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Question 391122: how do i find all the important information(vertex, co vertices ect.) to
(x-2)2 + (y-2)2 = 1
_____ _______
16 4
Answer by haileytucki(390)   (Show Source):
You can put this solution on YOUR website!
/= + - and ~= square root
((x-2)^(2))/(16)+((y-2)^(2))/(4)=1
Multiply each term by a factor of 1 that will equate all the denominators. In this case, all terms need a denominator of 16. The ((y-2)^(2))/(4) expression needs to be multiplied by ((4))/((4)) to make the denominator 16.
((x-2)^(2))/(16)+((y-2)^(2))/(4)*(4)/(4)=1
Multiply the expression by a factor of 1 to create the least common denominator (LCD) of 16.
((x-2)^(2))/(16)+((y-2)^(2)(4))/(16)=1
The numerators of expressions that have equal denominators can be combined. In this case, ((x-2)^(2))/(16) and (((y-2)^(2)(4)))/(16) have the same denominator of 16, so the numerators can be combined.
((x-2)^(2)+((y-2)^(2)(4)))/(16)=1
Simplify the numerator of the expression.
(x^(2)-4x+4y^(2)-16y+20)/(16)=1
Multiply each term in the equation by 16.
(x^(2)-4x+4y^(2)-16y+20)/(16)*16=1*16
Simplify the left-hand side of the equation by canceling the common factors.
x^(2)-4x+4y^(2)-16y+20=1*16
Multiply 1 by 16 to get 16.
x^(2)-4x+4y^(2)-16y+20=16
To set the left-hand side of the equation equal to 0, move all the expressions to the left-hand side.
x^(2)-4x+4y^(2)-16y+4=0
Use the quadratic formula to find the solutions. In this case, the values are a=1, b=-4, and c=4y^(2)-16y.
x=(-b\~(b^(2)-4ac))/(2a) where ax^(2)+bx+c=0
Use the standard form of the equation to find a, b, and c for this quadratic.
a=1, b=-4, and c=4y^(2)-16y
Substitute in the values of a=1, b=-4, and c=4y^(2)-16y.
x=(-(-4)\~((-4)^(2)-4(1)(4y^(2)-16y)))/(2(1))
Multiply -1 by each term inside the parentheses.
x=(4\~((-4)^(2)-4(1)(4y^(2)-16y)))/(2(1))
Simplify the section inside the radical.
x=(4\4~(-1(y^(2)-4y-1)))/(2(1))
Simplify the denominator of the quadratic formula.
x=(4\4~(-1(y^(2)-4y-1)))/(2)
First, solve the + portion of \.
x=(4+4~(-1(y^(2)-4y-1)))/(2)
Simplify the expression to solve for the + portion of the \.
x=2+2~(-1(y^(2)-4y-1))
Next, solve the - portion of \.
x=(4-4~(-1(y^(2)-4y-1)))/(2)
Simplify the expression to solve for the - portion of the \.
x=2-2~(-1(y^(2)-4y-1))
The final answer is the combination of both solutions.
x=2+2~(-1(y^(2)-4y-1)),2-2~(-1(y^(2)-4y-1))
The domain of an expression is all real numbers except for the regions where the expression is undefined. This can occur where the denominator equals 0, a square root is less than 0, or a logarithm is less than or equal to 0. All of these are undefined and therefore are not part of the domain.
(-1(y^(2)-4y-1))<0
Solve the equation to find where the original expression is undefined.
y<2-~(5) or y>2+~(5)
The domain of the rational expression is all real numbers except where the expression is undefined.
2-~(5)
The domain of the inverse of ((x-2)^(2))/(16)+((y-2)^(2))/(4)=1 is equal to the range of f(y)=2+2~(-1(y^(2)-4y-1)).
Range: 2-~(5)
Now, finding the x and y intercepts:
((x-2)^(2))/(16)+((y-2)^(2))/(4)=1
To find the x-intercept, substitute in 0 for y and solve for x.
((x-2)^(2))/(16)+(((0)-2)^(2))/(4)=1
Remove the parentheses around the expression 0.
((x-2)^(2))/(16)+((0-2)^(2))/(4)=1
Combine all similar expressions.
((x-2)^(2))/(16)+((-2)^(2))/(4)=1
Squaring an expression is the same as multiplying the expression by itself 2 times.
((x-2)^(2))/(16)+((-2)(-2))/(4)=1
Multiply -2 by -2 to get 4.
