I am self-teaching myself precalculus.My biggest sticking point is to move lines within a diagram, especially, functions in the coordinated plane. For instance, doing horizontal and vertical shifts of functions. Could there be a base line skill that I am lacking? Now I have two theories for this:

(1) As the diagram becomes more abstract we are no longer, "naming the parts of the diagram" but the parts are implied, like the book calls it graph of a function as opposed to the line representing the function. Therefore, I might get lost. (2) A lack of concentration to enable myself to move the line around, thereby, having difficulty manipulating the line around the coordinate plane. Maybe a specific spatial skill?

I am passionate about finding conceptual issues in learning and I really hope you guys can help me! Or point me out to literature that touches on this topic.

  • $\begingroup$ I am not completely sure I understand what you are asking, or if this question is on-topic for MESE. But if you have an equation for a particular graph, and you want to move the graph $a$ units to the right and $b$ units up, you can do this by replacing $x$ everywhere in the equation by $x-a$ and replacing $y$ everywhere with $y-b$. $\endgroup$ – mweiss Dec 3 '15 at 21:31

I'm not 100% sure I understand the question, but here's my hunch. The base skill to be learned in moving lines within a diagram is two-fold:

  1. Learning to the see the graph of a function in your head, based just on the equation. This is done be recognizing variables within an equation as points on a graph.

For example, in the slope-intercept form of a line y = mx + b, the m variable gives you a sense of how steep the line is, which direction it leans, and the y-intercept tells you where the line crosses the y-axis. You can get a good visualization of a line just from the equation.

In a parabola y = a(x-h)2 + k, the sign of the a variable tells you if the parabola opens up or down, and the vertex is at point (h, k).

In a circle, (x-h)2 + (y-k)2 = r2, you get a sense of the size of circle from the r variable, and the center is at point (h, k).

If you can look at an equation, and say (for example) "that's an ellipse," that's a good thing. If you can look at the equation and say "that's an ellipse with the center in the 4th quadrant and the major axis at about a 45 degree angle from the x-axis," then you will do well in class! You will have a feel for the properties of an equation, know how it behaves if one of the variables is perturbed.

  1. A 2nd reason to become efficient at moving lines is the idea of "translating coordinate systems." In physics and in programming, you may get an equation based on one coordinate system, but it's helpful to visualize a graph of the same shape, but in a coordinate system with a different origin. This is done more often than you realize.

In both of the above cases, if you can achieve fluency through studying theory alone, you're a better man than me. For me (and maybe most people) it's takes drilling. That's why your textbooks have you go through these exercises.

Congratulations on teaching yourself precalculus! I am awed and impressed!

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