Monday 27 March class

Topics:

• The Quadratic Formula and what we can learn from it about solutions to quadratic equations

• Graphs of quadratic functions

Summary of what the Quadratic Formula tells us about the solutions to quadratic equations: These observations come from the basic observation that the quantity under the radical sign (the radicand) is the part of the quadratic formula that can give rise to solutions which are irrational or complex. That is why we give that radicand a special name:

is called the discriminant of the quadratic equation .

What the discriminant tells us about the roots (solutions) : if all of the coefficients are integers, then

• If , there will be two solutions, both of which are real numbers.

• In the case where  and it is a perfect square, then the solutions will be rational numbers. [This means also that you could have solved the equation by factoring and using the Zero Product Property.]

• In the case where  and it is not a perfect square, then the solutions will be irrational numbers.

• If , there will be one solution, a real number.

• If , there will be two non-real complex solutions, which are conjugate to each other.

Graphs of quadratic functions:

We first considered the basic graph of . This graph is a parabola with vertex at (0,0), and it is symmetric by reflection over the y-axis. Please note that the graph also contains the points (1,1) and (-1, 1). You can see the graph of here (in desmos)

This is an example of a function : the idea is that we think of inputting a number for x, and the formula outputs a number ($latex x^{2}) which is the value of the function for that x. So x is the input and y is the output. (These are sometimes called the “independent variable” and the “dependent variable”.)

We then considered what would happen if we change the function by adding a number c to its output. So we are considering the graphs of functions of the form . By using desmos to play around with different values for c, we found that this moves the graph upward or downward by the amount c, depending on whether c is positive or negative. The graph with sliders to change the value of c is here (in desmos).

We next considered what would happen if we change the function by adding a number c to its input, in other words, we substitute (x+c) in place of x in the formula for the function. So we are considering the graphs of functions of the form . By using desmos to play around with different values for c, we found that this moves the graph to the left or to the right by the amount c, depending on whether c is positive or negative: it moves to the left if c is positive, and to the right if c is negative. The graph with sliders to change the value of c is here (in desmos).

Finally, we considered what would happen if we change the function by multiplying the output by a number a, so we are considering the graphs of functions of the form . By using desmos to play around with different values for a which were positive, we found that this stretches or compresses the graph toward the x-axis, depending on whether a is greater than 1 or a is between 0 and 1 . If a is negative, the effect is the same (stretching or compressing), but also the graph is reflected (“flipped”) over the x-axis, so it opens downward.  The graph with sliders to change the value of a is here (in desmos).

Putting these all together, we come up with a quadratic function of the form (using different letters now)

Compared to the basic parabola , this will be moved up or down by k units, right or left by h units, and it will have vertical stretching or compression due to the a. If a is positive, the parabola will open upward; if a is negative, the parabola will open downward. It vertex will be at the point (h, k). Here is the graph in desmos with sliders so you can change the values of a, h, and k and see what happens.

[Note that we took care of the problem that a positive number added to the input was moving us to the left, by writing (x-h), so instead of adding a positive number we are subtracting a negative number h in that case.]

Our object will be to take a quadratic function which is in the form and use completing the square to rewrite that equation in the standard form . Then we will easily be able to read the vertex of the parabola from the equation and also we will be able to sketch the graph very quickly and accurately by hand.

Please note that the method we use to rewrite the equation is NOT the method used in the textbook! In order to make it a bit easier to complete the square, we are first isolating the terms which have x in them on one side of the equation, and only at the end we go and solve for y again.

The WeBWorK assignments are related to the two things above.

Homework:

• Make sure that you have done all of the homework problems assigned last time on using the Quadratic Formula. (p. 595 #5-25 odd). Now go back to each of those problems, compute just the discriminant , and see how the number and nature of the solutions correspond to the discriminant as described in the summary above.

• Do the WeBWorK before you do the problems in the textbook below! There are two parts to the assignment, and the one on vertices will walk you step-by-step through the procedure for putting the equation into standard form . We did some of this in class.

• In the textbook, do the following problems: p. 625 #17-27 odd. Make sure that you are using completion of the square in order to find the vertex, not any other method. See Examples 1, 2 in Section 7.5, or see here. (link to PurpleMath – she completes the square using the same method we do, and explains how to read off the vertex and what the effect of a is on the graph.)

Remember that you can use the Piazza discussion board to ask questions if you get stuck on any of the WeBWorK or the other homework problems. Please be as specific as possible about what you are stuck on and don’t forget to put the problem itself into your question, as we may not have the textbook with us when we go to answer it!