Make sure to see the recommendation for ordering your work on the various homework assignments, at the end of this post!
Here is the classwork from the start of class, on vertical lines and then on graphing parabolas. If you have not finished with any or all of this, part of your homework is to complete it.
When you work on these, please make sure that you carefully read all of the notes and instructions, and do all parts of the parabola problems in the order that is listed. Because this was an active learning assignment, all of the notes and instructions are necessary information that you need to complete the problems, and you will also need to use your head.
MAT1275graphingParabolasPictures
New topic: The Distance Formula, and Circles. (Lesson 11 in the textbook)
The distance formula comes from the Pythagorean Theorem. I will link an extended discussion of the example I used in class to motivate this formula: it will be very easy to remember the formula once you understand its connection to the Pythagorean Theorem
Distance formula: The distance $d$ between two points $\left(x_1 , y_1 \right)$ and $\left(x_2 , y_2 \right)$ is given by
$d = \sqrt{\left(x_1 – x_2 \right)^{2} + \left(y_1 – y_2 \right)^{2}}$
or, in words, the distance is the square root of the sum of the difference of the x’s, squared, plus the difference of the y’s, squared.
Note that it does not matter at all what order you take those differences, because the results will be squared and so will always come out non-negative. Don’t be distracted by the subscripts!
Circles: are connected to the distance formula!
A circle consists of all the points in the plane which are a certain fixed distance $r$ away from a special point called the center of the circle.
Basic but very important example: the unit circle. We will be living intimately with this circle for weeks on end later in the course!
The unit circle is the set of all points in the plane which are at distance 1 away from the origin $(0,0)$. We say that 1 is the radius of the unit circle and $(0,0)$ is its center.
We can find the equation of the unit circle by using the distance formula: if $(x,y)$ is a point which is on the unit circle, then the distance from $(x,y)$ to $(0,0)$ must be 1:
$1 = \sqrt{\left(x – 0\right)^{2} + \left(y – 0 \right)^{2}}$
Square both sides and simplify: this gives the equation
$x^2 + y^2 = 1$, the equation of the unit circle.
More generally, if a circle has center $(h,k)$ and radius $r$, then its equation is $\left(x – h\right)^{2} + \left(y – k \right)^{2} = r^{2}$
So given the equation in that form, we can read off the center and compute the radius.
If the equation is not already in that form, we have to complete squares in order to put it into that form to find the center and radius. There is no alternative! That’s why I told you that you had to learn to complete squares when we solved quadratic equations and when we worked with parabolas, even though in those two cases there were other ways to get the information we were looking for. In some cases there is no other way than to use completion of squares! (So the smart people have already been practicing. Be one of them, if you aren’t already.)
Recommendations for the homework assignments:
• First complete the classwork linked at the top of this page (which we worked on in class) if you have not already done so. Make sure that you read ALL the notes and instructions carefully. If for some reason you go to a tutor with this classwork, make sure the tutor is aware it was an active learning assignment and they should not just hand you formulas. The Quiz on Thursday will be on graphing parabolas.
• Then complete the WeBWorK “DistanceFormula” which is due by Wednesday night! There are only two problems in it, but one of them you may have a question about. If so, please use the “Ask for help” button at the bottom of the problem page!
• Then start on (and complete if possible) the WeBWorK “Circles”. Do all but the last problem at this point. For the last problem we need one more ingredient, so we will work on this on Thursday.
