# Quadratics

### Grade 10 Unit

## What is a Parabola

## Vertex Form: y=a(x-h)2+k

**All Examples/Pictures are posted at the bottom of the topic.**

**Axis of Symmetry:** Every Parabola has an axis of symmetry which is the line that runs down its center. The H in the equation represents where point x is on the vertex and weather it will go left or right.

**Optimal Value:** The optimal value is the k in the equation and it is the highest or lowest value in a parabola.

**Transformations:**

Vertical Reflection: A reflection in which a plane figure flips over vertically.

Horizontal Translation: This moves the equation left or right, This is the H value in the equation and if it is less than 0 then it will move right and if it is more, it will move left.

Vertical Translation: This is the k value in the equation and if it is more than 0 it will move up and if less than it goes down.

Vertical Stretch: This is the a value in the equation and if it is more than one it will get steeper.

**X-Intercepts/Zero's:** You need this when you need to find the x intercept or zero's by subbing y=0 and isolate for x.

**Step Pattern: **This pattern shows us how steep the parabola gets. It goes over 1 up 1 and over 2 up 4.

**Example of a Word Problem: **Engineers built an arch bridge across the Hudson river. The arch bridge makes a parabola shape that has the equation y= -0.1(x-5) 2+ 12, where x and y are measured in meters. If the bridge makes contact with both banks at a height of 4 meters, how long is the distance between two banks of the Hudson river?

We find the graph of the arch bridge to be:

The vertex of the arch bridge is (5,12), which is point A. The bridge makes contact with both banks at a height of 4 meters (B and C), that means that the distance between two banks is the distance from B to C.

4= -0.1(x-5) 2+ 12

x = -4 or 14

Thus, the total distance between the two banks is 14 – (-4) = 18 meters.

## Standard Form: y=(x-r)(x-s)

**All Examples/Pictures are posted at the bottom of the topic.**

**The x-int are found by using the quadratic formula.**

**Axis of Symmetry: **You find the 2 zeroes by using quadratic formula and then you add the 2 x values and divide them by 2 to get the axis of symmetry.

**Optimal Value: **Once you find the AoS, substitute it in the equation to find the y-coordinate of the vertex.

**Factoring(Common):** Common factoring is finding what is common between the two equations, then dividing the whole thing by what is common. Then with what is left will go in bracket and what was used to divide, will now be used to multiply with what is in brackets.

**Factoring(Simple Trinomial): **The formula is **x²+bx+c **

**Factoring(Complex Trinomial): **A complex factor doesn't start with a co-efficient of x^2.

**Factoring(Perfect Squares): **If a equation starts with a square and ends with one it will be a perfect square trinomial most likely.

**Factoring(Difference of Squares): **This is when you start and end with a square but have no middle term. It cancels out. If you expand you would get the same expression as the first one.

## Factored Form: y=(x-r)(x-s)

**All Examples/Pictures are posted at the bottom of the topic.**

**The 2 zeros are in the brackets which is r and s and are always opposite of what you see.**

**Axis of Symmetry: **You have to add the 2 x points and divide them by 2. **x = (r+s)/2**

**Optimal Value: **You must have the 2 zeros and the Axis of Symmetry by using the formula above and sub in the x point into the equation to get the Optimal Value.