Quadratic Functions and their Basic Properties

There are many Functions that are not linear. For example, consider the function defined by the equation y=x2. Draw the graph of this function by first completing the table of values below, and connecting the points that you obtain.x -3 -2 -1 0 1 2 3y

Since the equation is not linear, do not connect the points with straight line segments, but with smooth curves. When you are done, you will have something like this:

This curve is called a parabola, whose shape is similar to the capital letter U with its sides more spread out. Curves like this represent quadratic functions. Quadratic functions are functions defined by equations of the form y=ax2+bx+c, where a, b, and c are constants and a≠0. Why is a required to be nonzero? What are the values of a, b, and c for the quadratic function defined by y=x2?

Equations of Quadratic FunctionThe equation y=ax2+bx+c may be written in the alternative form y=a(x-h)2+k. The following example shows how to transform the equation y=ax2+bx+c, which is called the standard form of the equation of a quadratic function, into the equivalent form y=a(x-h)2+k, which is sometimes called the vertex form of the equation. The point (h, k) is called the vertex of the graph of the function. The conversion uses the method of completing the square.

Example:Convert the equation y=2x2-4x+3 to the standard form.1. Factor out a=2 from the first two terms. y=2(x2-2x)+32. Complete the square of the quadratic polynomial inside the parentheses by adding 1 (1 is the square of -1, which is one-half of the coefficient -1 of the linear term -2x). Since this polynomial is multiplied by the factor a=2 outside, you actually added 2; so balance this by adding -1 to the constant term 3 outside the parentheses.y = 2(x2-2x+1)+3 - 2 = 2(x-1)2+1The equation is now in the vertex form, with a=2, h=1, and k=1.

To find the coordinates (h, k) of the vertex, with the quadratic function given:The vertex is the lowest (highest) point on the graph of a quadratic function when a 0 (a 0).˃ ˂1. When a 0, the minimum value of ˃the function is k.2. When a 0, the maximum value of ˂the function is k.

Intercepts of the Graphs of Quadratic FunctionIf the equation of a quadratic function is in the standard form y=ax2+bx+c, then1. the parabola intersects the y-axis at the point (0, c)The coordinates are obtained by letting x=0 and solving y in the equation y=ax2+bx+c.2. the x-coordinates of the point(s) where the parabola intersects the x-axis is (are) called the x-intercept(s) of the parabola. Since at these points, the value of y is zero, the x-intercepts are the roots of the associated quadratic equation ax2+bx+c=0.

Do all parabolas have x-intercepts? How many x-intercepts can a parabola have?When does a parabola have two x-intercepts? one x-intercepts? no x-intercept? Can a parabola have more than two x-intercepts? Why or why not?

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