によって Liam McNabb 2時間前.
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If you have the diameter and need the radius you can use diameter/2 to find the radius
If the diameter is 6 the radius would be 3
The diameter is the distance of one point on a circle to the opposite point (must pass through midpoint
Equation of any circle: (x - h)² + (y - k)² = r²
The h and k are the center of the radius, h is always the first number and k is the always second one
If the origin is at (7,8) and the radius is 11 the equation would be (x - 7)² + (y - 8)² = 121 ²
Equation of a circle with center at origin: x² + y² = r²
If the origin is (0,0) and the radius is 6 the equation would be x² + y² = 36
If points (x , y) are on the circle that means: x² + y² = r² If points (x , y) are outside the circle that means: x² + y² > r² If points (x , y) are inside the circle that means: x² + y² < r²
106 > 100
25 + 81 = 100
Is the point (−5,9) inside, outside or on the circle of x² + y² = 100
The radius is important because it is used in both circle equations
The radius of a circle is the line that connects from one part of the circle to the very middle
4. Put them together into y = mx + b to get the equation of the altitude
3. Find the y-intercept using the point from the vertex
2. Find the perpendicular slope (that is the slope of the altitude)
1. Find the slope of the opposite side from the vertex
y = -1/6x + 19.6
y-int = 19.6
Perpendicular slope = -1/6
Slope of opposite side = 6
The altitude is the line from a vertex that is perpendicular to the opposite side
5. Put them all into y = mx + b to get your right bisector equation
4. Calculate the y-intercept of the right bisector
3. Find the perpendicular slope (opposite of normal slope)
2. Find the slope of the original line
1. Find the midpoint of the original line
Points: z(16,17) x(12,8) y(18,8)
y = -1/6x + 10.5
y-int of right bisector = 10.5
Perpendicular Slope = -1/6
Slope of xy = 6
Midpoint of xy = 15 , 8
The line segment that intersects through the mid point of another line segment and makes a 90° angle
Put them together into y = mx + b to get the equation of the median
3. Calculate the y-intercept
2. Find the slope of the vertex to the midpoint
1. Find the midpoint of the opposite side
Points: a(16,17) b(12,8) c(18,8)
Final equation: y = 1/8x + 15
y-int = 15
17 = 1/8(16) + b
z
Midpoint of BC = 15 , 8
The line segment that connects a vertex to the midpoint of the opposite side
8 , 13
16/2 , 26/2
This would become: 4 + 12 / 2 , 7 + 19 /2
The middle of a line
Rise and Run can be found by doing: Run = x2 - x1 Rise = y2 -y1
Points: (4 , 7) and (12 , 19)
4.47cm
√20
√ 8 + 12
This would become: √ (12-4)² + (19 -7)²
The length of a line
This is one of the most important parts of this unit as you need this to find almost everything
In this case you would add the top to the bottom because the signs are different on our variables we are trying to cancel out. If the signs were the same we would subtract instead
This would result in us having one equation, this equation being: 8x = 28. From there divide each side by your coefficient to find your final answer
In this case you would isolate one of the variables. If you did this in the first equation it would become x = -3.5y + 17
You would then substitute that into the other equation to find your answer
In this case you would use the bottom equation and substitute it for 'y' in the top equation, making it, 2x + 5 ( 3x - 3 ) = 19
From there you would have the 'x' value, with that you can then find 'y' with the same method
The slope in this case the first slope would be 2/3 and the first point you can plot is (0,-1). The second slope is -1/0 and the second point is (0,4). You would then graph these lines and find the point of intersection.
For the final step you would switch the sign on the 2 to make it (2,5)
Whatever your 'a' value is it what you multiply the step pattern by, your step pattern helps determine how many units you go up for every one unit moved horizontally. Example 2 become 2, 6, 10
1, 3, 5
If the 'a' in y = ax² is a value above 0 that means a parabola has a vertical stretch but if 'a' is less than 0 that means the parabola has a vertical compression
Red is stretched and blue is compressed
If the 'a' in y = ax² is a negative '-a' that means the parabola is reflected over the x-axis
y = (x + h)² means there is a horizontal translation right (vertex moves to the right) y = (x - h)² means there is a horizontal translation left (vertex moves to the left)
y = x² + k means there is a positive translation (vertex moves up) y = x² - k means there is a negative translation (vertex moves down)
Real solutions mean zeroes (make sure to not square root it)
To solve you substitute the values (very simple once you know the formula)
ax² + bx + c = 0
For this perfect square you must put your two square rooted outside numbers/variables and put them into a bracket with a ² sign. Make sure to make the sign on the inside based on the sign of the second number/variable
An example of this could be 9x² + 42x + 49
This is correct because 2(3x)(7) = 42x
Perfect square trinomials are slightly different then normal perfect squares, there will be 3 terms, the first and last terms have to perfect squares but the middle term has to be twice the product of the square roots of the first and last term
Once you have a perfect square you make it into a positive and a negative bracket then factor to the fullest
An example could be x² - 16 or 4x² - 4
These would become (x - 4) (x + 4)
A perfect square is when you have a variable that you can get the square root of (any variable with an exponent) and another number that has to be a negative that you can also get the square root of.
