If the parabola is opening upwards, this is the minimum value. If the parabola is opening downards, this is the maximum value
### Axis of Symmetry
- A vertical line of symmetry for the parabola.
- This can be determined in many ways:
- Using factored form $`y=a(x-r)(x-s)`$, the axis of symmetry is at $`\dfrac{r+s}{2}`$
- Using standard form $`y=ax^2+bx+c`$, the axis of symmetry is at $`\dfrac{-b}{2a}`$
- Using vertex form, $`y=a(x-h)^2+k`$, the axis of symmetry is simply the $`x`$ coordinate of the vertex, so $`h`$.
### Direction Of Opening
- To put it bluntly, the parabola opens upward if the $`a`$ value is positive, and downwards if the $`a`$ value is negative.
- Think of a postive $`a`$ value as a **happy face**, and a negative $`a`$ value as a **sad face**.
### Step Property
- This is a property that can be used to quickly graph a quadratic function.
- The step property for an $`a`$ value of $`1`$ is $`1, 3, 5, 7, \cdots 2n+1`$ for any $`n \ge 1`$. The step property for any other a value is $`1a, 3a, 5a, 7a, \cdots a(2n+1)`$ for any $`n \ge 1`$.
- The step property tells us the difference of values between each point starting from the vertex, meaing if the vertex is at $`(x, y)`$, <br> the next two points would be $`(x-1, y+n), (x+1, y+n)`$, where $`n`$ is the $`n^{th}`$ step property number.
## Functions
`Function`: A relation in which no y-coordinates share an x-coordinate (e.g., circles are not functions)
### Vertical Line Test (VLT)
We can easily tell if a relation is a function by using the vertical line test. If a single straight line of $`x=n`$ for any $`n`$ has more than $`1`$ point on the function, then that relation is not a function. Bluntly, a function cannot have a vertical straightline touching any $`2`$ of its points.
