7I - Rules of polynomial functions
Families of polynomials:
A family of polynomial 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:
A family of polynomial 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:
- The equation \(y=ax^3+bx^2+cx-3\) describes a family of cubic graphs that all have a \(y\)-intercept of \((0,-3)\).
- The equation \(y=(ax-b)(x-2)^2\) describes a family of cubic graphs that all have a turning point at \((2,0)\).
- The equation \(y=ax^4+bx^3+cx^2+dx\) describes a family of quartic graphs that all pass through the origin, \((0,0)\).
7I - Example 1: Families of polynomial functions
A family of cubic functions all contain a point of inflection at \((2,4)\), the general equation is \(y=a(x-2)^2+4\). Find the value of \(a\) for the member of the family that passes through the point \(-2,8\).
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7I - Example 1: Video solution
7I - Example 1: Practice
Question 1: ABC Question 2: ABC 7I - Example 1: Solutions
Question 1: ABC Question 2: ABC |
7I - Example 2: Families of polynomial functions
A cubic passing through the origin contains two other distinct intercepts that are of equal distance from the origin. The family can be described by the general equation \(y=ax(a^2-b)\). Find the values of \(a\) and \(b\) if the graph passes through the points \((-2,-20)\) and \((1,16)\).
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7I - Example 2: Video solution
7I - Example 2: Practice
Question 1: ABC Question 2: ABC 7I - Example 2: Solutions
Question 1: ABC Question 2: ABC |
Determining the equation of a polynomial graph:
When determining the equation of a polynomial graph it is useful to keep in mind the different general forms explored in the families of cubic functions and quartic functions. The most important piece of information is the degree of the polynomial.
When determining the equation of a polynomial graph it is useful to keep in mind the different general forms explored in the families of cubic functions and quartic functions. The most important piece of information is the degree of the polynomial.
7I - Example 3: Determining the rule for a cubic function given four points (CAS)
With the assistance of CAS, find the equation of a cubic that passes through the points \((-3,-11)\), \((-2,11)\), \((1,5)\), and \((3,121)\). Express your answer in the form \(y=ax^3+bx^2+cx+d\) where \(a\), \(b\), \(c\) and \(d\) are real numbers.
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7I - Example 3: Video solution
7I - Example 3: Practice
Question 1: ABC Question 2: ABC 7I - Example 3: Solutions
Question 1: ABC Question 2: ABC |