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Update Unit 3: Quadratic Functions.md

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Grade 10/Math/MPM2DZ/Unit 3: Quadratic Functions.md
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@ -126,11 +126,28 @@ There are 3 main types of transformations for a quadratic function.
- The step property also gets affected.
## Forms of Quadratic Functions
- A quadratic relation in the form $`y=a(x-r)(x-s)`$ is said to be in factored form. The zeroes are $`x=r`$ and $`x=s`$.
-The axis of symmetry can be determined by using the formula $`\dfrac{r + s}{2}`$
-The axis of symmetry is also **x-coordinate** of the vertex.
### Standard Form
|Quadratic Form|Function|Zeroes|Vertex|Axis of Symmetry|
|:-------------|:-------|:-----|:-----|:---------------|
|**standard**|$`y=ax^2+bx+c`$|$`x= \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}`$|Plug in Axis of **symmetry** and solve for $`y`$|$`x = \dfrac{-b}{2a}`$|
|**vertex**|$`y=a(x-h)^2+k`$|Set $`y=0`$, solve for $`x`$|$`(h, k)`$|$`x = h`$|
|**factored**|$`y=a(x-r)(x-s)`$|$`(r, 0), (s, 0)`$|Plug in Axis of **symmetry** and solve for $`y`$|$`x = \dfrac{r+s}{2}`$|
### Vertex Form
- **Notice** that the $`a`$ value **stays the same** in all the forms.
- By using **completing the square**, we can find that the vertex is at $`(\dfrac{-b}{2a}, c - \dfrac{b^2}{4a})`$, where $`a =\not 0`$
### Factored Form
## Partial Factoring
- If $`y=ax^2+bx+c`$ cannot be factored, then we can use partial facotring to determine the vertex.
- We set $`y =c`$, then we basically now stated that $`0 = ax^2 + bx`$. Since there is no constant value, we can factor the equation to becoming $`x(ax + b)`$, from where we can solve for the values of $`x`$.
- In respect of the axis of symmetry, using the $`2 \space x`$ values we can find the axis of symmetry, by using the formula $`\dfrac{r+s}{2}`$, since the axis of symmetry works for any 2 opposite points on the parabola.
- With this, we can easily find the vertex of a quadratic equation.
## Quadratic Inequalities
- Quadratic inequalites can be solved **graphically** and **algebraically**. Since we know how to graph quadratic relations, we can solve quadtratic inequalites **graphically**
- An example of an **algebraic** solution of an quadratic inequality would be the **number line method** we learned in the previous unit.
- To solve it **graphically**, we will need to put the equation into factored form, then finding out the x-intercepts. With these, we can find out the side of the graph the solution is on.
- If the solution is greater than $`0`$, then it would be denoting the values that are above the x-axis, and below the y-axis for the opposite.
## Quadratic Inequalities