# Graph f(x)=3x^2-5x-6

f(x)=3×2-5x-6
Find the properties of the given parabola.
Rewrite the equation in vertex form.
Complete the square for 3×2-5x-6.
Use the form ax2+bx+c, to find the values of a, b, and c.
a=3,b=-5,c=-6
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=-52(3)
Multiply 2 by 3.
d=-56
Find the value of e using the formula e=c-b24a.
Simplify each term.
Raise -5 to the power of 2.
e=-6-254⋅3
Multiply 4 by 3.
e=-6-2512
e=-6-2512
To write -6 as a fraction with a common denominator, multiply by 1212.
e=-6⋅1212-2512
Combine -6 and 1212.
e=-6⋅1212-2512
Combine the numerators over the common denominator.
e=-6⋅12-2512
Simplify the numerator.
Multiply -6 by 12.
e=-72-2512
Subtract 25 from -72.
e=-9712
e=-9712
Move the negative in front of the fraction.
e=-9712
e=-9712
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
3(x-56)2-9712
3(x-56)2-9712
Set y equal to the new right side.
y=3(x-56)2-9712
y=3(x-56)2-9712
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=3
h=56
k=-9712
Since the value of a is positive, the parabola opens up.
Opens Up
Find the vertex (h,k).
(56,-9712)
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.
(56,-8)
(56,-8)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=56
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=-496
y=-496
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Up
Vertex: (56,-9712)
Focus: (56,-8)
Axis of Symmetry: x=56
Directrix: y=-496
Direction: Opens Up
Vertex: (56,-9712)
Focus: (56,-8)
Axis of Symmetry: x=56
Directrix: y=-496
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)2-5⋅0-6
Simplify the result.
Simplify each term.
Raising 0 to any positive power yields 0.
f(0)=3⋅0-5⋅0-6
Multiply 3 by 0.
f(0)=0-5⋅0-6
Multiply -5 by 0.
f(0)=0+0-6
f(0)=0+0-6
f(0)=0-6
Subtract 6 from 0.
f(0)=-6
f(0)=-6
-6
-6
The y value at x=0 is -6.
y=-6
Replace the variable x with -1 in the expression.
f(-1)=3(-1)2-5⋅-1-6
Simplify the result.
Simplify each term.
Raise -1 to the power of 2.
f(-1)=3⋅1-5⋅-1-6
Multiply 3 by 1.
f(-1)=3-5⋅-1-6
Multiply -5 by -1.
f(-1)=3+5-6
f(-1)=3+5-6
f(-1)=8-6
Subtract 6 from 8.
f(-1)=2
f(-1)=2
2
2
The y value at x=-1 is 2.
y=2
Replace the variable x with 2 in the expression.
f(2)=3(2)2-5⋅2-6
Simplify the result.
Simplify each term.
Raise 2 to the power of 2.
f(2)=3⋅4-5⋅2-6
Multiply 3 by 4.
f(2)=12-5⋅2-6
Multiply -5 by 2.
f(2)=12-10-6
f(2)=12-10-6
Simplify by subtracting numbers.
Subtract 10 from 12.
f(2)=2-6
Subtract 6 from 2.
f(2)=-4
f(2)=-4
-4
-4
The y value at x=2 is -4.
y=-4
Replace the variable x with 3 in the expression.
f(3)=3(3)2-5⋅3-6
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)2-5⋅3-6
Use the power rule aman=am+n to combine exponents.
f(3)=31+2-5⋅3-6
f(3)=31+2-5⋅3-6
f(3)=33-5⋅3-6
f(3)=33-5⋅3-6
Raise 3 to the power of 3.
f(3)=27-5⋅3-6
Multiply -5 by 3.
f(3)=27-15-6
f(3)=27-15-6
Simplify by subtracting numbers.
Subtract 15 from 27.
f(3)=12-6
Subtract 6 from 12.
f(3)=6
f(3)=6
6
6
The y value at x=3 is 6.
y=6
Graph the parabola using its properties and the selected points.
xy-120-656-97122-436
xy-120-656-97122-436
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex: (56,-9712)
Focus: (56,-8)
Axis of Symmetry: x=56
Directrix: y=-496
xy-120-656-97122-436
Graph f(x)=3x^2-5x-6