Curve Sketching

Summary

  • Gradient Intercept Form of an equation of a line helps plot the graph: y\quad =\quad mx\quad +\quad c
  • If we are to sketch a curve, we find the gradient by differentiating the equation to find the derivative.
  • An asymptote of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity.
  • Revise the rules for graphical transformations.

Firstly, let’s quickly recall from the article “coordinate geometry” , the equation of a straight line.

We studied 3 different forms in which an equation of a line can be written as:

1. Gradient Intercept Form

Equation of a line with gradients m and y intercept c is:

y\quad =\quad mx\quad +\quad c

2. Gradient Point Form

If a line with gradient m, passes through a point (x1, y1), then its equation is:

y\quad -\quad { y }_{ 1 }\quad =\quad m(x\quad -\quad { x }_{ 1 })

3. Two Point Form

Equation of a straight line passing through two points (x1 ,y1) and (x2, y2) is:

\frac { y\quad -\quad { y }_{ 1 } }{ { y }_{ 2 }\quad -\quad { y }_{ 1 } } \quad =\quad \frac { x\quad -\quad { x }_{ 1 } }{ { x }_{ 2 }\quad -\quad { x }_{ 1 } }

If we are given the gradient intercept form, we can find the gradient of the line and also the point c which is the point at which line intersects the y axis. If we are given these two values, we can easily draw a straight line on the graph by substituting different values of x in the equation and finding the corresponding values of y.

Example #1

Q. Show the line y\quad =\quad 2x\quad +\quad 5 on a graph.

Solution:

We know gradient m = 2 &

y intercept c = 5

Let’s take 5 random values of x and find the corresponding values of y.

x-2-1012
y13579

When x\quad =\quad -2

y\quad =\quad 2(-2)\quad +\quad 5\quad =\quad 1

When x\quad =\quad -1

y\quad =\quad 2(-1)\quad +\quad 5\quad =\quad 3

When x\quad =\quad 0

y\quad =\quad 2(0)\quad +\quad 5\quad =\quad 5

When x\quad =\quad 1

y\quad =\quad 2(1)\quad +\quad 5\quad =\quad 7

When x\quad =\quad 2

y\quad =\quad 2(2)\quad +\quad 5\quad =\quad 9

Now sketch the line y\quad =\quad 2x\quad +\quad 5

Moving on, it gets a little more complicated when we have to sketch curves. If we have an equation of a curve, we use the same method as above, following the steps below:

  1. First find the derivative of the equation which is the gradient of the curve.
  2. Substitute values of x in the equation and find the corresponding values of y.
  3. Find maximum or minimum points through double derivation.

Another important point to note is if an asymptote occurs. An asymptote of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity.

For example if we have a function f\left( x \right) \quad =\quad \frac { 1 }{ x\quad -\quad 2 } and when x = 2 then f(x) becomes \frac { 1 }{ 0 } which is equal to infinity and thus we get an asymptote at x = 2.

A very popular graph in which asymptotes exist is the graph of tan x.

Here we can observe an asymptote at x\quad =\quad \frac { \pi }{ 2 } and x\quad =\quad \frac { 3\pi }{ 2 } .

Recalling from the article “Functions”, the rules listed below are very important and must be remembered when sketching curves of different functions.

Any small change to the equation of a function could lead to a big change in its graph.

Graphical Transformations

Suppose we have a function y\quad =\quad f\left( x \right), some rules to remember when transforming graphs are:

  • The graph of y\quad =\quad f\left( x \right) \quad +\quad c where c is a constant, has the same shape of the function f(x) but is moved c units higher.
  • The graph of y\quad =\quad f\left( x \right) \quad -\quad c where c is a constant, has the same shape of the function f(x) but is moved c units lower.
  • The graph of y\quad =\quad f\left( x\quad +\quad k \right) , where k is a constant, has the same shape of the function but is moved k units to the left.
  • The graph of y\quad =\quad f\left( x\quad -\quad k \right) , where k is a constant, has the same shape of the function but is moved k units to the right.
  • The graph of y\quad =\quad f\left( -x \right) is simply a reflection of f(x) in the y axis.
  • The graph of y\quad =\quad -f\left( x \right) is a reflection of f(x) in the x axis.
  • The graph of y\quad =\quad af\left( x \right) is stretched by a scale factor a in the y axis.
  • The graph of y\quad =\quad f\left( ax \right) is stretched by a scale factor \frac { 1 }{ a } in the x axis.
Reference
  1. https://revisionmaths.com/advanced-level-maths-revision/pure-maths/geometry/curve-sketching
  2. https://www.intmath.com/trigonometric-graphs/4-graphs-tangent-cotangent-secant-cosecant.php