Today's soundtrack is Shai Linne: Storiez, a collection of (of course) stories about people from different places and times coming face-to-face with circumstances that test their faith.
This afternoon, I'm working on the assignment portion of the quadratic function lesson. Parts 1 and 2 of the lesson can be found here and here.
The very first question asked me to graph and identify the vertex of several equations. For the life of me, I could not figure out how to do so with my calculator. My first problem was that I could only see one intercept on my screen. I found an article on the Duke University website that advised trying either Zoom 0 or Zoom 6. The former showed me the vertex and all intercepts.
Next, the same article instructed that I press [2nd] [TRACE] to access the "CALC" function. Once there, we will choose either 3 for the minimum point, or 4 for the maximum point. Which one we need depends, of course, on whether ax² is positive or negative. Once we've chosen the right one, a little "x" will show up on one side of the vertex. Hit [ENTER], then use the arrow keys to slide the arrow to the other side of the vertex, then hit [ENTER] again. The screen will show a message saying "Guess?" Hit [ENTER] a third time to confirm. The calculator will then display the vertex!
To find the y-intercept, we type our calculation into the [Y=] section, go to CALC, choose VALUE, and indicate X=0. The y-intercept is then displayed. This one I was able to figure out myself, but I had to look up the method to find the x-intercept.
I found the method of finding the x-intercept in this guide on Montana State University's site. It's very similar to finding a vertex, and must be done once for each x-intercept.
For the next section, I decided to bunny-trail a bit. The workbook gave me an equation and asked me to fill in an x/y table, find the intercepts, the coordinates of the vertex, the equation of the axis of symmetry, the domain of the function, and the range of the function. But I realised something. At no point have I learned how to do this without a calculator. I don't understand how the quadratic equation actually translates into a graph. So I did a Google search and found a video: "How to Graph a Quadratic Function Without a Calculator"! BINGO!!
Our first step is to find the axis of symmetry. Next, we find the vertex. Once we know the vertex, we can fill in the table. Once we've got the table filled in, we can plot the points of the parabola on the graph.
To find the axis of symmetry, we use the equation x = -b/2a. The variables here correspond with the variable names that we assign to a quadratic equation in the quadratic formula.
Once we have found the x-axis of symmetry, we find our vertex point by identifying the y value. We do this by inserting the value of x back into our quadratic function and adding everything together. Interestingly, while doing so, I found that if ax² is a negative value, even if the value of x is negative, we should end up with a negative value after squaring; so we calculate it as -(ax²). We now have our vertex: the intersection of the x- and y- values.
To get the rest of the points of the parabola, we start filling in the rest of our table by solving for the other values of x inside of our quadratic function, then plotting those on the graph.
That was a super helpful lesson; I'm feeling much more comfortable with navigating quadratic functions now!