Find two numbers with the product of ac and the sum of b
Complex trinomials can also be in the forms: ax² y² + bxy + c or ax² +bxy + cy²
The two numbers here are -12 and -2, this is because -12 -2 = -14 (b) and -12 x -2 = 24 (ac)
ax² + bx + c as long as 'a' doesn't equal 1
Find two numbers that have a product of c and a sum of b. b= (_+_) c =( _ x _)
Simple trinomials can also be in the forms: x² y² + bxy + c or x² + bxy + cy²
This would become (x + 3) (x + 2) You can verify your answer is correct by using foil to get back to the original equation
The answer to this would be 3 and 2 because 3x2 = 6 and 3+2 = 5
x² + bx + c
The second factor is that if you multiply the first term by the last and multiply the 2 middle ones together you should get the same product
This is true because the right bracket are the highest common factors, and the right bracket are the result of the right bracket
The first factor is that the polynomials have 4 terms
This kind of multiplication can be applied to parabolas
A common abbreviation used to remember the order of how to multiply these is "FOIL" this stands for: First, Outer, Inner, Last.
(9x - 4) (x-6)
9x² - 54x - 4x + 24
Binomial Multiplication is multiplying multiple terms at a time
Factored form
The 's' and the 'r' in this equation tell us what the zeroes of the parabola are, whenever you have a value in one of these spots it is the opposite of that number, for example 3 would become -3
To find y-int substitute x for 0
(x - 4)² = (x - 4) (x - 4)
The sign of a determines rather the parabola opens up or down, if the a value is a negative it opens downwards (maximum value) but if it is a positive it opens upwards (minimum value)
y = a (x - r) (x - s)
How they are graphed
Parabolas are graphed using the format that first and second differences use, the first and second differences can be used to determine if your graph is correct or not.
Terminology
A parabola is a type of graph that is symmetrical and almost always shaped like a 'U' , all of the parts of a parabola are: the Zeros, where the parabola passes over the x- axis (this is also known as a root or an x-intercept), the Line of Symmetry, divides the parabola into to equal halves (straight vertical line), the Minimum or Maximum is the highest (maximum) or lowest (minimum) point on a parabola, the Vertex where the parabola and the line of symmetry meet (this will always be at the minimum or maximum), the optimal value is the y value of the minimum or maximum, and finally the y-intercept is the point where a part of the parabola crosses over the y-axis
Second differences
With these numbers you will find another pattern except this time all the differences are the same, in this case all of the differences are +2. The second difference will always be one number if you have done your chart correctly.
To determine the second differences you must go left to right but instead of using the first values use the first differences, in this case our first differences are: -5, -3, -1, +1, +3, +5.
Second differences are deciphered the same way that the first differences are except you take out the first step
First differences
With these numbers you find the difference of each number left to right, in this case it would be, -5, -3, -1, +1, +3, +5. As you can see there is a pattern in this sequence of numbers, this helps confirm that your values are correct
To determine the first difference you must find the values of x for every y value. Once you have found these go down the line and find the difference of each x value, in this case it would be, x = 9, 4, 1, 0, 1, 4, 9
When having to determine first and second differences you will get an equation like y = x² and then you will be given some values that are usually y = -3, -2, -1, 0 1, 2, 3
y = ax² + bx + c
There are 3 simple polynomials with names, these are: Monomial, Binomial and Trinomial. All other polynomials don't have names but can be described by how many terms they have. You can decipher how many terms a polynomial has by the amount of expressions it has, 3x-4³+7y would be a trinomial because it has three expressions (3x, 4³ and 7y)
The degrees of a polynomial are determined by the monomial with the biggest exponent for example: x³ + 4² would be 3rd degree another example is x²y³ - 4² would be 5th degree
This triangle would be solved by: cosA =46² + 117² - 82² / 2(46)(117)
This triangle would be solved by: a² = 86² + 65² - 2(86)(65) cos19
This triangle would be solved by: Sin94/10.5 = Sin40/b (the 'x' can be solved after you find the opposite side to Sin40)
One way you can confirm if a triangle is similar or not is by dividing the larger triangles sides by the smaller triangles sides. If all equal the same number then the triangle is similar
Something to keep in mind is that you don't need all lengths or all angles, if you have two lengths you can still do the dividing trick and if you have two angles you can fill it out by doing: 180-x-y = θ
