Answer:
Standard Form Equivalent Form Extreme Values
y=x^2-6x+17 (x-3)^2+8 (3,8)
y=x^2+8x+21 (x+4)^2+5 (-4,5)
y=x^2-16x+60 (x-8)^2-4 (8,-4)
Step-by-step explanation:
1) Standard form:
y=x^2-6x+17
Equivalent Form:
Can be found using completing the square method.
[tex]y=x^2-6x+17\\y=x^2-2(x)(3)+(3)^2-(3)^2+17\\y=(x-3)^2-9+17\\y=(x-3)^2+8[/tex]
So, Equivalent form is: (x-3)^2+8
Extreme value:
Extreme values are basically the minimum and maximum value of the function.
Minimum Value will be found by finding derivative of the function:
The derivate is: 2x-6
Now, put the derivate equal to zero: 2x-6 = 0
[tex]2x=6\\x=6/3 \\x=3[/tex]
Maximum value can be found by putting minimum value in the given function:
Put x = 3 and solve:
[tex](3)^2-6(3)+17\\9-18+17\\9-1\\=8\\[/tex]
So, the extreme values is: (3,8)
2) Standard form:
y=x^2+8x+21
Equivalent Form:
Can be found using completing the square method.
[tex]y=x^2+8x+21\\y=x^2+2(x)(4)+(4)^2-(4)^2+21\\y=(x+4)^2-16+21\\y=(x+4)^2+5[/tex]
So, Equivalent form is: (x+4)^2+5
Extreme value:
Extreme values are basically the minimum and maximum value of the function.
Minimum Value will be found by finding derivative of the function:
The derivate of x^2+8x+21 is: 2x+8
Now, put the derivate equal to zero:
[tex]2x+8 = 0\\2x=-8\\x=-8/2 \\x=-4[/tex]
So, minimum value is: -4
Maximum value can be found by putting minimum value in the given function:
Put x = -4 and solve:
[tex]x^2+8x+21\\=(-4)^2+8(-4)+21\\=16-32+21\\=5[/tex]
So, Maximum value is: 5
So, the extreme values is: (-4,5)
3) Standard form:
y=x^2-16x+60
Equivalent Form:
Can be found using completing the square method.
[tex]y=x^2-16x+60\\y=x^2-2(x)(8)+(8)^2-(8)^2+60\\y=(x-8)^2-64+60\\y=(x-8)^2-4[/tex]
So, Equivalent form is: (x-8)^2-4
Extreme value:
Extreme values are basically the minimum and maximum value of the function.
Minimum Value will be found by finding derivative of the function:
The derivate of x^2-16x+60 is: 2x-16
Now, put the derivate equal to zero:
[tex]2x-16 = 0\\2x=16\\x=16/2 \\x=8[/tex]
So, minimum value is: 8
Maximum value can be found by putting minimum value in the given function:
Put x = 8 and solve:
[tex]x^2-16x+60\\=(8)^2-16(8)+60\\=64-128+60\\=-4[/tex]
So, Maximum value is: -4
So, the extreme values is: (8,-4)
