SOLUTION:

Question 204503:
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
# 1
Start with the given function

Replace each "x" with "x+h"

FOIL (ie expand)

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Move onto the given difference quotient.

Plug in and

Distribute

Combine like terms.

Factor out the GCF "h" from the numerator.

Cancel out the common terms.

Simplify

So when

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# 2

Take note that when it is less than zero. Because when , this means that we simply plug in to get: .

So

Also, when it is greater than zero. Since when , we just plug in to get: .

So

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# 3

Start with the given equation.

Add 3 to both sides.

Subtract from both sides.

Rearrange the terms.

Divide both sides by to isolate y.

Break up the fraction and simplify.

We can see that the equation has a slope and a y-intercept .

Now to find the slope of the perpendicular line, simply flip the slope to get . Now change the sign to get . So the perpendicular slope is .

Now let's use the point slope formula to find the equation of the perpendicular line by plugging in the slope and the coordinates of the given point

.

Start with the point slope formula

Plug in , , and

Rewrite as

Distribute

Multiply

Subtract 4 from both sides.

Combine like terms.

So the equation of the line perpendicular to that goes through the point

is .

Here's a graph to visually verify our answer:

Graph of the original equation (red) and the perpendicular line (green) through the point

.

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# 5

Start with the given equation

Group like terms.

Take half of the x-coefficient -8 to get -4. Square -4 to get 16. Add this value to both sides.

Take half of the y-coefficient -6 to get -3. Square -3 to get 9. Add this value to both sides.

Combine like terms.

Rewrite 25 as

Now the equation is in the form (which is a circle) where (h,k) is the center and "r" is the radius

In this case, , , and

So the center is (4,3) and the radius is 5 units.

Here's the graph: