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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 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.
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) |
GC interprets some statements in ways having nothing to do with highlighting points in the plane.
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.
(a) | (b) |
(c) | (d) |
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