Category Archives: Sullivan Chapter 3

Difference Quotient: Quadratic Function

Example:  Find the difference quotient for f(x)=x^2-9x

The Difference Quotient:{f(x+h)-f(x)}/h

Solution:

The Difference Quotient Formula
{f(x+h)-f(x)}/h
Write the difference quotient for the given function
={(x+h)^2-9(x+h)-(x^2-9x)}/h
Apply the exponent and use the distributive property
={(x+h)(x+h)-9x-9h-x^2+9x}/h
Multiply
={x^2+xh+xh+h^2-9x-9h-x^2+9x}/h
Combine the like terms
={x^2+2xh+h^2-9x-9h-x^2+9x}/h
Combine the like terms.  Only terms with h should remain
={2xh+h^2-9h}/h
Divide h into each term
={2xh}/h+{h^2}/h-{9h}/h
Cancel the common h from each term
=2x+h-9

Difference Quotient: Rational Function

Example:  Find the difference quotient for f(x)={4x}/{x+5}

The Difference Quotient:{f(x+h)-f(x)}/h

Solution:

The Difference Quotient Formula
{f(x+h)-f(x)}/h
Write the difference quotient for the given function
={{4(x+h)}/{(x+h)+5}-{4x}/{x+5}}/h
Use the distributive property
={{4x+4h}/{x+h+5}-{4x}/{x+5}}/h
Simplify the complex fraction by multiplying the numerator and denominator by the common denominator
={{4x+4h}/{x+h+5}-{4x}/{x+5}}/h {{(x+5)(x+h+5)}/1}/ {{(x+5)(x+h+5)}/1}
Distribute the common denominator to each fraction in the numerator.
={{(4x+4h)(x+5)(x+h+5)}/{x+h+5}-{4x(x+5)(x+h+5)}/{x+5}} /{h(x+5)(x+h+5)}
Cancel the common factor
={{(4x+4h)(x+5)}-{4x(x+h+5)}} /{h(x+5)(x+h+5)}
 Multiply the expression in the numerator
={4x^2+20x+4xh+20h-4x^2-4xh-20x} /{h(x+5)(x+h+5)}
 Combine like terms
={20h} /{h(x+5)(x+h+5)}
 Cancel a common h from the numerator and denominator
={20} /{(x+5)(x+h+5)}

 

Difference Quotient: Rational Function

Example:  Find the difference quotient for f(x)=1/x^2

The Difference Quotient:{f(x+h)-f(x)}/h

Solution:

The Difference Quotient Formula
{f(x+h)-f(x)}/h
Write the difference quotient for the given function
={1/(x+h)^2-1/x^2}/h
Simplify the complex fraction by multiplying the numerator and denominator by the common denominator
={1/(x+h)^2-1/x^2}/h {{x^2(x+h)^2}/1}/{{x^2(x+h)^2}/1}
Distribute the common denominator to each fraction in the numerator.
={1/(x+h)^2-1/x^2}/h {{x^2(x+h)^2}/1}/{{x^2(x+h)^2}/1}
={{x^2(x+h)^2}/(x+h)^2-{x^2(x+h)^2}/x^2}/{h{x^2(x+h)^2}}
Cancel the common factor
={{x^2(x+h)^2}/(x+h)^2-{x^2(x+h)^2}/x^2}/{h{x^2(x+h)^2}}
={x^2-(x+h)^2}/{h{x^2(x+h)^2}}
Simplify by squaring the binomial
={x^2-(x+h)^2}/{h{x^2(x+h)^2}}
={x^2-(x+h)(x+h)}/{h{x^2(x+h)^2}}
={x^2-(x^2+xh+xh+h^2)}/{h{x^2(x+h)^2}}
Simplify by combining like terms and distributing the negative
={x^2-(x^2+2xh+h^2)}/{hx^2(x+h)^2}
={x^2-x^2-2xh-h^2}/{hx^2(x+h)^2}
 Combine like terms
={-2xh-h^2}/{hx^2(x+h)^2}
 Cancel a common h from the numerator and denominator
={-2x-h}/{x^2(x+h)^2}

The difference quotient for f(x)=1/x^2 is {-2x-h}/{x^2(x+h)^2}

Graphing an Exponential Equation by Transformations

Example: For the function below.  Graph using transformations.  Find the y-intercept.  State the horizontal asymptote and the domain and range.

f(x)=-2^{x+2}+2

First we must examine the base function y=2^x

Graph using plotting points.  We can use the standard set of x-values to find ordered pairs.

xy
-22^(-2)=1/4
-12^(-1)=1/2
02^0=1
12^1=2
22^2=4

The graph below shows the points plotted and the line that connects them.  This graph has a horizontal asymptote at y=0.  The domain is (- infty, infty)  and the range is (0, infty)

 

Analyze the transformations.

f(x)=-2^{x+2}+2

The +2 in the exponent shifts the graph left 2 units.

The – in the front of the base reflects the graph over the x-axis.

The +2 next to the base shifts the graph and the horizontal asymptote up two units.

You can see the graph after the transformations.

The horizontal asymptote is y=2.  The domain is (- infty, infty)  and the range is (- infty, 2)

To find the y-intercept we let x=0.

f(0)=-2^{0+2}+2

f(0)=-(2^{2})+2

f(0)=-(4)+2

f(0)=-4+2

f(0)=-2

Thus the y-intercept is (0,-2)