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Learning intentions:
In this section we will examine:
- Families of quadratics which share a common feature.
- Determining specific members of quadratic families.
Families of quadratic functions
A family of quadratic functions will all share a common feature, or common features, while other parameters are used to indicate that many different equations exist within the family. Examples include:
- Quadratics with a y-intercept of 4: y = ax2 +bx + 4
- Quadratics with an x-intercept of -3: y = a(x - b)(x + 3)
- Quadratics with a turning point at (-1, 4): y = a(x + 1) + 4
- Quadratics with two x-intercepts (one being the origin): y = ax2 + bx
The dynamic GeoGebra worksheet below illustrates the family of quadratic with x-intercepts of -2 and 3.
- Please click on the play button in the bottom left hand corner to animate!
4J - VIDEO EXAMPLE 1:
A family of quadratics all have a vertex existing on the line x = 3, the general equation for this family is y = a(x - 3)2 + k. Determine the equation of the member which has a range of [5, ∞) and passes through the point (5, 11).
4J - VIDEO EXAMPLE 2:
A family of quadratics with two x-intercepts, one being the origin, has the general equation
- Find the value of b (in terms of a) which gives the other x-intercept at (0, 4).
- If the quadratic goes through the point (1, 6), find the value of a. Hence, state the full equation of this member of the family.
Finding the rule of a quadratic function
When determining the rule of a quadratic you need to identify what information is provided. Possible information in written or graphical form, includes:
- The y-intercept
- The x-intercept(s)
- The turning point
- A point on the curve
Depending on the information provided, you may choose to use a quadratic form to enter the information into. Recall the three quadratics equations forms:
- The general form of a quadratic equation is:
- The turning point form of a quadratic equation (with turning point (h,k)) is:
- The factorised form of a quartic equation (x-intercepts occur at (b,0) and (a,0) is:
Other useful quadratic equations for specific contexts include:
When given two intercepts where one is the origin:
Download summary of finding the rule of a quadratic PDF.
When given a quadratic symmetric about the y-axis.
Remember: for every unknown (a, b, c) you need one point (or piece of information).
Any three distinct points will define a unique parabola; however, if one of the points is the turning point only one other point is required to define the unique parabola.
4J - VIDEO EXAMPLE 3:
Determine the rule for the quadratic which goes through the points (0, 1), (1, 1) and (2, 2).
4J - VIDEO EXAMPLE 4:
Determine the rule for the quadratic with a turning point at (1, 2) and passing through another point (-1, 9).
4J - VIDEO EXAMPLE 5:
Determine the rule for the quadratic with x-intercepts at x = 1 and x = 2 and passes through another point (0, 3).
4J - VIDEO EXAMPLE 6:
Determine the rule for the following quadratic graph:
Success criteria:
You will be successful if you can:
- Correctly match information given about a quadratic to a general quadratic equation.
- Determine the rule for a quadratic by substituting information into a general quadratic equation.
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