As Easy As 1-2-3
Basics of Quadratics (Review)
Basics of Parabolas
· May open upwards or downwards
· The zero is where the graph crosses the x-axis
· Zeroes are also called x-intercepts
· The axis of symmetry (AOS) divides the parabola into two equal halves
· Vertex is where the AOS and parabola meet. It is the parabola's maximum or minimum value
· Optimal value is y co-ordinate of the vertex
· Y-intercept is where the graph crosses y-axis
The 3 Quadratic Equations
An equation in vertex form can be used to solve for:
- The optimal value (The y value of the vertex)
-The axis of symmetry (The x value of the vertex)
- Stretch or Compression of parabola
- Direction of opening
- x intercepts/zeroes
In this form:
x=h is the axis of symmetry
If "a" is negative, then the graph opens downwards like an upside down "U".
If "a" < 1, the graph of the parabola widens. This just means that the "U" shape of parabola stretches out sideways.
If |a| > 1, the graph of the graph becomes narrower (The effect is the opposite of |a| < 1).
The Step Pattern
The first step in graphing a parabola is to understand the step pattern:
With the equation y=x^2, the pattern is (1,1),(2,4),(3,9).
In the step pattern the x value increases at a constant rate of 1. In the equation y=x^2, the x value is squared to get the y value, which gives you the pattern 1,4,9 etc.
If the a value in the equation is more than 1, for example y=4x^2, the y values in the step pattern are multiplied by the a value. In the case, the points would become (1,4),(2,16),(3,36).
Finding the Vertex
To find the vertex in an equation in vertex form, all you need to do is find the h and k values in the equation.
If there is no h or k value in the equation, then the value is 0.
In the equation y=x^2, there is no h or k value, meaning the vertex is at (0,0).
Finding x intercepts
To find the x intercepts with an equation in vertex form put y=0 and solve the equation.
An equation in factored form can be used to solve for:
-The x intercepts (r and s values)
-The axis of symmetry (x value of vertex)
-The optimal value (y value of vertex)
Finding the x intercepts
The x intercepts are the r and s values in a factored form equation. Just like the h value in vertex form, the r and s values are negative, making their values opposite. For example if the equation is y=(x+5)(x-6), the x intercepts would be (-5,0) and (6,0).
The optimal value
To find the optimal value, you to find the axis of symmetry. Once you have that, you put it into to your equation as the value of x and solve.
The axis of symmetry
To find the axis of symmetry, all you need to do is add the r and s values and divide by 2 as shown in the image above.
An equation in standard form can be used to solve for:
-The x intercepts
-The axis of symmetry
-The optimal value
A standard form equation can also be converted to vertex form by completing the square.
Factoring to turn to Factored Form include:
The factors of c have a sum of b