Quadratic Functions

By: Shera. P

Topics

  1. Properties of Quadratic Functions
  2. Maximum/ Minimum Values
  3. Inverse
  4. Operations with Radicals
  5. Solving Quadratic Functions
  6. Determining Zeros
  7. 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

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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
There are 4 different methods you can use in order to determine the maximum/ minimum value of a quadratic function. Those methods include:
  1. Completing the square
  2. Factoring
  3. Partial Factoring
  4. The other way
Completing the square:
  1. Factor the a-value out from the first two terms in the function
  2. Divide the second term by two and square it
  3. Add and subtract the new squared number
  4. Bring the negative number outside of the bracket by multiplying it with the a-value
  5. 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:
  1. Factor the a-value out of the function
  2. Factor the trinominal within the brackets
  3. Use the x-intercepts to fins the axis of symmetry by adding them together and diving by two
  4. Sub the newly found axis of symmetry back into the function to find the y-coordinate of the vertex
Partial Factoring:
  1. Partial factor the first two terms in the function
  2. Determine the x-intercepts from the factored form
  3. Add the x-intercepts together and divide by two to find the axis of symmetry
  4. Sub the axis of symmetry into the function to find the y-coordinate
The other way:
  1. Use the formula x= -b/2a
  2. Sub the appropriate value into the equation to find the axis of symmetry
  3. 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
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Example of algebraically finding 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:
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5.Solving Quadratic Equations:

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

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 = 2roots
- 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

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
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Finding Where Two Lines Intersect
Practice Question:
1. Find the point of intersection algebraically.
f(x) =3x^2 -4x +3
g(x)= -5x+4