Generalizing "for" Lists and Intervals

If you want to graph three points along the line \(x=1\), you could individually plot \((1,1)\), \((2,1\)), and \((3,1)\). ‘For’ gives a more succinct option within a single expression line. Using ‘for,’ graph the same three points by typing \((a,1)\) for \(a=[1,2,3]\). Changing the point to \((a,a)\) for \(a=[1,2,3]\) will plot three points along a line: \((1,1)\), \((2,2)\), and \((3,3)\).

Wondering how to graph a 3x3 grid using ‘for’? To graph a grid, try \((a,b)\) for \(a=[1,2,3], b=[1,2,3]\). In this case, ‘for’ uses the Cartesian product, generating every possible ordered pair where the \(x\)-value is from list \(a\) and the \(y\)-value is from list \(b\).

Try it! Open the graph by clicking on the image and try it out. How would you create a 4x3 grid using the lists \(a\) and \(b\)?

With lists, you can plot points closer and closer together by reducing the interval between the numbers.

Using the example from above, if you want more than three points to appear, shorten the distance between the numbers in the list. For example, graphing \((a,a)\) for \(a=[1,1.1…3]\) will plot 21 points of the form \((a,a)\) that are 0.1 units apart. To plot even closer points, change the list to something like \(a=[1,1.01…3]\), which would display 201 points of the form \((a,a)\) that are 0.01 units apart.

You can keep reducing the distance to create the appearance of a solid line. Or, you can graph a line using ‘for’ and an interval. For example, \((a,a)\) for \(1\le a\le3\) will graph a line segment of all the points of the form \((a,a)\) between \(1\) and \(3\). To graph the full line, type \((a,a)\) for \(-\infty\le a\le\infty\). This will show all the points that satisfy the condition where the \(x\)-value is equal to the \(y\)-value.

Try it! Open the graph by clicking on the image and try it out. What other lines can you graph using 'for'?

Let's reason about the path that the vertex of a parabola in standard form would follow. Starting with \(f(x)=ax^2+bx+c\), our parabola will have a vertex \(V=(-\frac{b}{2a},f(-\frac{b}{2a}))\).

We'll start by looking at how the value of \(b\) affects our parabola. In the example, you'll see sliders at \(a=1\), \(b=0\), and \(c=1\).

We can start by choosing values we want to test for \(b\). For example, we can set \(b=[-5...5]\), an 11 element list with all of the integers from \(-5\) to \(5\). Then, if we type \(V\) for \(b=[-5...5]\), we will see the 11 different places the vertex would be for each of those \(b\) values.

One way to start building up an intuition for parametric equations is to visualize the path over a list of discrete points before visualizing that same path as a continuous curve within an interval. For example, the parametric equation of a unit circle is \((\cos(t), \sin(t))\). While the special parameter t can be used to graph this parametric curve, the interval can also be defined inline using any parameter to represent the circumference of a circle.

Starting with a list of values \(c= [0, \frac{\pi}{4}…2\pi]\), plot \((\cos(c), \sin(c))\) for \(c = [0, \frac{\pi}{4}…2\pi]\) to visualize 8 equally spaced points around the unit circle.

With lists, plot points closer and closer together along the curve by increasing the number of elements in the list. To take this one step further, graph a continuous curve with an interval. Try typing \((\cos(c), \sin(c))\) for \(0 \le c \le 2\pi\) to graph the continuous unit circle.