Equation of a Circle: Where Is the Center and R Is the Radius

FORMULAS:

Equation of a Circle: where is the center and r is the radius.

Point-Slope Form of Equation of a Line: where is a point on the line and m is the slope.

Distance between two points is given by

And the Midpoint is

Solve the following equations. Complex answers must be written in the form of . (be able to identify type of equation and strategy for solving)

Add 27 to both sides.

Cube root both sides.

What is the solution set for this equation?x =

Subtract the 1 to get a zero on the right side.

Use the quadratic formula.

Do the work under the radical and reduce the radical.

Reduce all three whole numbers.

x = 1, -3

Square root both sides. Because it is an even root, you get plus/minus

The square root of 16 is 4, so you get

Now we have to solve for 4 and -4. We will solve for 4 first.

Square root both sides. Because it is an even root, you get plus/minus

This sets up two equations to solve. The positive and negative 2.

Subtract 1 from each equation.

Now we go back and solve for the -4.

Square root both sides. A negative under the radical, gives an i.

Subtract 1 from both sides.

x = 0, 3, -2

Zero is on the right already so factor the left. Start with GCF of .

Factor the trinomial by using values that add to be -1 and multiply to be -6

Set each piece equal to zero and solve.

Factor the equation just like you would a quadratic.

Set each parenthesis equal to 0.

Get rid of the 5 by subtracting or adding as necessary.

Get the by itself by dividing when necessary.

Square root both sides which gives plus/minus on the right.

Get rid of the by multiplying top and bottom by that . Since the second
root has a negative, pull out an i.

Since there are four parts to the left side, we can factor by grouping. To
do this, we factor the GCF out of the front two () and the GCF out of the second two (6).

Now I can GCF the entire left side. (x – 7) factors out of both which leaves the (). Each of these gets a parenthesis.

To get a zero, one of the parenthesis must be zero since they are multiplying. So I set each parenthesis piece equal to zero to solve.

Get the x by itself. In the first part, I subtract 6. In the second I add 7 which is a final answer for that part of the problem.

Square root both sides to get x by itself.

Since, I simplify that portion of the problem.

Now that each is solved for x, I have my answers.

Get zero on the right by adding 1.

Use the quad. formula.

Do the work under the radical and multiply the denominator.

The square root of 16 is 4. Because it is a -16, you also pull out an i.

Reduce the whole numbers by dividing them all by 2.

Factor the denominator so we can find the common denominator.

Get a common denom. by multiplying each fraction by what it is missing.

Now solve the numerator. The denom. gives domain restrictions.

Distribute through the parenthesis.

Get everything to one side.

Factor.

Solve the parenthesis for zero.

Make sure the answer does not fall within the restriction from the denom.

Domain: To get the domain, I know that the denominator can not be zero. So, I

solve for the equations x + 3 = 0 and x – 3 = 0.

3, but only 3 works D: or

Get the radical by itself by adding 3y and adding 10.

Square both sides to get rid of the radical.

Distribute (FOIL) the right side.

Get everything to one side by adding y and subtracting 4.

Substitute into the quadratic formula.

Work out part under the radical.

Store the value into the calculator to check which one works.

Domain: To get the domain, you can’t have a negative under an even

radical, so I set the radical greater than or equal to zero. Solve by subtracting the 4 and dividing by -1 (which flips the sign.)

or The first answer is in Set Notation, the second in Interval Notation.

Get the x by itself by subtracting 16.

Square root both sides to get x by itself. Square rooting gives plus/minus.

The square root of 16 is 4. Because it is a -16, you also pull out an i.

This won’t factor, so I use the quad. formula.

Solve for the radical and the denominator.

Since there is a negative under the radical, pull out an i.

This is the difference of squares, so I can factor it.

Now I set each part = 0 to solve.

Get rid of the 7 by adding or subtracting as necessary.

Divide by 3 to get the by itself.

Square root both sides to get x by itself.

Multiply top and bottom by to get rid of the radical in the

bottom. Since the first one had a negative under the radical, pull out an i.

Solve by completing the square

Divide by the a value to get a 1 in the front of the quadratic.

Send the final number to the other side by subtracting.

Take half of the middle number and put it in the squared parenthesis with

the x. Square that number and add it to the other side.

Add the numbers on the right.

Square root both sides. This will give a plus/minus to the right side.

Rationalize the denom. by multiplying top and bottom by the radical.

Get the x by itself by adding 1 to the other side. Combine.

Use the discriminate of the quadratic equation to find the type and number of solutions for

-3 so 2 complex (imaginary) solutions.

This is the part of the quadratic formula that is the discriminant

Substituting gives a -3. This means there are 2complexanswers. If you get a positive value, there are 2 rational solutions, a zero means one rational solution, and a negative means 2 complex or imaginary solutions.

Evaluate the following expressions and write the result in the form a + bi
(3 + 5i) – (2 – 6i)

Send the negative through the second parenthesis.

Combine like terms.

The problem can be rewritten as the parenthesis times itself.

FOIL the problem.

so substitute.

Combine like terms.

Multiple top and bottom by the conjugate (opposite) of the bottom.

FOIL the top and bottom.

so substitute.

Combine like terms. Separate the fraction and reduce.

