Quadratic Functions: the effect of "b"
- Whit Ford
A quadratic equation in "Standard Form" has three coefficients: a, b, and c. Changing either a or c causes the graph to change in ways that most people can understand after a little thought. However, changing the value of b causes the graph to change in a way that puzzles many. The graph below contains three sliders, one for each coefficient. You may click and drag them left or right to alter the value of each coefficient, and the graph will change to reflect the new value. Note that the vertex of the parabola is identified on the graph as point V, with its coordinates shown. Below the three sliders are a series of equivalent equations, each of which describes the graph being shown. The first equation is in "Standard Form", the second in "Vertex Form" (start with the Standard Form, then complete the square), and the remaining ones expand the Vertex Form for reasons that will be explained below. Use the red slider to vary the value of a, the coefficient of the squared term. Notice how the graph becomes wider or taller, and reflects vertically about the x-axis when a becomes negative. a is the vertical dilation factor for this function, as shown by the Vertex Form of the equation. Use the green slider to vary the value of c, the constant term. Notice how the graph slides straight up or down, without changing its shape at all. Changing c translates the graph vertically by adding a constant value to all y-coordinates on the graph, as shown by the Vertex Form of the equation. Use the blue slider to vary the value of the linear term b. Pay attention of the path that the point V travels as you move the b slider back and forth. Does this path surprise you? Please continue reading the text below the graph to explore why this behavior occurs.
All of the equations shown in the bottom of the applet are equivalent equations. To get from the first to the second, complete the square. To get from the second to the third, square the binomial in parentheses. To get from the third to the fourth, distribute and collect like terms. The last equation above can be compared to the first equation, Standard Form, since the two equations describe the same curve. Both forms of the equation have a single squared term, so their coefficients must be equal. Similarly, the coefficients of the linear terms must be equal, and the constant terms must be equal. The coefficients of both squared terms (in the first and last equations) are already equal to one another. Since neither involve b, we can ignore them as we explore why changes to b affect the graph as they do. Setting the coefficients of the linear terms (the "x" terms) from the first and last equations equal to each other produces: This shows that when b changes (but a does not), the only thing on the right side of the equation that can change to reflect the change in b is H. So, changing b by some percentage must cause H (the x-coordinate of the vertex) to change by the same percentage, but in the opposite direction. You can verify this is true by paying attention to the x-coordinate of the vertex as you change the value of b (try values of 1 and 2 for b). If the value of b doubles, the value of the x-coordinate of the vertex will also double. Setting the constant terms from the first and last equations equal to each other produces: If only b is being changed, then c and a are both constant in this equation and will not change as b is changed. As described in the previous paragraph, H will change by the same percentage as b, but in the opposite direction. In this equation, H is squared, so if a and c remain constant, K (the y-coordinate of the vertex) is the only quantity that can change to offset the squared change in H. When a is positive, a times the square of H will always be a positive quantity, so K must be a negative quantity proportional to the square of H. This explains the graph's movement along a parabolic trajectory as b is changed. Changing b causes H to change by the same percentage, which causes K to change in the opposite direction by an amount proportional to the change in the square of H. In other words, the y-coordinate of the vertex will move along a parabola as the value of b is changed. If you wish to use other applets similar to this, you may find an index of all my applets here: https://mathmaine.com/2010/04/27/geogebra/