# What are examples of using graphs to help solve word problems?

Jan 31, 2015

Here is a simple example of a word problem where graph helps.

From a point $A$ on a road at time $t = 0$ one car started a movement with a speed $s = U$ measured in some units of length per unit of time (say, meters per second).

Later on, at time $t = T$ (using the same time units as before, like seconds) another car started moving in the same direction along the same road with a speed $s = V$ (measured in the same units, say, meters per second).

At what time the second car catches on with the first, that is both will be on the same distance from point $A$?

Solution

It makes sense to define a function that represents a dependency of the distance $y$ covered by each car from time $t$.

The first car started at $t = 0$ and moved with a constant speed $s = U$. Therefore, for this car the linear equation expressing this dependency looks like $y \left(t\right) = U \cdot t$.

The second car started later by $T$ units of time. So, for the first $T$ units it covered no distance, so $y \left(t\right) = 0$ for $t \le T$. Then it starts moving with a speed $V$, so it's equation of movement will be $y \left(t\right) = V \cdot \left(t - T\right)$ for $t > T$. In this case a function is defined by two different formulas on two different segments of the argument $t$ (time).

Algebraically, the solution to this problem can be found by solving an equation
$U \cdot t = V \cdot \left(t - T\right)$
that results in
$t = \frac{V \cdot T}{V - U}$

Obviously, $V$ should be greater than $U$ (otherwise, the second car would never catch up with the first).
Let's use concrete numbers:
$U = 1$
$V = 3$
$T = 2$
Then the solution is:
$t = \frac{3 \cdot 2}{3 - 1} = 3$

If we are not so well versed in Algebra and equations to construct the equation above, we can use graphs of these two functions to visualize the problem.
The graph of a function $y \left(t\right) = 1 \cdot t$ looks like this:
graph{x [-1, 10, -1, 10]}
The graph of a function $y \left(t\right) = 0$ if $t \le 2$ and $y \left(t\right) = 3 \cdot \left(t - 2\right)$ if $t > 2$ looks like this:
graph{1.5x+|1.5x-3|-3 [-1, 10, -1, 10]}
If we draw both graphs on the same coordinate plane, the point they intersect (looks like $t = 3$ when both functions equal to $3$) would be the time both cars are at the same location. This corresponds to our algebraic solution $t = 3$.

In this and many other cases the graph might not provide an exact solution, but it helps a lot to understand the reality behind a problem.
Moreover, graphical representation of a problem would help to find a precise analytical approach to exact solution. In the example above this process of intersecting two graphs gives a strong hint to an equation used to algebraically solve the problem.