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Section 3.7
Understanding GC's DNA

Graphing Calculator (GC) has graphing in its DNA. (Hence, its name!)

Any statement you type having an x or y in it is a signal to GC to graph the statement in the x-y coordinate system (with exceptions described below).

However, "graph the statement" has a special meaning in GC.

To GC, "graph a statement" in its coordinate plane means to highlight all points whose coordinates make the statement true.

Type the following into GC. Press ctrl-Enter for a new line. Press Enter, or click Graph, to display a graph.

Think about why GC highlights the points it does before reading the explanations below. Keep in mind what "graph a statement" means to GC.

$x=2$
$y=4$
$y=x^2$
$x=y^2$
$x\lt y$
$-4\lt x \lt 4,\,1\lt y \lt 3$
$2x+1\gt x^2$
$2x+1\gt y \gt x^2$

Hide a statement's graph by clicking on its color bar and selecting X, like below, if the display becomes confusing.

Here are reasons GC highlighted the points it did for each statement. Compare your reasons with these.

You typed: GC Interpreted Your Statement To Mean:
$x=2$ Highlight all the points in the plane with coordinates (x,y) that have an x-coordinate of 2. Those points will be $(2,y)$ for every value of y. Think of the value of y varying while $x=2$. (see video)
$y=4$ Highlight all the points in the plane with coordinates $(x,y)$ that have a y-coordinate of 4. Those points will be $(x,4)$ for every value of x. Think of the value of x varying while $y=4$. (see video)
$y=x^2$ Highlight all the points in the plane with coordinates $(x,y)$ such that the y-coordinate is the same as the square of x. Those points will be $(x,x^2)$ for every value of x. Think of the value of x varying. (see video)
$x=y^2$ Highlight all the points in the plane with coordinates $(x,y)$ so that the x-coordinate is the same as $y^2$. Those points will be $(y^2,y)$ for every value of y. That is, the horizontal coordinate in every highlighted point will be $y^2$ when the vertical coordinate is y. Think of the value of y varying. (see video)
$x\lt y$ Highlight all the points in the plane having coordinates $(x,y)$ such that the x-coordinate of the point is less than the y-coordinate of the point. Think of the value of x varying. (see video)
$-4\lt x \lt 4,\,1\lt y \lt 3$ Highlight all the points in the plane having coordinates $(x,y)$ such that the highlighted point's x-coordinate is between -4 and 4 AND the highlighted point's y-coordinate is between 1 and 3 (see video).
$2x+1\gt x^2$ Highlight all the points in the plane having coordinates $(x,y)$ such that $2x+1$ is larger than $x^2$. Notice! There is no restriction on y, so every point having an x-coordinate such that $2x+1>x^2$ will be highlighted. Think of the value of x varying. (see video)
$2x+1\gt y \gt x^2$ Highlight all the points in the plane having coordinates $(x,y)$ such that $2x+1$ is larger than $x^2$ AND the the point's y-coordinate is between $2x+1$ and $x^2$. Notice! There is a restriction on y, so only points having an x-coordinate such that $2x+1>x^2$ AND a y-coordinate between $2x+1$ and $x^2$ will be highlighted. Think of the value of x varying. (see video)

Exceptions to GC's Compulsion to Highlight Points

GC interprets some statements in ways having nothing to do with highlighting points in the plane.

Examples

You type: GC Will Interpret Your Statement To Mean:
$a=4$ Whenever I see the letter a, substitute the number 4.
$f(s)=3s^2-\cos(2s)+1$ Okay. The definition of the function f is $f(s)=3s^2-\cos(2s)+1$. Whenever I see $f(\_\_)$, substitute what is inside the parentheses for the letter s in the definition of f and evaluate (compute a value). Remember, the values of variables vary!
$g(x)=e^{\cos(x)}-2$ Okay. The definition of the function g is $g(x)=e^{\cos(x)}-2$. Whenever I see $g(\_\_)$, substitute what is inside the parentheses for the letter x in the definition of g and evaluate (compute a value). The letter x in a function definition does NOT represent values on the x-axis. It is merely a placeholder, just like the letter s in the definition of f given above.

We will discuss the idea of function in Section 3.10 and expand the idea of function notation in Section 3.11.

Exercise Set 3.7

  1. In GC, enter statements that will highlight points as in each of a-d.
    (a) (b)
    (c) (d)
  2. In each of a-g,
    1. $x\lt \dfrac{y}{2}$
    2. $|y-3|=0.5$
    3. $x^2+y^2\lt 1$
    4. $0\lt y^2\lt x^3\lt 4$
    5. $|x|\lt y$
    6. $|x|\lt -y$
    7. $|x|\lt |y|$


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