Algebra students often have a difficult time understanding the relationship between a graph of a straight or a curved line and an equation. Because most algebra classes teach equations before graphs, it is not always clear that the equation describes the shape of the line. Therefore, curved lines are a special case in algebra; their equations may take on one of many forms, depending on the curved line you are dealing with.
In high school algebra, the kinds of curved lines that students are most likely to see are the graphs of quadratic equations. These equations take the form of f(x) = ax^2 + bx + c, and can be solved a variety of ways; students will often be asked to find the solutions, or the zeros, of these graphs, which are the points at which the graph crosses the x-axis. Before working with the graphs, however, students should be comfortable with the format of quadratic equations and may work on factoring them as well.
Graphing Quadratic Equations
Quadratic equations will graph as parabolas, or symmetrical curved lines that take on a bowl-like shape. These equations will have one point that is higher or lower than the rest, which is called the vertex of the parabola; the equations may or may not cross the x or y axis.
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A parabola that is graphed downwards, or that looks like an upside-down bowl, has a negative coefficient for the part of the equation ax^2. In this case, the vertex will be the highest point on the parabola. However, the axis of symmetry, or the perfect symmetry present in parabolic/quadratic equations with positive coefficients, will remain the same.
Other Curved Lines
Students may come across curved lines that are not quadratic equations; these expressions may have some other kind of exponent attached to the variable, such as x^3 or even higher expressions. To find the equation for a non-parabolic, non-quadratic line, students can isolate points on the graph and plug them into the formula y = mx+b, in which m is the slope of the line and b is the y-intercept.