Identifying Curves and Equations

What is a relation?

A relation is a relationship between sets of information. For example, think of the names of the people in your class and their heights. If I gave you a height (e.g. $162$162 cm), you could tell me all the names of the people who are this tall and there may be more than one person. Let's say someone came to your class looking for the person who was $162$162 cm tall, that description might fit four people! There's not one clear answer. This data could be expressed as a relation.

What is a function?

A function is a special type of relation, where each input only has one output. It is often written in the form $f(x)$f(x), where $x$x is the value you give it. For example, say we have the expression $f(x)=2x$f(x)=2x. Let's construct a table of values to record the results:

See how each $x$x value gives a unique $f(x)$f(x) value? This is what makes this data a function.

Distinguishing between relations and functions

The Vertical Line Test

If you can draw a line that crosses the graph in more than one place, then the relation is not a function. 

Here is an example of a relation that is not a function. 

If you can draw a vertical line that only crosses the graph in one place, then it is a function.

Here is one example of a function.

Here is another function.

 

Distinguishing without graphing

If you can write a relationship between $x$x and $y$y then we can see that there is a relation. However, if this relationship only yields one value of $y$y for each $x$x value, then it is a function.

 

There are a number of different types of functions and relations you need to be able to recognise, both in graph form and in function form.  

Linear

Graph

In a graph, you can recognise a linear relationship as a straight line.

Linear functions can be increasing (positive gradient), decreasing (negative gradient) or horizontal (zero gradient).

Equation

The equation of a linear function normally takes one of $2$2 forms.

$y=mx+b$y=mx+b   (gradient intercept form)

$ay+bx+c=0$ay+bx+c=0     (general form)

Of course it could also be presented in another way.

Linear equations have the highest power of $x$x, equal to $1$1. 

Quadratics

Graph

In a graph you can recognise a quadratic relationship by looking for the characteristic parabolic shape.  

Quadratics can be positive (concave up, like a smiling mouth), or negative (concave down, like a sad mouth).

They can also be on above, below or crossing the x-axis. Every quadratic will have a y-intercept.

Equation

The equation for a quadratic function can be identified by checking that the largest power of the $x$x variable is $2$2.  ($x^2$x2) 

Some common forms of the quadratic are:

Simple form:  $f(x)=ax^2$f(x)=ax2

General form:  $f(x)=ax^2+bx+c$f(x)=ax2+bx+c

Turning point form: $f(x)=a\left(x-h\right)^2+k$f(x)=a(x−h)2+k

Fully factorised form:  $f(x)=\left(ax+b\right)\left(cx+d\right)$f(x)=(ax+b)(cx+d)

Cubics

Graph

These ones are positive cubics.  They are all increasing at the end values. In formal mathematics we say that it has increasing extrema behaviour.

These ones are negative cubics.  They are all decreasing at the end values. In formal mathematics we say that it has decreasing extrema behaviour.

Equation

The equation for a cubic curve can be identified by checking that the largest power of the $x$x variable is $3$3.  ($x^3$x3)

Some common forms of the cubic are:

Simple form: $f(x)=ax^3$f(x)=ax3

General form:  $f(x)=ax^3+bx^2+cx+d$f(x)=ax3+bx2+cx+d

h,k form: $f(x)=a\left(x-h\right)^3+k$f(x)=a(x−h)3+k

Fully factorised form: $f(x)=\left(x+a\right)\left(x+b\right)\left(x+c\right)$f(x)=(x+a)(x+b)(x+c)

Hyperbolas

Graph

Hyperbolas have $2$2 parts and $2$2 asymptotes.  The asymptote is a line that the curve gets really close to, but never touches or crosses it.

Positive hyperbolas exist in the Q1 and Q3 quadrants as formed by the asymptotes.   

Negative hyperbolas exist in the Q2 and Q4 quadrants as formed by the asymptotes.   

Equation

Hyperbolas are a special function that have the simple form: $f(x)=\frac{1}{x}$f(x)=1x​

We can vary the position of the hyperbola by applying transformations (you will learn formally about transformations later).

Another forms of hyperbolic equation is h,k form:  $f(x)=\frac{a}{x-h}+k$f(x)=ax−h​+k

Exponential

Graph

Exponential graphs look like this: 

They have one part, unlike the hyperbola which is made up of $2$2 parts.  

There is one horizontal asymptote that the function either lives entirely below or above of.  

Equation

At this stage we don't need to do too much to exponential equations, apart from recognising them in a simple form.

An exponential equation is when the x variable occurs in the index.

For example, $y=2^x$y=2x and $y=-4^x$y=−4x are examples of positive and negative exponentials respectively.  

Circles

Graph

The graph of a circle, is pretty easy to identify.  As long as we can recognise the shape of a circle!

A circle can have a centre at any coordinate point and any size radius.  

Equation

Circle equations are a little more complicated than the ones above, but in some forms they are easily identifiable.

$x^2+y^2=r^2$x2+y2=r2   is a circle with centre at the origin and radius of $r$r

$\left(x-h\right)^2+\left(y-k\right)^2=r^2$(x−h)2+(y−k)2=r2    is a circle with centre $\left(h,k\right)$(h,k) and radius $r$r.

Worked Examples

Question 1

Question 2

Question 3