When students enter the classroom today, they should see a quadratic equation written on the board along with the instructions to make a sketch of the graph before I'm finished taking attendance.
y = x^2 - 5x + 4
After they've had a couple of minutes to do this, I'll move to the front of the room and ask them to describe the graph. We'll identify the vertex, y-intercept, and roots and I'll make a quick sketch for all to see. Next, I'll ask the class to describe to me how the graph will change if we change the final constant. (This produces a vertical translation.) Then I'll ask how the coefficient on the quadratic term affects the graph. (This determines the end behavior and the "steepness" of the curve.)
Here are some teacher's notes for your reference.
Now, I get to the point: Can anyone tell me how the graph will be affected if I change the linear coefficient? Probably not, so I'll open up this GeoGebra applet so that we can investigate. Watch this video for some details on what you'll see.
After getting an intuitive sense of how the linear coefficient affects the graph, I'll ask this question: What values of b produce real roots? Based on what we've already seen, the class should recognize that there will be real roots as long as the absolute value of b is greater than or equal to 4. We can (and should) confirm this with the discriminant of the quadratic formula.
Part of this discussion should focus on where the 4 comes from. A common confusion here is that the 4 came simply from the c-value. However, looking more deeply we see that b really depends on the value of 4ac. In this case, a happens to be 1, and in this case, c happens to also be 4. In other words, the boundary value of b (the one that stands between 2 real roots and 2 complex roots) depends on the product of a and c, and will always be equal to the square root of 4ac.
As usual, students will be required to work individually at first. (See Strategy Video for Individual Time) In this case, I want to see that everyone is able to use the Quadratic Formula to correctly compute a pair of solutions and that they understand how they are supposed to graph them.
Next, the students should start working in teams. (See Strategy Video for Group Time) The best strategy in this case is to have the students divide up the various b-values so that at least two students are independently calculating each pair of solutions, but so that none of the students have to do too much repetitive calculation. Typically, I'd ask for simplified radical form, but in this case decimal form will make graphing the solutions easier. I'd also advise them not to graph the solutions until they've had them checked by someone else, otherwise they'll end up doing a fair bit of erasing! In order to keep the different solutions distinct and to make their "motion" more evident, it might help to use a different color for each pair. (MP 6)
As class ends, I do not expect anyone to have completely finished this investigation. I also think these problems are not the sort of thing students should be expected to do on their own. So, tonight's homework will be to put in an honest 15 minutes of work. Try to make some progress so that you can return to class tomorrow either with new questions to ask or new answers to share.
Alternatively, you might consider a single homework problem like this:
"Given f(x) = 2x^2 + bx + 9, find the values of b for which the function will have real roots."
Based on the discussion at the beginning of class, everyone should be able to answer this question on their own.