# Section 3.7Understanding 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.

Reflection 3.7.1.

Type the following statements into GC. Press Enter, or click Graph, to display a graph. Press ctrl-Enter for a new line. Hide a statement's graph by clicking on its color bar and selecting X, like below, if the display becomes confusing.

After graphing a statement, explain why GC highlights the points it does. Keep in mind what "graph a statement" means to GC.

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

Here are reasons GC highlighted the points it did for each statement in Reflection 3.7.1. Compare your explanations 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) $x\lt 2y$ Highlight all the points in the plane having coordinates $(x,y)$ such that the x-coordinate of the point is less than twice 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. Enter statements in GC that cause it to highlight points as in each of a-d.
 (a) (b) (c) (d)
2. In each of a-g,
• Type the statement but DO NOT press Enter
• Explain how GC will interpret the statement
• Predict what GC will produce
• Press Enter
• Explain why GC produced what it did
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. $|x|\lt |y|$