For most students, being asked to graph a quadratic equation is not a big deal. For example, given the equation y=(x^2)+1, students can plug in values for x to get the corresponding y values, plot their (x,y) points, and connect the dots. Easy. Some students may even be able to explain that this graph is just the graph of the parent function (y=x^2) translated up one unit. However, students who know this do not necessarily understand the effects of each of the coefficients in the general quadratic equation y=a(x^2)+bx+c. In particular, teachers often avoid (or at most gloss over) the direct effects that the coefficient b has on a quadratic equation, because the way in which b affects the graph is less obvious than the effects of a or c. Using Desmos as our technology of choice, let’s investigate how b affects the quadratic equation.
In order to get a clear picture of how b changes our graph, we will let a=1 and c=0. Starting with b=1, we can incrementally increase b by one unit, noticing that as b increases, the vertex of our graph traces the graph y=-x^2. The same is true for when we incrementally decrease b by one unit, allowing us to see the following pattern:
Given y=(x^2)+bx, the vertex of the graph is ( [-b/2] , [-(b^2)/4] ).
This is illustrated in the graph below, where the green points indicate the location of the vertex for the graph when b=-4 to b=4.
It should be noted that when a=-1 and c=0, and b is changed in the same way as described above, the vertex of the graph y=-(x^2)+bx traces the graph y=x^2. This produces the following pattern:
Given y=-(x^2)+bx, the vertex of the graph is ( [b/2] , [(b^2)/4] ).
In order to fully understand the effects of b on a quadratic function, we next need to graph these two equations:
For the sake of clarity, we will let a=1 or a=-1. Now, as we change b, we see that doing so changes the vertex of the graph y=a(x^2)+bx such that the line y=bx is tangent to the graph at the origin (0,0). See example graphs with a=1 and a=-1 below:
Now that we have seen the isolated effects of b on the quadratic equation, let us consider equations in which c does not equal zero. We will graph the following two equations:
Keeping a=1 or a=-1, we see that as b and c both change, the vertex of the graph changes such that the line y=bx+c is tangent to the graph y=a(x^2)+bx+c at (o, c). This can be seen in the images below.
Finally, we will examine the case when a does not equal one or negative one. For now, c=0. Choosing a fixed value for b, we can record the coordinates of the vertex as we change the value of a. Through this process, we see a pattern:
Given y=a(x^2)+bx+c, the vertex of the graph is ( x, f(x) ) where x=[-b/(2a)] and f(x)=-a(x^2)+c
Realizing that the x coordinate of the vertex is also our line of symmetry, we can now graph any quadratic equation.
Through this process, we have seen yet another example of how Desmos can serve as an effective technology for discovering graphical relationships in the classroom. Drawing these same conclusions is much harder when using technology such as the TI-84 graphing calculator, because the TI-84 does not have sliders with which a student could adjust coefficients of general equations. At best, a student using a TI-84 for this investigation might notice general trends of the effects of changing b but be unable to draw precise conclusions.
With Desmos as free and accessible tool for teachers, students should soon “b” able to explain the effects of constants in the quadratic equation!