# Graph y=-3x^2-6x+9

y=-3×2-6x+9
Find the properties of the given parabola.
Rewrite the equation in vertex form.
Complete the square for -3×2-6x+9.
Use the form ax2+bx+c, to find the values of a, b, and c.
a=-3,b=-6,c=9
Consider the vertex form of a parabola.
a(x+d)2+e
Substitute the values of a and b into the formula d=b2a.
d=-62(-3)
Simplify the right side.
Cancel the common factor of 6 and 2.
Factor 2 out of 6.
d=-2⋅32⋅-3
Cancel the common factors.
Factor 2 out of 2⋅-3.
d=-2⋅32(-3)
Cancel the common factor.
d=-2⋅32⋅-3
Rewrite the expression.
d=-3-3
d=-3-3
d=-3-3
Cancel the common factor of 3 and -3.
Factor 3 out of 3.
d=-3(1)-3
Move the negative one from the denominator of 1-1.
d=-(-1⋅1)
d=-(-1⋅1)
Multiply.
Multiply -1 by 1.
d=1
Multiply -1 by -1.
d=1
d=1
d=1
Find the value of e using the formula e=c-b24a.
Simplify each term.
Raise -6 to the power of 2.
e=9-364⋅-3
Multiply 4 by -3.
e=9-36-12
Divide 36 by -12.
e=9+3
Multiply -1 by -3.
e=9+3
e=9+3
e=12
e=12
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
-3(x+1)2+12
-3(x+1)2+12
Set y equal to the new right side.
y=-3(x+1)2+12
y=-3(x+1)2+12
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=-3
h=-1
k=12
Since the value of a is negative, the parabola opens down.
Opens Down
Find the vertex (h,k).
(-1,12)
Find p, the distance from the vertex to the focus.
Find the distance from the vertex to a focus of the parabola by using the following formula.
14a
Substitute the value of a into the formula.
14⋅-3
Simplify.
Multiply 4 by -3.
1-12
Move the negative in front of the fraction.
-112
-112
-112
Find the focus.
The focus of a parabola can be found by adding p to the y-coordinate k if the parabola opens up or down.
(h,k+p)
Substitute the known values of h, p, and k into the formula and simplify.
(-1,14312)
(-1,14312)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=-1
Find the directrix.
The directrix of a parabola is the horizontal line found by subtracting p from the y-coordinate k of the vertex if the parabola opens up or down.
y=k-p
Substitute the known values of p and k into the formula and simplify.
y=14512
y=14512
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Down
Vertex: (-1,12)
Focus: (-1,14312)
Axis of Symmetry: x=-1
Directrix: y=14512
Direction: Opens Down
Vertex: (-1,12)
Focus: (-1,14312)
Axis of Symmetry: x=-1
Directrix: y=14512
Select a few x values, and plug them into the equation to find the corresponding y values. The x values should be selected around the vertex.
Replace the variable x with -2 in the expression.
f(-2)=-3(-2)2-6⋅-2+9
Simplify the result.
Simplify each term.
Raise -2 to the power of 2.
f(-2)=-3⋅4-6⋅-2+9
Multiply -3 by 4.
f(-2)=-12-6⋅-2+9
Multiply -6 by -2.
f(-2)=-12+12+9
f(-2)=-12+12+9
f(-2)=0+9
f(-2)=9
f(-2)=9
9
9
The y value at x=-2 is 9.
y=9
Replace the variable x with -3 in the expression.
f(-3)=-3(-3)2-6⋅-3+9
Simplify the result.
Simplify each term.
Multiply -3 by (-3)2 by adding the exponents.
Multiply -3 by (-3)2.
Raise -3 to the power of 1.
f(-3)=(-3)(-3)2-6⋅-3+9
Use the power rule aman=am+n to combine exponents.
f(-3)=(-3)1+2-6⋅-3+9
f(-3)=(-3)1+2-6⋅-3+9
f(-3)=(-3)3-6⋅-3+9
f(-3)=(-3)3-6⋅-3+9
Raise -3 to the power of 3.
f(-3)=-27-6⋅-3+9
Multiply -6 by -3.
f(-3)=-27+18+9
f(-3)=-27+18+9
f(-3)=-9+9
f(-3)=0
f(-3)=0
0
0
The y value at x=-3 is 0.
y=0
Replace the variable x with 0 in the expression.
f(0)=-3(0)2-6⋅0+9
Simplify the result.
Simplify each term.
Raising 0 to any positive power yields 0.
f(0)=-3⋅0-6⋅0+9
Multiply -3 by 0.
f(0)=0-6⋅0+9
Multiply -6 by 0.
f(0)=0+0+9
f(0)=0+0+9
f(0)=0+9
f(0)=9
f(0)=9
9
9
The y value at x=0 is 9.
y=9
Replace the variable x with 1 in the expression.
f(1)=-3(1)2-6⋅1+9
Simplify the result.
Simplify each term.
One to any power is one.
f(1)=-3⋅1-6⋅1+9
Multiply -3 by 1.
f(1)=-3-6⋅1+9
Multiply -6 by 1.
f(1)=-3-6+9
f(1)=-3-6+9
Subtract 6 from -3.
f(1)=-9+9
f(1)=0
f(1)=0
0
0
The y value at x=1 is 0.
y=0
Graph the parabola using its properties and the selected points.
xy-30-29-1120910
xy-30-29-1120910
Graph the parabola using its properties and the selected points.
Direction: Opens Down
Vertex: (-1,12)
Focus: (-1,14312)
Axis of Symmetry: x=-1
Directrix: y=14512
xy-30-29-1120910
Graph y=-3x^2-6x+9