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Quadratic Function Definition

What is a quadratic function?
Video covers the characteristics of a quadratic function.

Video Guide
0:21 What is the vertex ?
0:31 What is the axis of symmetry?
1:13 What is the domain and the range?
2:16 Explanation of interval increase and interval decrease
2:44 What is the rate of change?
3:55 What is the max and min?
4:10 How do you find the y intercept and the x intercept?
5:17 What is end behavior?


The graph of a quadratic function is in the shape of a parabola. A parabola can take many shapes but is basically in the shape of a U. See an example below.
parabala
quadratic function f(x)=a(x-h)^2 +k
Quadratic Function form =
Let's break down the equation in order to understand what each part tells us.
quadratic function f(x)=a(x-h)^2 +k
  • If a is NEGATIVE then the graph reflects across the x-axis.
  • If |a| is less than 1, the graph SHRINKS.
  • If |a| is greater than 1, the graph STRETCHES.

  • h is TRICKY!
  • If h is POSITIVE then the graph moves LEFT.
  • If h is NEGATIVE then the graph moves RIGHT.
  • Axis of Symmetry: x = h
  • Special Case

  • If k is POSITIVE then the graph moves UP.
  • If k is NEGATIVE then the graph moves DOWN.
  • Vertex: (h, k)

1. Put the equation in standard form. A quadratic equation has two forms.
standard form y=ax^2 +bx +c
vertex form y=a(x-h)^2
2. Identify the values of a, b, &c

3. Find the axis of symmetry: 

4. Construct a table of values for x and y

5. Plot the points and connect them with a U- shaped curve and arrows

Several examples of Graphing Quadratic Equations

0:15 Graph y=2(x-2)^2-5
This graph is a change of direction of a parabola.
Remember: "The opposite of H and the same sign as K"
3:08 This example is not in vertex form so you begin by finding your vertex. To convert from standard form to vertex use -b/2a
Graph y=x^2+6x +2

Sample problems changing a quadratic equation from standard form to vertex form.
The two forms are:
Standard equals y= ax^2 + bx +c
Vertex form equals y=a(x-h)^2 +k

0:47 Convert y=-2x^2 +34x -8 (standard form to vertex form)
3:34 Plug the numbers into the vertex formula to finish converting from the standard form to the vertex form.
Given an equation in vertex form how do you switch the equation to standard form?

Video Guide
0:48 Given the equation y=-(x-8)^2 +2 which is in vertex form,switch to standard form. This problem involves distributing a negative which involves changing all of the signs.
vertex formula -b/2a,f(-b/2a)
x=-b/2a axis of symmetry
Let's look at quadratic equations and what transformations are occurring to the parent function.y=x^2

Video Guide
0:24 y=2(x-9)^2 +3 The skeleton equation equals y=a(x-K)^2 +K
a=stretch h=left right shift,k=up down shift

1:55 What are the transformations of: y=-4/7(x)^2 -6

3:05 What if you are given descriptions and you have to write an equation? Once you write your descriptions you plug these into the vertex equation

Graphing a Quadratic Function

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What is a Quadratic Function?
Graphing a Quadratic Function.
Changing a Quadratic Equation from Standard form to Vertex form.
Changing a Quadratic Function from Vertex form to Standard Form.
Quadratic Equation transformations and the parent function.

Standard Form

Vertex Form

Finding the Vertex

Axis of Symmetry