# Graph f(x)=-3x^2+6x-9

f(x)=-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 -1 by 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=-6
e=-6
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
-3(x-1)2-6
-3(x-1)2-6
Set y equal to the new right side.
y=-3(x-1)2-6
y=-3(x-1)2-6
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=-3
h=1
k=-6
Since the value of a is negative, the parabola opens down.
Opens Down
Find the vertex (h,k).
(1,-6)
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,-7312)
(1,-7312)
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=-7112
y=-7112
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Down
Vertex: (1,-6)
Focus: (1,-7312)
Axis of Symmetry: x=1
Directrix: y=-7112
Direction: Opens Down
Vertex: (1,-6)
Focus: (1,-7312)
Axis of Symmetry: x=1
Directrix: y=-7112
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 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
Subtract 9 from 0.
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.
Raise -1 to the power of 2.
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
Simplify by subtracting numbers.
Subtract 6 from -3.
f(-1)=-9-9
Subtract 9 from -9.
f(-1)=-18
f(-1)=-18
-18
-18
The y value at x=-1 is -18.
y=-18
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
Subtract 9 from 0.
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.
Raise 3 to the power of 2.
f(3)=-3⋅9+6(3)-9
Multiply -3 by 9.
f(3)=-27+6(3)-9
Multiply 6 by 3.
f(3)=-27+18-9
f(3)=-27+18-9
f(3)=-9-9
Subtract 9 from -9.
f(3)=-18
f(3)=-18
-18
-18
The y value at x=3 is -18.
y=-18
Graph the parabola using its properties and the selected points.
xy-1-180-91-62-93-18
xy-1-180-91-62-93-18
Graph the parabola using its properties and the selected points.
Direction: Opens Down
Vertex: (1,-6)
Focus: (1,-7312)
Axis of Symmetry: x=1
Directrix: y=-7112
xy-1-180-91-62-93-18
Graph f(x)=-3x^2+6x-9