Answer:
Part 11) The vertex is the point (4,32)
Part 12) The vertex is the point (1,-5)
Part 13) The vertex is the point (-2,5)
Part 14) The vertex is the point (-1,-1)
Part 15) The vertex is the point (1,8)
Part 16) The vertex is the point (3,-26)
Part 17) The vertex is the point (-5,-32)
Step-by-step explanation:
Part 11) we have
[tex]y=-x^2+8x+16[/tex]
we know that
The quadratic equation written in vertex form is equal to
[tex]y=a(x-h)^2+k[/tex]
where
a is the leading coefficient
(h,k) is the vertex
Convert to vertex form
[tex]y=-x^2+8x+16[/tex]
Factor -1
[tex]y=-(x^2-8x)+16[/tex]
Complete the square
[tex]y=-(x^2-8x+16)+16+16[/tex]
[tex]y=-(x^2-8x+16)+32[/tex]
Rewrite as perfect squares
[tex]y=-(x-4)^2+32[/tex]
The vertex is the point (4,32)
Part 12) we have
[tex]y=3x^2-6x-2[/tex]
we know that
The quadratic equation written in vertex form is equal to
[tex]y=a(x-h)^2+k[/tex]
where
a is the leading coefficient
(h,k) is the vertex
Convert to vertex form
[tex]y=3x^2-6x-2[/tex]
Factor 3
[tex]y=3(x^2-2x)-2[/tex]
Complete the square
[tex]y=3(x^2-2x+1)-2-3[/tex]
[tex]y=3(x^2-2x+1)-5[/tex]
Rewrite as perfect squares
[tex]y=3(x-1)^2-5[/tex]
The vertex is the point (1,-5)
Part 13) we have
[tex]y=-2x^2-8x-3[/tex]
we know that
The quadratic equation written in vertex form is equal to
[tex]y=a(x-h)^2+k[/tex]
where
a is the leading coefficient
(h,k) is the vertex
Convert to vertex form
[tex]y=-2x^2-8x-3[/tex]
Factor -2
[tex]y=-2(x^2+4x)-3[/tex]
Complete the square
[tex]y=-2(x^2+4x+4)-3+8[/tex]
[tex]y=-2(x^2+4x+4)+5[/tex]
Rewrite as perfect squares
[tex]y=-2(x+2)^2+5[/tex]
The vertex is the point (-2,5)
Part 14) we have
[tex]y=2x^2+4x+1[/tex]
we know that
The quadratic equation written in vertex form is equal to
[tex]y=a(x-h)^2+k[/tex]
where
a is the leading coefficient
(h,k) is the vertex
Convert to vertex form
[tex]y=2x^2+4x+1[/tex]
Factor 2
[tex]y=2(x^2+2x)+1[/tex]
Complete the square
[tex]y=2(x^2+2x+1)+1-2[/tex]
[tex]y=2(x^2+2x+1)-1[/tex]
Rewrite as perfect squares
[tex]y=2(x+1)^2-1[/tex]
The vertex is the point (-1,-1)
Part 15) we have
[tex]y=-5x^2+10x+3[/tex]
we know that
The quadratic equation written in vertex form is equal to
[tex]y=a(x-h)^2+k[/tex]
where
a is the leading coefficient
(h,k) is the vertex
Convert to vertex form
[tex]y=-5x^2+10x+3[/tex]
Factor -5
[tex]y=-5(x^2-2x)+3[/tex]
Complete the square
[tex]y=-5(x^2-2x+1)+3+5[/tex]
[tex]y=-5(x^2-2x+1)+8[/tex]
Rewrite as perfect squares
[tex]y=-5(x-1)^2+8[/tex]
The vertex is the point (1,8)
Part 16) we have
[tex]y=3x^2-18x+1[/tex]
we know that
The quadratic equation written in vertex form is equal to
[tex]y=a(x-h)^2+k[/tex]
where
a is the leading coefficient
(h,k) is the vertex
Convert to vertex form
[tex]y=3x^2-18x+1[/tex]
Factor 3
[tex]y=3(x^2-6x)+1[/tex]
Complete the square
[tex]y=3(x^2-6x+9)+1-27[/tex]
[tex]y=3(x^2-6x+9)-26[/tex]
Rewrite as perfect squares
[tex]y=3(x-3)^2-26[/tex]
The vertex is the point (3,-26)
Part 17) we have
[tex]y=x^2+10x-7[/tex]
we know that
The quadratic equation written in vertex form is equal to
[tex]y=a(x-h)^2+k[/tex]
where
a is the leading coefficient
(h,k) is the vertex
Convert to vertex form
[tex]y=x^2+10x-7[/tex]
Complete the square
[tex]y=(x^2+10x+25)-7-25[/tex]
[tex]y=(x^2+10x+25)-32[/tex]
Rewrite as perfect squares
[tex]y=(x+5)^2-32[/tex]
The vertex is the point (-5,-32)
