Quadratics

Quadratics

1. Quadratic Functions

   • Definition
     • A Quadratic function is any function that can be written in the form y=ax^2+bx+c
       • where a ,b ,and c are real numbers and a≠0.
   • Differences
     • First Differences
       • The differences between y-values for a constant change in x-values are called the first differences.
     • Second Differences
       • When the first differences have a constant rate of change that is the second different
       • All quadratic functions have a constant second difference
     • Example
       • y=x^2
         • The First difference is increasing by two so therefore the second difference is 2
   • Identifying quadratic functions
     • Ordered pairs or table
       • If you can find the Second Difference and it is constant then it is a quadratic function
     • With equations
       • If it is written out or can be written out as y=ax^2+bx+c
     • In a Graph
       • In a Graph it is represented as a parabola
   • Graphing Quadratic Functions
     • When graphing quadratic functions you need to make a table with at least 3 points, usually 5 points are best
       • Then you graph the points connecting them with a smooth curve
     • parabola
       • When graphed the shape is a parabola
       • If a>0 then the parabola is open upwards
       • If a<0 then the parabola is open downwards
       • Vertex
         • The highest or lowest point on a parabola is called the vertex
         • If the parabola opens upwards the ordered pair at the vertex is its minimum
         • If the parabola opens downwards the ordered pair at the vertex is its maximum
   • Domain and range
     • Domain
       • The Domain Is all real numbers
     • Range
       • If the parabola if facing upwards the eveyrthing above the minimum is the range
       • If the parabola is facing downwards then everything below the maximum is the range

2. Zero of a function

   • Zeros
     • The Zero of a function is an x-value that makes the function equal to 0
     • The Zero of a function is the x-intercept of the function
     • A quadratic function may have 1 ,2 ,or no zeros
   • The axis of symmetry
     • The axis of symmetry is a vertical line that can split a parabola in halve
     • The axis of Symmetry always passes through the vertex of the parabola
     • Finding the axis of symmetry
       • With one zero you can simply use the x-coordinate of the vertex
       • If the function has two zeros then find the average of both
       • If there are no zeros then you can use the formula x=-(b/2a)
         • B and a come from the equation y=ax^2+bx+c
   • How to find the vertex of a parabola
     • Step 1: Find the axis of symmetry by using zeros or the formula
     • Step 2: Find the y-coordinate by substituting the x-coordinate into the equation
     • Step 3: write vertex as ordered pair

3. How to graph a function

   • The y-intercept of a quadratic function is always c
     • The x-coordinate of the y-intercept is 0
   • Find the axis of symmetry
   • Find the vertex
   • Find the y--intercept
   • Find two more points by using random points below the axis of symmetry
   • Graph the points, the y-intercept, and the vertex
   • Finally Using the axis of symmetry create the reflection of the points then use slight curves to connect the points

4. How to solve a word problem example

   • Find the axis of symmetry
   • Find the vertex
   • The vertex is the answer

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