• Physics Graphs Basics 2 – Nature of Gradients

The gradient (slope) of a graph tells how the y-quantity on the graph changes with respective to the x-quantity.

Gradient = Δy / Δx

Only 2 points are required to calculate the gradient.

The gradient of a graph may either be

• Constant – this would correspond to a straight line. The gradient may be calculated by considering 2 points on the line.
• Changing – this would correspond to a curve (non-straight line). The gradient at a point on the curve may be obtained by first drawing a tangent (a straight line that touches only that point on the curve) to the curve at that point. The gradient is then calculated by considering 2 points on the tangent on the cure at that point.

Consider the points 1 (x₁, y₁) and 2 (x₂, y₂)

Gradient = (y₂ – y₁) / (x₂ – x₁)

Note that the gradient for any line may also be calculated mathematically by differentiating the function describing the line. At any point, the gradient is obtained by evaluating the differentiated function at the coordinates of the point. But we won’t be considering this here.

Though the gradient at a specific point is more usually required, it is also important to know how the gradient of a graph changes as the x- and y-coordinates change. In physics, this allows us to better understand how the situation involved is changing (or not changing).

So, below, we will look at the general shapes of possible graphs and identify the information they give about the nature of the gradient.

Graphs with constant gradient

Let’s first consider straight lines where the gradients are constant.

Consider graph (a),

Note that x_B > x_A and y_A = y_B.

Either Gradient = Δy / Δx = (y_B – y_A) / (x_B – x_A) = 0 / (x_B – x_A) = 0

Or Gradient = Δy / Δx = (y_A – y_B) / (x_A – x_B) = 0 / (x_A – x_B) = 0

Gradient of graph (a) is ZERO

Consider graph (b),

Note that x_C = x_D and y_C > y_D.

Gradient = Δy / Δx = Δy / 0 = ± ∞ (depending on how the value are taken, it can be either + or –)

Gradient of graph (b) is INFINITE.

Consider graph (c),

Note that x_E > x_F and y_E > y_F.

Either Gradient = Δy / Δx = (y_E – y_F) / (x_E – x_F) = (+ve value) / (+ve value) = +ve

Or Gradient = Δy / Δx = (y_F – y_E) / (x_F – x_E) = (-ve value) / (-ve value) = +ve

Gradient of graph (c) is POSITIVE.

Consider graph (d),

Note that x_H > x_G and y_G > y_H.

Either Gradient = Δy / Δx = (y_G – y_H) / (x_G – x_H) = (+ve value) / (-ve value) = -ve

Or Gradient = Δy / Δx = (y_H – y_G) / (x_H – x_G) = (-ve value) / (+ve value) = -ve

Gradient of graph (d) is NEGATIVE.

The above are the 4 types of (straight lines) graphs with constant gradients.

Now, let’s consider 2 straight lines of the same type in graph (c). Which one of them will have a greater positive value?

There is a simple way to determine which of the 2 graphs above will have a greater value of gradient.

First consider the graphs (a) and (b). It is seen that a horizontal line has a gradient of zero while a vertical line has a gradient of infinity (largest value). So, these 2 lines can served as references for other straight lines.

If a line is closer to a horizontal line than to a vertical, then its gradient is closer to zero – its gradient has a small value. Similarly, if a line is closer to a vertical line, then its gradient is closer to infinity – its gradient has a larger value.

Thus, it can be determined that the gradient of graph (e) is larger than that of graph (f).

The same logic applies to graphs of the same type as graph (d), except that now, the value would have a negative sign.

So, it can be determined that the value of the gradient of graph (g) [which is closer to negative infinity] is more negative than that of graph (h) [which is closer to zero but is also negative].

Graphs with changing gradients

Now, let’s look at graphs with changing gradients – i.e. they are not straight lines. Of course there is an infinite possibility of such curves, but we will look at those with general regular shapes – the others may be deduced from them.

As stated before, the gradient at a point on a curve is found by calculating the gradient of the tangent at that point. A tangent a point on a curve is a straight line that touches only that point on the curve, and no other points on the curve.

So, basically, we will need to examine the gradients of straight lines (tangents) at different points on the curve. These would correspond to the different possibilities, already explained above, for straight lines. 

Consider graph (i),

To understand how the gradient changes, we need to consider the gradients of the tangents at different points on the curve.

Consider the 3 ‘tangents’ drawn. From what have been explained for straight lines, the gradient of the ‘blue’ tangent has a large negative value. The gradient of the ‘red’ tangent is less than the ‘red’ one but greater than the ‘green’ tangent. The gradient of all the tangents drawn are negative.

So, the value of the gradient of the curve keeps on decreasing as the value of the x-axis increases while the sign is always negative.  {If we consider both the value and the sign, some may say that the gradient is increasing. But let’s take it as have been explained first, that is, gradient decreases while the sign is always negative.}

Gradient of graph (i) DECREASES (from a large negative value towards zero)

Consider graph (j),

The gradients of all the tangents drawn are negative. The values of the gradients are as follows:

Gradient of ‘blue’ tangent < Gradient of ‘red’ tangent < Gradient of ‘green’ tangent

Gradient of graph (j) INCREASES (from zero towards larger negative values)

Consider graph (k),

The gradients of all the tangents drawn are positive. The values of the gradients are as follows:

Gradient of ‘blue’ tangent < Gradient of ‘red’ tangent < Gradient of ‘green’ tangent

Gradient of graph (k) INCREASES (from zero towards larger positive values)

Consider graph (l),

The gradients of all the tangents drawn are positive. The values of the gradients are as follows:

Gradient of ‘blue’ tangent > Gradient of ‘red’ tangent > Gradient of ‘green’ tangent

Gradient of graph (l) DECREASES (from a large positive value towards zero)

   
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