數學~函數~有式~(40點以上)

1.)
f(x)=x^2-2x-5
f(k+1)=k
(k+1)^2-2(k+1)-5=k^2-2k-5
k^2+2k+1-2k-2-5=k^2-2k-5
k^2+2k+1-2k-2=k^2-2k
2k+1-2=0
2k-1=0
k=1/2

2a.)
g(x)= 6/(x-2 )?
g(4)=6/(4-2 )
g(4)=3

2b.)
g(3)-g(-4)?
=[6/(3-2 )]-[6/(-4-2 )]
=6-(-1)
=6+1
=7

3a)
f(x+1)=x^2+4x+k
f(1+1)=1^2+4(1)+k
f(2)=1+4+k
8=5+k
k=3

(b)
f(-1) =(-1)^2+(4)(-1)+3
f(-1) =1-4+3
f(-1) =0

(c)
Put x=y-1 in to f(x+1)
f(y+1)=x^2+4x+3
f([y-1]+1)=(y-1)^2+4(y-1)+3
f(y)=y^2-2y+1+4y-4+3
f(y)=y^2+2y
f(x)=x^2+2x

(d)
f(x)=3
x^2+2x=3
x^2+2x-3=0
(x+3)(x-1)=0
x=-3 or 1

4a)
y=ax^2+bx+1
5=a(2^2)+2b+1
5=4a+2b+1
2=2a+b........................1
11=a([-1]^2)+b(-1)+1
11=a-b+1
10=a-b..........................2
1+2:
10+2=2a+b+a-b
12=3a
a=4
a-b=10
b=-6
so a=4,b=-6

4b)
y=4x^2-6x+1
y=4(x^2-3x/2)+1
y=4(x^2-3x/2+9/16)-4[9/16]+1
y=4(x-3/4)^2-5/4
y的極小值=-5/4,所對應的x值=3/4

4c)
頂點=(3/4,-5/4)
對稱軸:x=3/4

5a)
y=3-(4+x)^2
極大值是3

5b)
y=-x^2+6x-25
y=-(x^2-6x)-25
y=-(x^2-6x+9-9)-25
y=-(x-3)^2+9-25
y=-(x-3)^2+16
極大值是16