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Update Unit 3: Quadratic Functions.md
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# Unit 3: Quadratic Functions
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## Definitions
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- `Linear relation`: A relation with a constant rate of change
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- `Linear relation`: A relation with a constant rate of change, or with a constant first difference
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- `Quadratic relation`: A relation where the 2nd differences are constant
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- `Non-linear relation`: A relation with a variable rate of change
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- `Degree` of a polynomial: Highest exponent that appears in any term of a polynomial
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- `a-value`: Step property
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- `Function`: A relation in which no y-coordinates share an x-coordinate (e.g., circles are not functions)
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## Parabolas
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A standard graph of a parabola $`y=x^2`$ would look something like this:
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<img src="https://study.com/cimages/multimages/16/2f02bcfd-854d-486a-ab39-30961b3337c4_yx2.jpg" width="300">
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There are several things you should know about a parabola:
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- X-intercepts
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- Y-intercepts
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- Vertex
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- Optimal value
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- Axis of Symmetry
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- Direction Of Opening
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- Step Property
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### X-intercepts
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- These are the zeroes of the quadratic function, or the solutions you found when solving for a quadratic function in factored form.
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- These are the values of $`x`$ where $`y=0`$.
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- 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 $`(r, 0), (s, 0)`$.
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### Y-intercepts
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- These are the values of $`y`$ where $`x=0`$.
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### Vertex
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- This highest/lowest value of $`y`$ that the parabola takes.
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- This point tells us alot of things, including the `axis of symmetry` and the `maximum/minimum/optimal` value.
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- With vertex form in $`y=a(x-h)^2 + k`$, we know that the vertex is at $`(h, k)`$.
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### Optimal value
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- This is basically the $`y`$ value of the vertex, and is useful for maximum/minimum word problems
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- If the parabola is opening upwards, this is the minimum value. If the parabola is opening downards, this is the maximum value
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### Axis of Symmetry
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- A vertical line of symmetry for the parabola.
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- This can be determined in many ways:
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- Using factored form $`y=a(x-r)(x-s)`$, the axis of symmetry is at $`\dfrac{r+s}{2}`$
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- Using standard form $`y=ax^2+bx+c`$, the axis of symmetry is at $`\dfrac{-b}{2a}`$
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- Using vertex form, $`y=a(x-h)^2+k`$, the axis of symmetry is simply the $`x`$ coordinate of the vertex, so $`h`$.
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### Direction Of Opening
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- To put it bluntly, the parabola opens upward if the $`a`$ value is positive, and downwards if the $`a`$ value is negative.
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- Think of a postive $`a`$ value as a **happy face**, and a negative $`a`$ value as a **sad face**.
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### Step Property
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- This is a property that can be used to quickly graph a quadratic function.
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- 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 $`n \ge 1`$.
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- 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.
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## Functions
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`Function`: A relation in which no y-coordinates share an x-coordinate (e.g., circles are not functions)
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### Vertical Line Test (VLT)
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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.
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### Ways of Representing Functions
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#### 1. Table Of Values
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|$`x`$|$`y`$|
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|:----|:----|
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|$`2`$|$`5`$|
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|$`5`$|$`7`$|
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|$`6`$|$`5`$|
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#### 2. Coordinates in a Set
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$`f = \{(-2, 1), (0, 1), (3, 1), (4, 1), (7, 1)\}`$
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#### 3. Graph
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<img src="https://d2jmvrsizmvf4x.cloudfront.net/5ptsdt6SVmzgiF5QPfeS_2016-01-11_203936.jpg" width="300">
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#### 4. Mapping (Bubble Diagrams)
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<img src="https://www.varsitytutors.com/assets/vt-hotmath-legacy/hotmath_help/topics/mapping-diagrams/diagram.gif" width="300">
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#### 5. Equation
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$`y=x^2-5`$
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#### 6. Function Notation
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$`f(x) = x^2 - 5`$
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### Domain And Range
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- A `domain` is the set of $`x`$-values, and the `range` is the set of $`y`$-values.
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- To represent domain and range, we use **set notation** to represnt it or simply by listing the $`x`$ and $`y`$ values.
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- We use listing method for representing points, such as:
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- <img src="https://s3-eu-west-1.amazonaws.com/functionsandgraphs/graph+with+all+points+except+F.png" width="400">
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- $`D = \{-3, -1, 0, 1, 2, 2.5 \}`$
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- $`R = \{-2, -1, 0.5, 3, 3, 5 \}`$
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- Make sure the values or sorted in order.
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- We use **set notation** or **interval notation** to represent a continous graph, such as:
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- <img src="http://www.analyzemath.com/high_school_math/grade_11/quadratic/find_quadratic_f(x).gif" width="300">
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- $`D = \{x \mid x \in \mathbb{R} \}`$
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- $`R = \{y \mid y \ge 2, y \in \mathbb{R} \}`$
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## Transformations
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There are 3 main types of transformations for a quadratic function.
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### Vertical Translation
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- When we graph the quadratic relation $`y=x^2+k`$, the vertex of the parabola has coordinates $`(0, k)`$
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- When $`k \gt 0`$, the graph of $`y=x^2`$ is **vertically** translated **up** $`\mid k \mid`$ units.
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- When $`k \le 0`$, the graph of $`y=x^2`$ is **vertically** translated **down** $`\mid k \mid`$ units.
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### Horizontal Translation
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- When we graph the quadratic relation, $`y=(x-h)`$ the vertex of the parabola has coordinates $`(h, 0)`$
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- When $`h \gt 0`$, the graph of $`y=x^2`$ is **horizontally** translated **left** $`\mid k \mid`$ units.
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- When $`h \le 0`$, the graph of $`y=x^2`$ is **horizontally** translated **right** $`\mid k \mid`$ units.
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### Vertical Stretch/Compression
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- If $`a \lt 0 \rightarrow`$, the graph is **reflected** over the x-axis.
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- If $`\mid a \mid gt 1 \rightarrow`$ vertical stretch.expansion by a **factor of** $`\mid a \mid`$.
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- If $`0 \lt \mid a \mid \lt 1 \rightarrow`$ vertical compression by a factor of $`\mid a \mid`$
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- The step property also gets affected.
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## Forms of Quadratic Functions
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