From e850089484809ffb07edc8f297948c4c1363ddf0 Mon Sep 17 00:00:00 2001 From: James Su Date: Wed, 30 Oct 2019 02:12:34 +0000 Subject: [PATCH] Update Unit 3: Quadratic Functions.md --- .../MPM2DZ/Unit 3: Quadratic Functions.md | 25 ++++++++++++++++--- 1 file changed, 21 insertions(+), 4 deletions(-) diff --git a/Grade 10/Math/MPM2DZ/Unit 3: Quadratic Functions.md b/Grade 10/Math/MPM2DZ/Unit 3: Quadratic Functions.md index 6e73166..6f98233 100644 --- a/Grade 10/Math/MPM2DZ/Unit 3: Quadratic Functions.md +++ b/Grade 10/Math/MPM2DZ/Unit 3: Quadratic Functions.md @@ -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 \ No newline at end of file