Descriptive Line Graphs

Now that we have discussed displacement, velocity, and acceleration I want to look at one of the most common ways this information is displayed, in a line graph. There is a ton of information that we can gather from these graphs so this whole lesson is dedicated to just that.

First, let’s quickly talk about line graph anatomy so we can have a clear picture of what I am referring to as I go through the next couple of ideas There are three major components: axes, intercepts, a slope, and occasionally a trend line equation that describe a line graph as shown below.

The axes tell us what we are measuring and define the scale of measurement. While the slope defines how our data points relate to one another. Lastly, the intercepts define the value of one variable when the other is zero. For example, in a v versus t graph, the y-intercept tells us the initial velocity since at this point t=0.

A trendline equation sums up all of this information in a nice tidy fashion. Where y=mx+b m is the slope, b the y-intercept, and the x-intercept can be solved by setting y equal to zero.

Pick a Point

Most commonly we will be asked to pick out a specific point on a graph and line it up with the other axis in order to find the correct answer. For example, we might be asked: What power must our car generate 10 seconds into traveling in order to maintain its constant velocity at that time if it generates 500N of force?

When approaching these questions we are going to need to pick out a specific point and determine what other value aligns with the given information. For example, the above questions asks us to determine the power needed to maintain a constant velocity at t = 10 seconds. When we look at the 10-second mark on our graph it corresponds to a velocity of 20 m/s.

So long as we know that power is equal to force times velocity we can easily determine the power by multiplying the force in the question stem by 20 m/s which we determined from the graph. (I know we haven’t covered power yet so don’t sweat it if you didn’t know that equation!)

[latexpage]
\[P=Fv\]
\[P=500N \times 20\frac{m}{s} \to P=10,000W\]

Therefore our car will need to generate 10,000 Watts of power to maintain its current speed.

Area Within the Curve

Another important value to understand is the area within the curve (AWC), which dimensionally represents the product of the axes variables. For example, we might be asked what is the total displacement that our car underwent in the first 25 seconds of travel?

Since this graph is a velocity versus time graph the area will represent displacement.

[latexpage]
\[
v \times t=x
\]
or from a units perspective
\[\frac{m}{s} \times s= m
\]

Therefore the area within the curve represents displacement. In order to answer the question, we need to define the borders of our shape and then plug and chug.

Since the question asks for the distance traveled in the first twenty-five seconds the base of our shape extends from 0 to 25 seconds. From here we will outline any included axis and lines formed by the data itself.

Now we need to define the height of our shape. To do this drop down a line extending from the highest point of the shape to the bottom of the graph. Try and place this line where the shape changes height since this often defines several of the shapes of our larger shape as we will see.

Here we can see that there is a weird polygon, a rectangle, and triangle.

The area of weird polygons are generally pretty challenging to calculate so we need to break them into more standard shapes. To do this draw another line straight across at a velocity of 10.

Now all of our shapes are pretty standard and it should be pretty easy to calculate the overall area of this shape by summing all of the individual areas together.

Shapes To Know

This means that we need to know how to calculate the area of both shapes. While there are a vast array of geometric shapes that the graphs can form the MCAT tends to stick with rectangles and triangles.

[latexpage]

\[
Area_{Rectangle} = base \times height
\]
\[Area_{Triangle} = \frac{1}{2}base \times height
\]

Slope

Additionally, the slope of a line graph describes the quotient of the two axes and is determined by taking the y-axis variable and dividing it by the x-axis variable or rise over run. For example, we might be asked what is the acceleration of our car from five to ten seconds?

Since this is a velocity versus time graph the slope is equal to velocity over time or acceleration. If we want to know the acceleration from five to ten seconds then we would need to determine the slope of the line connecting these points by determining the change in the y-axis (velocity) and dividing it by the change in the x-axis (time).

Slopes To Know

The identity and overall shape of the slope are typically more important than the actual numerical value. With this in mind, you should be able to pick out and understand whether or not the slope is positive or negative and linear or exponential in addition to what this means for the dimension in question.

Linear graphs look like straight lines when they point upwards and to the right the slope is positive and when they point downwards and to the right their slope is negative. A linear slope demonstrates that the slope the variable represents doesn’t change over time. For example, if these graphs were velocity-time graphs the slope would represent acceleration. Furthermore, the acceleration would be constant because the slope doesn’t change throughout the graph.

Exponential slopes on the other hand are curved and represent a changing slope. In this set of graphs below the slope is positive because overall the line points up and to the right. Again if these were velocity-time graphs the slope would represent acceleration. Therefore the graph on the left-hand side shows that as time goes on the acceleration is increasing while on the right-hand graph the acceleration is slowing down as time progresses.

In contrast the graphs below while exponential have negative slopes. This means if these graphs are velocity-time graphs the slope represents negative acceleration or more plainly deceleration. On the left the deceleration is slowing as time goes on while on the right the deceleration is increasing rapidly.

Intercepts

Lastly, there are the intercepts. What these represent is very graph dependent and frankly they are sometimes completely unhelpful. To see how intercepts work I am going to step out of our physics content here for a moment and discuss chemistry. Take for example a graph displaying Gibb’s free energy (∆G) versus temperature (T).

Say we wanted to know at what temperature range a reaction will be spontaneous. To determine this from the graph alone we would need to know at what temperature is ∆G equal to zero and then determine whether increasing or decreasing the temperature makes ∆G negative since a negative ∆G value corresponds to a spontaneous reaction. If you don’t know this yet no big deal we will cover all of this material later.

∆G will be equal to zero at the x-intercept so the x-intercept value is the temperature at which ∆G is equal to zero. Therefore we can define temperature values above this number as non-spontaneous and temperature values below this as spontaneous. We would go through the same process for the y-intercept except it wouldn’t tell us much beyond the ∆G value of the reaction at absolute zero.

There is a lot to digest here a so so many ways the MCAT can present reasoning based questions about these graphs. Just make sure to think back to these principles as invariably one of them will apply and allow you to get to the correct answer.