# 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 =**

**Δ**

**y /**

**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 (x

_{1}, y_{1}) and 2 (x_{2}, y_{2})**Gradient =**

**(y**

_{2}– y_{1}) / (x_{2}– x_{1})
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.

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.

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Previous: Physics Graphs Basics 1 - Coordinates of Points

__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.

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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Previous: Physics Graphs Basics 1 - Coordinates of Points

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