Difference Quotient

We have discussed functions and relations on our page Functions and Relations, now we will discuss value of function for given x. 

To calculate value of given function at x=a, we plug in a for x in the expression. 

And then simplify the expression to get the value.
For example, if we have $f(x)=2x^2+3x+5$

Let's calculate value of function f(x) at x=2. 

We have $f(2)=2\cdot 2^2+3\cdot2+5=8+6+5=19$

Similarly, we can plug any value for x, even we can substitute any algebraic expression for x in the expression.
If we have $f(x)=2x^2+5x-7$
Let's calculate $x=2+h$, we get 

$f(x)=2x^2+5x-7$

$f(2+h)=2(2+h)^2+5(2+h)-7$

         $= 2(4+4h+h^2)+10+5h-7$

         $=8+8h+2h^2+10+5h-7$

Adding like terms, we get
$f(2+h)=2h^2+13h+11$

and $f(2)=(2\cdot 2^2+5\cdot 2-7)=11$

And we can calculate, $f(2+h)-f(2)$

$f(2+h)-f(2)\\=(2h^2+13h+11)-11\\=2h^2+13h+11-11\\=2h^2+13h$

Now we can divide it by h, and we get 

$\dfrac{f(2+h)-f(2)}{h}=\dfrac{2h^2+13h}{h}=2h+13$

$\dfrac{f(2+h)-f(2)}{h}=2h+13$

This expression is called Difference Quotient. 

Difference Quotient is helpful in finding and understanding a derivative at a point.
It's used in limit and derivative topics extensively. 

Try these problems, find the difference quotient for each problem.
1. $f(x)=5x^2+2x-9$  at x=3

2. $f(x)=3x^2-2x+9$ at x=2

3. $f(x)=x^2-2x+3$  at x=1

Sometimes, we say that difference quotient is the slope of tangent to the curve of f(x) at x=a where a be any real number. 

Slope of tangent to any curve f(x) at x=a  is $slope=\dfrac{f(a+h)-f(a)}{h}$. 

Then, we can obtain equation of tangent using point slope form as  $y=m(x-a)+f(a)$ and here point is (a, f(a)) and slope m is the difference quotient and $m=\dfrac{f(a+h)-f(a)}{h}$