Functions (Graphs and Behaviour)

Hong Kong

Stage 4 - Stage 5

Lesson

We need to be able to put all we know about functions together so that we can recognise any one of them when they're given to us in a mixed set.

Let's recap the types of functions we need to recognise.

- Linear
- Quadratic
- Cubic
- Exponential
- Logarithmic
- Reciprocal
- Trigonometric

And how should we be able to recognise them?

- In equation form
- In table/numerical form
- In graphical form

The trick here is to familiarise yourself with all the different ways a particular function can be written.

Let's start with the easiest to recognise functions first.

Trigonometric functions are the easiest to recognise from their equation because they will have sin, cos or tan in their equation!

Logarithmic functions are also very easy to recognise because they have $\ln$`l``n` or $\log$`l``o``g`in their equation.

Exponential functions are also quite easy because they have $x$`x` as the power. Even easier still is if it's a natural exponential of base $e$`e` because then you'll see $e^x$`e``x`.

Linear functions are the next easiest and that's because you're probably most familiar and confident with them. They will appear either in the gradient/intercept form of $y=mx+c$`y`=`m``x`+`c` or the general form of $ax+by=d$`a``x`+`b``y`=`d`

Reciprocal or Hyperbola functions are quite easy because they are identified as having x in the denominator and appear as $y=\frac{a}{\left(x+h\right)}+v$`y`=`a`(`x`+`h`)+`v`. Sometimes it can be more tricky when they are written as $xy=a$`x``y`=`a`.

Quadratic functions come in the following three forms:

- general form $y=ax^2+bx+c$
`y`=`a``x`2+`b``x`+`c` - factor form $y=a(x+d)(x+e)$
`y`=`a`(`x`+`d`)(`x`+`e`) - turning point or vertex form $y=a(x+h)^2+v$
`y`=`a`(`x`+`h`)2+`v`

Cubic functions come in the following four forms:

- general form $y=ax^3+bx^2+cx+d$
`y`=`a``x`3+`b``x`2+`c``x`+`d` - factor form $y=a(x+e)(x+f)(x+g)$
`y`=`a`(`x`+`e`)(`x`+`f`)(`x`+`g`) - repeated root form $y=a(x+e)^2(x+f)$
`y`=`a`(`x`+`e`)2(`x`+`f`) - point of inflection form $y=a(x+h)^3+v$
`y`=`a`(`x`+`h`)3+`v`

Let's remind ourselves what each of our functions generally look like.

Linear Function

Here you can review how to find the equation of a linear function from the graph.

Quadratic Function

Here you can review how to find the equation of a quadratic function from the graph in factored form or turning point form.

Cubic Function

Here you can review how to find the equation of a cubic function from the graph in factored form, repeated root form or point of inflection form.

Exponential Function

Here you can review how to find the equation of an exponential function from the graph.

Logarithmic Function

Here you can review how to find the equation of a logarithmic function from the graph.

Reciprocal Function

The video below shows you how to find the equation of a reciprocal function from the graph.

Trigonometric Function

Here you can review how to find the equation of a trigonometric function from the graph for sine, cosine and tangent.

Linear Functions

Quadratic Functions

Exponential Function

Reciprocal Function (Hyperbola)

Which of the following functions is an appropriate model for the data shown on the graph?

Loading Graph...

linear, $f\left(x\right)=mx+b$

`f`(`x`)=`m``x`+`b`Aquadratic, $f\left(x\right)=ax^2+bx+c$

`f`(`x`)=`a``x`2+`b``x`+`c`, $a>0$`a`>0Blogarithmic, $f\left(x\right)=a\log x+c$

`f`(`x`)=`a``l``o``g``x`+`c`Cquadratic, $f\left(x\right)=ax^2+bx+c$

`f`(`x`)=`a``x`2+`b``x`+`c`, $a<0$`a`<0D

Consider the given graph.

Loading Graph...

Which function does this graph represent?

exponential

Acubic

Bquadratic

Clinear

Dhyperbola

EFill in the gaps:

A cubic function with point of inflection at $\left(h,k\right)$(

`h`,`k`) has a general equation of the form:$\editable{}=a\left(x-\editable{}\right)^3+\editable{}$=

`a`(`x`−)3+where $a$

`a`is a constant.Using the result of the previous part, determine the equation of the cubic function that has been graphed.

Consider the function that has been graphed.

Loading Graph...

Which function does this graph represent?

Reciprocal

AExponential

BLogarithmic

CCubic

DDetermine the equation of the base $10$10 logarithmic function that has been graphed.