Standard Forms

If you consider the examples quadratic equations offered in the previous sections, you find that the standard form of the equation can prove useful as a way to easily discern the basic features of the parabola the equation generates. To recapitulate, consider again the extended form of the equation:

a (x – h)2 + k

If you know the value of a, then you can determine how wide or narrow the parabola is likely to appear. If a is positive, the parabola opens upward. If a is negative, the parabola opens downward. If the value of h is positive, then the vertex of the parabola shifts to the right, or positive, direction on the x axis. If it is negative, then the vertex of the parabola shifts to the left, or negative, direction on the x axis. If the value of k is positive, then the vertex shifts upward on the y axis. If the value of k is negative, then the vertex shifts downward on the y axis.

The equation in this form proves so valuable that it is worthwhile knowing how to convert quadratic equations so that they appear in this form. Toward this end, consider again the standard form of a quadratic equation:

ax2 + bx + c

To alter an equation you find in this form so that you can discern its component variables, you perform an operation known as completing the square. The next section covers in detail the procedure for completing the square of a quadratic equation. For now it remains important to focus on the notion that the extended form of the equation consists of a restatement of the standard form of the equation. Table 9.1 provides a summary of the features of the extended form of the equation. Subsequent sections of this chapter discuss features not covered in the previous sections.

Table 9.1. Features of the Standard Form
ItemDiscussion
ax2 + bx + cStandard form of a quadratic equation. You can set the constants b and c to 0, resulting in an expression of the form ax2, but you may not set the constant a to 0. By definition, the quadratic equation is an equation of the second degree, so if you eliminate the term with the coefficient corresponding to the second-degree variable, you change the equation so that it no longer corresponds to the definition of a quadratic equation.
a (x – h)2 + kThis is the form a quadratic equation can assume if you rewrite it by completing the square.
(h, k)This coordinate pair establishes the position of the vertex. If h is negative, then the vertex lies to the left of the y axis. If h is positive, then the vertex lies to the right of the y axis. If the variable k is positive, then the vertex is above the x axis. If the variable k is negative, then it lies below the x axis.
hThe value of h defines the line of symmetry for the parabola. If this is a positive value, then the line of symmetry shifts to the right of the y axis. If the value is negative, then the line of symmetry shifts to the left of the y axis.
aThe value of a determines how sharply the parabola rises. If the value is greater than 1, then the parabola narrows and rises more precipitately. If the value is less than 1, then the parabola becomes wider and rises less precipitately.
kThe constant k establishes the y intercept for the parabola. If the value of k is positive, then the vertex moves upward relative to the x axis. If the value of k is negative, then the vertex shifts downward.

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