Polynomial function (n≤2)
Learning objectives
- To introduce the notion of polynomial function.
- To know how to characterize a quadratic function and an affine function.
- To understand the influence of the function’s parameters on its graphical representation.
- To interpret the graph of a function.
- To be able to express a polynomial function of the second degree in different forms (standard, factored, vertex) knowing the coordinates of certain characteristic points.
A polynomial function of degree 2 is also called a quadratic function. Its graph is a parabola whose axis of symmetry is parallel to the y-axis.
A quadratic function can be expressed in three formats:
- Its general expression f(x) = a.x2 + b.x + c, is known as the "standard" form. a, b and c are the parameters of the function where c caracterizes the height of the parabola (where it intercepts the y-axis). f (0) = c.
- The "factored" form f(x) = a.(x - z1).(x - z2) reveals the "zeros" of the parabola (i.e. z1 and z2 are the two points where f(z1) = f(z2) = 0)
- f(x) = a.(x - h)2 + k is the "vertex" form. x = h is where on the x-axis the parabola changes direction. x = h is the axis of symmetry. h is the horizontal shift and k the vertical shift of the parabola. This is why the point (h,k) is called the "turning point".
This animation helps to understand the influence of each parameter on the form and plot of the function.
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