JOURNEY THROUGH QUADRATIC RELATIONS
By: Bansari Shah
What's Quadratics & How's it Useful
Basically the word quadratics comes from 'Quad' meaning square since the variables get squared ( ).
Table of Contents
Key information about Parabola
- Basic Information Facts About Parabolas
Intro to Parabolas
- Key Features of Quadratic Relations
First & Second Differences
Types of equations
- Key Facts
- Finding An Equation When Given the Vertex
- Graphing from Vertex Form
- Isolating for x
- Solving Using Quadratic Equation
- Graphing from Standard Form
- Completing the square: vertex to standard form
- Factors & zeros
- Expanding - factored form to standard form
- Common Factoring
- Simple Trinomial
- Complex Trinomial
- Difference of Squares
- Perfect Square
- Factoring by grouping
- Number Problems
- Optimization Problems
- Revenue Problems
- Unit Analysis
Basic Important Facts About Parabolas
- Parabolas can open either up or down
- The zero of a parabola is where the graph crosses the x-axis
- 'Zeros' can also be called "x-intercepts " or "roots"
- The axis of symmetry divides the parabola into two equal halves.
- The vertex of a parabola is the point where the axis of symmetry and the parabola meet. It is the point where the parabola is at the minimum or maximum value.
- The optimal value is the value of the y co-ordinate of the vertex.
- The y-intercept of a parabola is where the graph crosses the y-axis.
Key Features of Quadratic Relations
First & Second Differences
Types of Equations
The axis of symmetry : (x = h)
The optimal value is (y = k)
Note: When writing down the vertex, remember that the h value is always the opposite. If the h value is positive then when you write it down it would be negative and if the h value is negative then when you write it down it would be positive.
When in a vertex form each letter is responsible for a transformation.
'a' stretches the parabola vertical if this value is a whole number. If the value is a fraction than that would be a vertical compression. Furthermore, the 'a' value tells us whether the parabola will open up or down. If the "a" value is negative then the parabola will open down & if the 'a' value is positive then the parabola will open up.
(NOTE: if the 'a' value is negative, the vertex will be a maximum value & if the 'a' value is positive, the vertex will be a minimum value.)
'h' moves the vertex of the parabola either left or right. When 'h' is negative move right and when its positive move left.
(NOTE: it affects the horizontal translation)
'k' moves the vertex of the parabola up or down. If the 'k' value is negative the parabola opens down. (NOTE: affects the vertical translation. Also if the "a" value is negative there will be a reflection)
Finding An Equation When Given Vertex
Graphing From Vertex Form
- Find the vertex, since the formula is given in vertex form. (h,k)
- Find the y-intercept. In order to find the y-intercept you need to plug in the value of x to be 0.
- Find the x-intercept(s). To find the x-intercept, you need to plug in the value of y to be 0. You can solve for "x" by using the quadratic formula.
- Graph the parabola using the points which you found in steps 1- 3.
Isolating For x
Solving Using Quadratic Equation
Graphing From Standard Form
There are 3 possible solutions of a discriminant.
- When the discriminant is less than zero, there is no solution.
- When the value of discriminant is more than zero, there will be 2 solutions.
- When the discriminant is 0, there will be only one solution. Whether you add or subtract it from 'b', you will get the same answer.