((x-2)^(2))/(16)+(4)/(4)=1
Reduce the expression (4)/(4) by removing a factor of 4 from the numerator and denominator.
((x-2)^(2))/(16)+1=1
Multiply each term by a factor of 1 that will equate all the denominators. In this case, all terms need a denominator of 16.
((x-2)^(2))/(16)+1*(16)/(16)=1
Multiply the expression by a factor of 1 to create the least common denominator (LCD) of 16.
((x-2)^(2))/(16)+(1*16)/(16)=1
Multiply 1 by 16 to get 16.
((x-2)^(2))/(16)+(16)/(16)=1
The numerators of expressions that have equal denominators can be combined. In this case, ((x-2)^(2))/(16) and ((16))/(16) have the same denominator of 16, so the numerators can be combined.
((x-2)^(2)+(16))/(16)=1
Simplify the numerator of the expression.
(x^(2)-4x+20)/(16)=1
Multiply each term in the equation by 16.
(x^(2)-4x+20)/(16)*16=1*16
Simplify the left-hand side of the equation by canceling the common factors.
x^(2)-4x+20=1*16
Multiply 1 by 16 to get 16.
x^(2)-4x+20=16
To set the left-hand side of the equation equal to 0, move all the expressions to the left-hand side.
x^(2)-4x+4=0
In this problem -2*-2=4 and -2-2=-4, so insert -2 as the right hand term of one factor and -2 as the right-hand term of the other factor.
(x-2)(x-2)=0
Combine the two common factors of (x-2) by adding the exponents.
(x-2)^(2)=0
Set each of the factors of the left-hand side of the equation equal to 0.
x-2=0
Since -2 does not contain the variable to solve for, move it to the right-hand side of the equation by adding 2 to both sides.
x=2
To find the y-intercept, substitute in 0 for x and solve for y.
(((0)-2)^(2))/(16)+((y-2)^(2))/(4)=1
Remove the parentheses around the expression 0.
((0-2)^(2))/(16)+((y-2)^(2))/(4)=1
Combine all similar expressions.
((-2)^(2))/(16)+((y-2)^(2))/(4)=1
Squaring an expression is the same as multiplying the expression by itself 2 times.
((-2)(-2))/(16)+((y-2)^(2))/(4)=1
Multiply -2 by -2 to get 4.
(4)/(16)+((y-2)^(2))/(4)=1
Reduce the expression (4)/(16) by removing a factor of 4 from the numerator and denominator.
(1)/(4)+((y-2)^(2))/(4)=1
The numerators of expressions that have equal denominators can be combined. In this case, (1)/(4) and ((y-2)^(2))/(4) have the same denominator of 4, so the numerators can be combined.
(1+(y-2)^(2))/(4)=1
Simplify the numerator of the expression.
(y^(2)-4y+5)/(4)=1
Multiply each term in the equation by 4.
(y^(2)-4y+5)/(4)*4=1*4
Simplify the left-hand side of the equation by canceling the common factors.
y^(2)-4y+5=1*4
Multiply 1 by 4 to get 4.
y^(2)-4y+5=4
To set the left-hand side of the equation equal to 0, move all the expressions to the left-hand side.
y^(2)-4y+1=0
Use the quadratic formula to find the solutions. In this case, the values are a=1, b=-4, and c=1.
y=(-b\~(b^(2)-4ac))/(2a) where ay^(2)+by+c=0
Use the standard form of the equation to find a, b, and c for this quadratic.
a=1, b=-4, and c=1
Substitute in the values of a=1, b=-4, and c=1.
y=(-(-4)\~((-4)^(2)-4(1)(1)))/(2(1))
Multiply -1 by each term inside the parentheses.
y=(4\~((-4)^(2)-4(1)(1)))/(2(1))
Simplify the section inside the radical.
y=(4\2~(3))/(2(1))
Simplify the denominator of the quadratic formula.
y=(4\2~(3))/(2)
First, solve the + portion of \.
y=(4+2~(3))/(2)
Simplify the expression to solve for the + portion of the \.
y=2+~(3)
Next, solve the - portion of \.
y=(4-2~(3))/(2)
Simplify the expression to solve for the - portion of the \.
y=2-~(3)
Solve the equation.
y=2+~(3),2-~(3)
These are the x and y intercepts of the equation ((x-2)^(2))/(16)+((y-2)^(2))/(4)=1.
x=2, y=2+~(3),2-~(3)
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