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Stage 4 - Stage 5

# Choosing the correct function model

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
• Cubic
• Exponential
• Logarithmic
• Reciprocal
• Trigonometric

And how should we be able to recognise them?

• In equation form
• In table/numerical form
• In graphical form

## Recognising a Function from its Equation

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

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 $\ln$ln or $\log$login 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 $e^x$ex.

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=mx+c or the general form of $ax+by=d$ax+by=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 $xy=a$xy=a.

Quadratic functions come in the following three forms:

• general form $y=ax^2+bx+c$y=ax2+bx+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=ax3+bx2+cx+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

## Recognising a function from its graph

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.

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.

## Recognising a Function from a table of values

Linear Functions

Exponential Function

Reciprocal Function (Hyperbola)

#### Worked Examples

##### QUESTION 1

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

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

A

quadratic, $f\left(x\right)=ax^2+bx+c$f(x)=ax2+bx+c, $a>0$a>0

B

logarithmic, $f\left(x\right)=a\log x+c$f(x)=alogx+c

C

quadratic, $f\left(x\right)=ax^2+bx+c$f(x)=ax2+bx+c, $a<0$a<0

D

##### QUESTION 2

Consider the given graph.

1. Which function does this graph represent?

exponential

A

cubic

B

C

linear

D

hyperbola

E
2. Fill 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.

3. Using the result of the previous part, determine the equation of the cubic function that has been graphed.

##### QUESTION 3

Consider the function that has been graphed.

1. Which function does this graph represent?

Reciprocal

A

Exponential

B

Logarithmic

C

Cubic

D
2. Determine the equation of the base $10$10 logarithmic function that has been graphed.