# Graph 3x^2+12x-18 3×2+12x-18
Find the properties of the given parabola.
Rewrite the equation in vertex form.
Complete the square for 3×2+12x-18.
Use the form ax2+bx+c, to find the values of a, b, and c.
a=3,b=12,c=-18
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=122(3)
Simplify the right side.
Cancel the common factor of 12 and 2.
Factor 2 out of 12.
d=2⋅62⋅3
Cancel the common factors.
Factor 2 out of 2⋅3.
d=2⋅62(3)
Cancel the common factor.
d=2⋅62⋅3
Rewrite the expression.
d=63
d=63
d=63
Cancel the common factor of 6 and 3.
Factor 3 out of 6.
d=3⋅23
Cancel the common factors.
Factor 3 out of 3.
d=3⋅23(1)
Cancel the common factor.
d=3⋅23⋅1
Rewrite the expression.
d=21
Divide 2 by 1.
d=2
d=2
d=2
d=2
Find the value of e using the formula e=c-b24a.
Simplify each term.
Raise 12 to the power of 2.
e=-18-1444⋅3
Multiply 4 by 3.
e=-18-14412
Divide 144 by 12.
e=-18-1⋅12
Multiply -1 by 12.
e=-18-12
e=-18-12
Subtract 12 from -18.
e=-30
e=-30
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
3(x+2)2-30
3(x+2)2-30
Set y equal to the new right side.
y=3(x+2)2-30
y=3(x+2)2-30
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=3
h=-2
k=-30
Since the value of a is positive, the parabola opens up.
Opens Up
Find the vertex (h,k).
(-2,-30)
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
Multiply 4 by 3.
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.
(-2,-35912)
(-2,-35912)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=-2
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=-36112
y=-36112
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Up
Vertex: (-2,-30)
Focus: (-2,-35912)
Axis of Symmetry: x=-2
Directrix: y=-36112
Direction: Opens Up
Vertex: (-2,-30)
Focus: (-2,-35912)
Axis of Symmetry: x=-2
Directrix: y=-36112
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 -3 in the expression.
f(-3)=3(-3)2+12(-3)-18
Simplify the result.
Simplify each term.
Raise -3 to the power of 2.
f(-3)=3⋅9+12(-3)-18
Multiply 3 by 9.
f(-3)=27+12(-3)-18
Multiply 12 by -3.
f(-3)=27-36-18
f(-3)=27-36-18
Simplify by subtracting numbers.
Subtract 36 from 27.
f(-3)=-9-18
Subtract 18 from -9.
f(-3)=-27
f(-3)=-27
-27
-27
The y value at x=-3 is -27.
y=-27
Replace the variable x with -4 in the expression.
f(-4)=3(-4)2+12(-4)-18
Simplify the result.
Simplify each term.
Raise -4 to the power of 2.
f(-4)=3⋅16+12(-4)-18
Multiply 3 by 16.
f(-4)=48+12(-4)-18
Multiply 12 by -4.
f(-4)=48-48-18
f(-4)=48-48-18
Simplify by subtracting numbers.
Subtract 48 from 48.
f(-4)=0-18
Subtract 18 from 0.
f(-4)=-18
f(-4)=-18
-18
-18
The y value at x=-4 is -18.
y=-18
Replace the variable x with -1 in the expression.
f(-1)=3(-1)2+12(-1)-18
Simplify the result.
Simplify each term.
Raise -1 to the power of 2.
f(-1)=3⋅1+12(-1)-18
Multiply 3 by 1.
f(-1)=3+12(-1)-18
Multiply 12 by -1.
f(-1)=3-12-18
f(-1)=3-12-18
Simplify by subtracting numbers.
Subtract 12 from 3.
f(-1)=-9-18
Subtract 18 from -9.
f(-1)=-27
f(-1)=-27
-27
-27
The y value at x=-1 is -27.
y=-27
Replace the variable x with 0 in the expression.
f(0)=3(0)2+12(0)-18
Simplify the result.
Simplify each term.
Raising 0 to any positive power yields 0.
f(0)=3⋅0+12(0)-18
Multiply 3 by 0.
f(0)=0+12(0)-18
Multiply 12 by 0.
f(0)=0+0-18
f(0)=0+0-18
f(0)=0-18
Subtract 18 from 0.
f(0)=-18
f(0)=-18
-18
-18
The y value at x=0 is -18.
y=-18
Graph the parabola using its properties and the selected points.
xy-4-18-3-27-2-30-1-270-18
xy-4-18-3-27-2-30-1-270-18
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex: (-2,-30)
Focus: (-2,-35912)
Axis of Symmetry: x=-2
Directrix: y=-36112
xy-4-18-3-27-2-30-1-270-18
Graph 3x^2+12x-18     