Reduce and take the negative out of the radical by pulling out an i.

FOIL the problem.

so substitute and combine like terms.

If P(0, 1) and Q(2, 3) are two points
Find the distance between P and Q

Substitute into the distance formula.

Work out what is under the radical.

Reduce the radical.

Find the midpoint(1, 2)

Substitute into the midpoint formula and write the answer as a point.

Find the equation of the circle for which the segment PQ is a diameter

Substitute into the circle formula with the midpoint being the center

and half of the diameter being the radius. Solve for the radius.

Find and and 1

Divide 103 by 4 and get a remainder of 3. Likewise, divide 72 by 4 and

get a remainder of 0. Follow this pattern:

R0 = 1R1 = iR2 = -1R3 = -i

Find the center and radius of each circle and sketch the graph.
C: (2, -1) r: 2

The center comes from the opposite of the values in the parenthesis.

C: (2, -1) r: 2The radius comes from the square root of the final number.

Since the center is (2, -1), create a point there. The radius is 2, so create a
point two to the right, left, above, and below the center. Connect the points with a circle.

C: (-1, 3)r:

Put the x’s and the y’s together. Send the whole number to the other side.

Complete the square by taking half of the x and y value into the

parenthesis and adding the square of each to the opposite side.

C: (-1, -(-3)) The center is the opposite of the parenthesis. The radius is the square root

of the other side.

To graph, the center is (-1,3), put a point there. The radius is which is about 1.7. Create a point to the left, right, above and below the center 1.7 away from the center. Connect those points with a circle.

Find an equation of the circle that satisfies the given conditions: Center at and through (2, 0).

D = Since the circle goes through (2, 0), there is a radius from the center to this point. We must find the distance between the points to get r.

D = Solve the radical.

Now that we have the center and radius, substitute into the circle formula.

Square the 3.

Use the distance formula to prove that a triangle with vertices at (1, 2), (4, 2), and (1, 6) is a right triangle.

Distance from (1, 2) to (4, 2)First I find the distance between the first two points using the distance

formula which is . Solve for the distance.

Remember that when you square a negative it becomes positive.

Distance from (4, 2) to (1, 6)Find the distance between the next two points.

Distance from (1, 2) to (1, 6)Find the distance between the first and last points.

Hypotenuse: 5 Legs: 3, 4If it is a right triangle, the longest side will be the hypotenuse or c.

This is true Substitute it into the Pythagorean Theorem () and see if you

So this is a right triangle.Get a true statement. If you do, it is a right triangle.

Find an equation for the line with the given property.
It passes through the point P(-1, -2) and is parallel to the line

m = -2Solve for y to get it intoform. The slope is now -2.

Parallel lines have the same slope. Substitute into point slope formula.

Simplify the problem.

It has y-intercept 1 and is perpendicular to the line

Solve for y to get it into form. The slope is now.

Perpendicular lines have negative reciprocal slopes, so change the sign

and flip the fraction upside down. The y-intercept is 1, so substitute it into

b and the slope into m.

The following lines have slopes 5, 0, -3 and UD. Match each line to its slope value.

Slope of line A is ______0______A horizontal line has a slope of 0.

Slope of line B is ______UD______A vertical line has an undefined slope

Slope of line C is ______5______An increasing line has a positive slope.

Slope of line D is ______-3______A decreasing line has a negative slope.

A new car is worth $25,000 when it is purchased it 1998. By 2004 it has depreciated in value to $13,504. Assuming the rate of depreciation is linear:
If t = 0 in 1998, model the value of the car as a linear function in the form where f(x) is the value in dollars and t is time.

(0, 25000) and (6, 13504)Create points from the information. Since 1998 is the value 0, the

zero matches to 25000. 2004 is six years later, so the x-value of

the second point is 6 with the y-value of 13504 for monetary value

The slope (m) comes from the formula .

Substitute into point slope form which is .

Solve for y by adding the 25000 to the other side.

Substitute the f(x) for y.

How much will the car be worth in 2009?

2009 is t = 11Since t = 0 in 1998, t = 11 in 2009.

Substitute the 11 for the t in the function.

Value is $3,924Solve the right side of the equation.

Write the linear equation in slope-intercept form, find its slope, x- intercept, y-intercept and sketch its graph

Slope: x-int: 7.5 y-int: -5

Subtract the 2x to the other side.

Divide both sides by -3 to get the y by itself.

Slope: y-int: -5Now we are in the form so the slope is m and the y-int is b.

To get the x-int, substitute a 0 for the y value.

x-int: 7.5Solve for x by adding 5 and dividing by .

Biologists have observed that the chirping rate of crickets of a certain species is linearly related to the temperature. If a cricket produces 120 chirps per minute at F and 168 chirps per minute at F, Find the linear equation that describes the temperature t and the number of chirps per minute m. Use your linear equation to estimate the temperature if the crickets are chirping at 150 chirps per minute.

(120, 70) and (168, 80)Points can be made from the data.

Use the slope formula to find the slope (m).

I can substitute my slope into the equation y = mx + b.

I can substitute a point in for x and y.

Multiply the right.

b = 45Solve for b by subtracting 25.

Since the y value is time and the x is chirps, I can substitute t and c.

cpmSubstitute the chirps into c and get your chirps per minute.