Decoding Visually

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dots to pairs

The graph below is the graph of $y = m(x)$. One of the dots has been highlighted. This dot visually encodes a function pair. How do we decipher the pair represented by this dot?

$\blacktriangleright $ desmos graph

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We need the coordinates of this dot. To obtain these, we move perpendicularly from the dot to the axes.

alternate text: Cartesian plane. Horizontal axis labeled x. Vertical axis labeled y=m(x). Graph begins with a solid dot at (-3,-1.4), moves down to (-1.6, -4), then moves up to (1.6,2), then moves down to a solid dot at (4,-3.2). The dot on the graph at (-1,-3.5) is highlighted. There is a vertical line segment added to the graph, which connects the highlighted point to (-1,0). There is a horizontal line segment added to the graph, which connects the highlighted point to (0,-3.5).

$\blacktriangleright $ desmos graph

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From there we can identify the domain number, $-1$, and its range partner (function value), $-3.52$. We now know that $m(-1)=-3.5$.

According to the graph for $y = m(x)$ above, $m(2) = \answer [tolerance=0.3]{1.8}$

According to the graph for $y = m(x)$ above, $m(0)$ is positivenegative.

According to the graph for $y = m(x)$ above, $m(1.8)$ $<$$>$ $m(3.2)$.

Below is the graph for $k = T(w)$. Use this graph to answer the following questions.

alternate text: Cartesian plane. Horizontal axis labeled w. Vertical axis labeled k=T(w). The graph consists of two line segments. The first connects two solid dots at (-7,6) and (-1,-6). The second line segment connects a hollow dot at (3,-4) and a solid dot at (8,1).

$\blacktriangleright $ desmos graph

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$T(-5) = \answer [tolerance=0.3]{2}$
$T(-1) = \answer [tolerance=0.3]{-6}$
$T(4) = \answer [tolerance=0.3]{-3}$
$T(8) = \answer [tolerance=0.3]{1}$

Which of the expressions below best describes the domain of $T$?
$[-7,8]$ $[-6,6]$ $[-7,-1] \cup [3,8]$ $[-7,-1] \cup (3,8]$
Which of the expressions below best describes the range of $T$?
$[-6,6]$ $(-4, 1]$ $[-6,6] \cup (-4,1]$ $[-6,6] \cup [-4,1]$

Below is the graph for $z = w(t)$.

$\blacktriangleright $ desmos graph

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According to the graph, how many solutions are there to the equation $w(t) = 2$?
$0$ $1$ $2$ $4$
According to the graph, how many solutions are there to the equation $w(t) = 0$?
$0$ $1$ $2$ $4$
According to the graph, how many solutions are there to the equation $w(t) = -3$?
$0$ $1$ $2$ $4$

Well-Defined Functions

To be a function, our set of pairs must satisfy one rule:

Each domain number must be in exactly one pair.

Graphically, this means two dots on the graph of a function cannot be right above one another. A vertical line cannot intersect the graph of a funciton in two points.

Vertical Line Test

The curve with points whose coordinates satisfy $y=f(x)$ represents $y$ as a function of $x$ on a set $S$ if and only if the vertical line $x=a$ intersects the curve $y=f(x)$ at exactly one point for every $a \in S$. This is called the vertical line test.

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more examples can be found by following this link
More Examples of Function Graphs

2026-06-26 02:37:51