Graphs and Effects of Parameter Changes

The two-dimensional Cartesian coordinate system may be used to graph a variety of equations in the form of straight and curved lines. One way to graph an equation is to determine a number of different values for the variables and plot them on the graph. It can be helpful, however, to understand how a change in the parameters of an equation affects the resulting line.

Graphs of Straight Lines

The graph of the simple equation y = 1x, or y = x, is graphed in (a) below. The line passes through every coordinate point where x = y, such as (2, 2) or (−3, −3).

Notice what happens to the graph of the equation when the parameters of the equation are changed by using a different coefficient for x, such as y = 3x. As part (a) shows, the new line has a steeper slope. If the coefficient is further increased, the slope will become even steeper, and the line will become closer to vertical.

Conversely, if the coefficient for x is decreased, a different change in the line occurs. As part (b) above shows, the line for has a more gradual slope than the line for y=x. If the coefficient were to be further decreased, the line would become flatter, and it would appear more like a horizontal line.

If the coefficient is a negative number, an interesting line results. For instance, the graph of the equation y = −3x is shown in part (b) above. Comparing this line with the graph for y = 3x shows that the graph of y = −3x moves down from the left to right (a negative direction) and the graph of y = 3x moves up from left to right (a positive direction). These two lines therefore are described as having a negative and a positive slope, respectively. From this example, it can be seen that the coefficient of x in the equation of a line indicates the slope of the line.

Lines Not Passing through the Origin. All of the lines described above pass through the origin (0, 0), but many lines do not. The line for the equation y = 3x + 2 will pass above the origin, as shown in part (a) below. The line has the same slope as y = 3x, but has been shifted two units up the y -axis. The two lines are thus parallel because they have the same slope.

The equation y = 3x − 2, also graphed in (a), passes through the y –axis at −2. The slope of the line is 3, which is the same the slope for y = 3x. But compared to y = 3x, the line for y = 3x − 2 has been shifted two units down the y -axis.

Understanding the role of the coefficient of x in indicating the slope, and the role of the constant in indicating the point where the line will intersect the y -axis, can make it easier to graph an equation. The equation y = 2x + 4 will pass through the y –axis at 4 and has a slope of 2, as shown in part (b) above.

The equation y = 2x + 4 may be expressed in different forms. For example, x =½ y − 2. The form y = 2x + 4, written in general terms as y = mx + b, is called the slope intercept form. In this form, the coefficient m is the slope, and the constant b indicates where the line intersects the y –axis. The intersection point is called the y -intercept. The graph of the equation y = 3x − 4 has a slope of 3 and intersects the y -axis at −4.

Graphs of Curved Lines

Changes in the parameters of higher-degree equations (that is, higher than first-degree) will result in patterns of changes in their graphs similar to those for straight lines. For example, the graph of y = x ², as shown in part (a) below, is a curve known as a parabola. The coefficient indicates whether the parabola opens facing up or facing down, and whether it is narrow or broad.

Changing the coefficient of x ² to a negative value results in a parabola that is a mirror-image of the parabola for y = x ². The parabola for y = x ² opens upward, and the parabola for y = −x ² opens downward.

If the coefficient of x ² is changed so the equation is y = 3x ², notice the changes to the graph of the equation, as depicted in part (b) below. The parabola still passes through the origin, but it is narrower than y = x ². If the coefficient of the graph were increased even more, then the parabola would become even narrower. Conversely, if the coefficient of the original equation were decreased, the resulting parabola would still pass through the origin, but would become broader, as depicted in (b) below as y = ½x ²

Compare the parabola in (b) below for y = 3x ² to the parabola for y = −3x ². As with the equations y = x ² and y = -x ², one parabola is the mirror image of the other. The two parabolas open in opposite directions.

Curves Not Passing through the Origin. Changing a parameter of the equation y = x ² to y = x ² + 2 will change the point at which the parabola crosses the y -axis. In this case, the parabola does not pass through the origin, but passes through the y -axis at 2, as shown below. The parabola has been shifted two units up the y -axis. Similarly, the parabola of y = x ² − 5 is shifted 5 units down the y -axis, and passes through the y -axis at −5.

In the general equation for a parabola, y = ax ² + b, the coefficient a indicates the shape of the parabola and in which direction the parabola opens, upward or downward. The value of b indicates where the parabola intersects the y -axis.

More complex equations and their graphs will also show patterns that result from changes in the parameters of the equations. Knowing how changes in the various parameters of an equation affect the graph of the equation is helpful for drawing, interpreting, and applying equations and their graphs.

see also Graphs.

Arthur V. Johnson II

Bibliography

Bellman, Allan, Sadie Bragg, Suzanne Chapin, Theodore Gardella, Bettye Hall, William Handlin, and Edward Marfune. Advanced Algebra: Tools for a Changing World. Needham, MA: Prentice Hall, 1998.