SOLUTION: Quadratic Equations Complete the square to rewrite each relation in the form y=a(x-h)^2+k. Then, sketch a graph of each parabola, labelling the vertex, the axis of symmetry, and t

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equation Customizable Word Problems -> SOLUTION: Quadratic Equations Complete the square to rewrite each relation in the form y=a(x-h)^2+k. Then, sketch a graph of each parabola, labelling the vertex, the axis of symmetry, and t      Log On


   

Question 177829: Quadratic Equations
Complete the square to rewrite each relation in the form y=a(x-h)^2+k. Then, sketch a graph of each parabola, labelling the vertex, the axis of symmetry, and two other points.
a)y=x^2+4x+10
b)y=-x^2+6x-8
c)y=-2x^2-12x-7
Thank you very much
Answer by jojo14344(1513) About Me  (Show Source):
You can put this solution on YOUR website!

a)y=x%5E2%2B4x%2B10
Completing the square:
y=%28x%5E2%2B4x%2B4%29%2B10-4
highlight%28y=%28x%2B2%29%5E2%2B6%29
Vertex: (-2,6)
Axis of Symmetry=-2
Solving the X-intercepts:
x=%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29--->wheresystem%28a=1%2Cb=4%2Cc=10%29
Solving the Discriminant: b%5E2-4ac=4%5E2-4%2A1%2A10=16-40=-24---> since negative, it has no solutions for real numbers.
Y-Intercept: f%28x%29=0
y=0%5E2%2B4%280%29%2B10
y=10
Continuing,
x=%28-4%2B-i%2Asqrt%28-24%29%29%2F%282%2A1%29=%28-4%2B-i%2A4.8989795%29%2F2
Or see the graph,

b)y=-x%5E2%2B6x-8----->y=-1%28x%5E2%2B6x-8%29
Completing the square:
y=-1%28x%5E2%2B6x%2B9%29%2B8-9
highlight%28y=-1%28x-3%29%5E2-1%29------->a%28x-h%29%5E2%2Bk wheresystem%28h=x=3%2Ck=y=highlight%281%29%29, highlighted: a%2Ak=%28-1%29%28-1%29=1, oks!
Vertex: (3,1)
Axis of Symmetry: =3
Remembersystem%28a=-1%2Cb=6%2Cc=-8%29
Solving for Discriminant: b%5E2-4ac=6%5E2-4%2A%28-1%29%28-8%29=36-32=4
Then,

x=%28-6%2B-2%29%2F-2
x=%28-6%2B2%29%2F-2=-4%2F-2=2
x=%28-6-2%29%2F-2=-8%2F-2=4
X-Interceptssystem%28x=2%2Cx=4%29
Y-Intercept: f%28x%29=0
y=-%280%29%5E2%2B6%280%29-8
y=-8
We'll see the graph,

c)y=-2x%5E2-12x-7----->y=-2%28x%5E2%2B6x%2B7%2F2%29
Completing the square:
y=-2%28x%5E2%2B6x%2B9%29%2B%287%2F2%29-9
y=-2%28x%2B3%29%5E2%2B%287-18%29%2F2
highlight%28y=-2%28x%2B3%29%5E2-%2811%2F2%29%29-->it follows a%28x-h%29%5E2%2Bk wheresystem%28h=x=-3%2Ck=y=11%29
Vertex: (-3,11)------->Again, y=a%2Ak=-2%2A%28-11%2F2%29=11
Axis 0f Symmetry= -3
For X-Intercepts:
Remembersystem%28a=1%2Cb=6%2Cc=7%2F2%29
WE 1st solve for the discriminant: b%5E2-4ac=6%5E2-4%2A1%2A%287%2F2%29=36-%2828%2F2%29=36-14=22
Then,
x=%28-6%2B-4.69%29%2F2
x=%28-6%2B4.69%29%2F2=-1.31%2F2=highlight%28-0.655%29
x=%28-6-4.69%29%2F2=-10.69%2F2=highlight%28-5.345%29
Y-Intercept: f%28x%29=0
y=-2%280%29%5E2-12%280%29-7
y=-7
We see the graph:

Thank you,
Jojo