# Quadratic Functions

### By: Shera. P

## Topics

- Properties of Quadratic Functions
- Maximum/ Minimum Values
- Inverse
- Operations with Radicals
- Solving Quadratic Functions
- Determining Zeros
- Linear- Quadratic Systems

## 1. Properties of Quadratic Functions

3 ways Quadratic Functions can be expressed:

- f(x) = ax^2 + bx +c [standard form]
- f(x) = a(x-h)^2 +k [vertex form]
- f(x) = a(x-r)(x-s) [factored form

- table of values can tell us whether we have a linear or quadratic function by looking at the first and second differences

- if all the first differences are equal we know that it is a linear function

-if all that second differences are equal we know that it is a quadratic function

## 2. Maximum/ Minimum Values

**Key Facts:**

- if a > 0 the quadratic will have a minimum value, the parabola will open up
- if a < 0 the quadratic will have a maximum value, the parabola will open down
- the y-coordinate of the vertex helps you determine the max or min value of the function

**Completing the square****Factoring****Partial Factoring****The other way**

**Completing the square:**

- Factor the a-value out from the first two terms in the function
- Divide the second term by two and square it
- Add and subtract the new squared number
- Bring the negative number outside of the bracket by multiplying it with the a-value
- Simplify

By completing the square you are putting the quadratic in vertex form to make it easier to determine the max value.

Completing the Square - Solving Quadratic Equations

**Factoring:**

- Factor the a-value out of the function
- Factor the trinominal within the brackets
- Use the x-intercepts to fins the axis of symmetry by adding them together and diving by two
- Sub the newly found axis of symmetry back into the function to find the y-coordinate of the vertex

**Partial Factoring:**

- Partial factor the first two terms in the function
- Determine the x-intercepts from the factored form
- Add the x-intercepts together and divide by two to find the axis of symmetry
- Sub the axis of symmetry into the function to find the y-coordinate

**The other way:**

- Use the formula x= -b/2a
- Sub the appropriate value into the equation to find the axis of symmetry
- Sub the new value into the function to find the y-coordinate

## 3.Inverse

Key Terms:

**Inverse of a function:**the reverse of the original function, so it it undoes what the original function has done

**Key Facts:**

- the equation of the inverse can be determined by interchanging the x and y
- you can find the graph of the inverse by switching the x and y coordinates of the original function
- the domain of the original function is the range of the inverse, the range of the original is the domain of the inverse

f(x) = 2x^2 -1

y = 2x^2 -1

x =2y^2 -1

x+1= 2y^2 [square root both sides]

y= x+1 /2

f^-1(x) = x+1/2 [square root]

## 4. Operations with Radicals

Key Terms:

**radical:**a square root of a number**entire radical:**coefficient of the radical is one**mixed radical:**coefficient of the radical is other than one

Rules:

- coefficients multiply with other coefficients
- number under radical multiplies with other numbers under radicals
- when adding or subtracting radicals they have to be like radicals, they have the same number under the radical sign
- mixed radicals are in simplest form when the smallest number is under the radical

Simplifying radicals

Practice Problems:

## 5.Solving Quadratic Equations:

There are two ways you can solve quadratic equations. You can graph it or you can use the quadratic formula.

A ball is thrown into the air, height is h(t) in metres, time is t in seconds. h(t) = 2(t+3)(t-9), after how many seconds will the ball reach max/min height? what is this height?

h(t) = 2(t+3)(t-9)

zeros: x= -3, x=9

opening: up

x= (-3+9)/2

=3

h(3) = 2(3+3)(3-9)

= -72

vertex= (3, -72)

The value provided are all components that will help to find the graph of the function.

__Example of graphing:__A ball is thrown into the air, height is h(t) in metres, time is t in seconds. h(t) = 2(t+3)(t-9), after how many seconds will the ball reach max/min height? what is this height?

h(t) = 2(t+3)(t-9)

zeros: x= -3, x=9

opening: up

x= (-3+9)/2

=3

h(3) = 2(3+3)(3-9)

= -72

vertex= (3, -72)

The value provided are all components that will help to find the graph of the function.

## 6. Determining the zeros

- you can find the number of zeros by graphing or solving the function
- you can find the number of roots by looking at the discriminant

- if discriminant = 0 = 1 root

- if discriminant < 0 = 0 roots

- you can also look at the position of the vertex to find the number of roots

Example of factoring to find roots:

f(x) = x^2 -6x - 16

= (x-8)(x+2)

x= 8 or -2

f(x) = x^2 -6x - 16

= (x-8)(x+2)

x= 8 or -2

## 7. Linear Quadratic Systems

- a line is able to intersect a parabola at one, two, or no point
- in order to find the point of intersection you need to set the two equations to equal one another
- not all point of intersections found will apply to the question, you need to look and see which answers make the most sense

Finding Where Two Lines Intersect

Practice Question:

1. Find the point of intersection algebraically.

f(x) =3x^2 -4x +3

g(x)= -5x+4

1. Find the point of intersection algebraically.

f(x) =3x^2 -4x +3

g(x)= -5x+4