# Graph 16(t^2-3t-12) 16(t2-3t-12)
Find the properties of the given parabola.
Rewrite the equation in vertex form.
Simplify 16(x2-3x-12).
Apply the distributive property.
y=16×2+16(-3x)+16⋅-12
Simplify.
Multiply -3 by 16.
y=16×2-48x+16⋅-12
Multiply 16 by -12.
y=16×2-48x-192
y=16×2-48x-192
y=16×2-48x-192
Complete the square for 16×2-48x-192.
Use the form ax2+bx+c, to find the values of a, b, and c.
a=16,b=-48,c=-192
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=-482(16)
Simplify the right side.
Cancel the common factor of 48 and 2.
Factor 2 out of 48.
d=-2⋅242⋅16
Cancel the common factors.
Factor 2 out of 2⋅16.
d=-2⋅242(16)
Cancel the common factor.
d=-2⋅242⋅16
Rewrite the expression.
d=-2416
d=-2416
d=-2416
Cancel the common factor of 24 and 16.
Factor 8 out of 24.
d=-8(3)16
Cancel the common factors.
Factor 8 out of 16.
d=-8⋅38⋅2
Cancel the common factor.
d=-8⋅38⋅2
Rewrite the expression.
d=-32
d=-32
d=-32
d=-32
Find the value of e using the formula e=c-b24a.
Simplify each term.
Raise -48 to the power of 2.
e=-192-23044⋅16
Multiply 4 by 16.
e=-192-230464
Divide 2304 by 64.
e=-192-1⋅36
Multiply -1 by 36.
e=-192-36
e=-192-36
Subtract 36 from -192.
e=-228
e=-228
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
16(x-32)2-228
16(x-32)2-228
Set y equal to the new right side.
y=16(x-32)2-228
y=16(x-32)2-228
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=16
h=32
k=-228
Since the value of a is positive, the parabola opens up.
Opens Up
Find the vertex (h,k).
(32,-228)
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⋅16
Multiply 4 by 16.
164
164
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.
(32,-1459164)
(32,-1459164)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=32
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=-1459364
y=-1459364
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Up
Vertex: (32,-228)
Focus: (32,-1459164)
Axis of Symmetry: x=32
Directrix: y=-1459364
Direction: Opens Up
Vertex: (32,-228)
Focus: (32,-1459164)
Axis of Symmetry: x=32
Directrix: y=-1459364
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 1 in the expression.
f(1)=16(1)2-48⋅1-192
Simplify the result.
Simplify each term.
One to any power is one.
f(1)=16⋅1-48⋅1-192
Multiply 16 by 1.
f(1)=16-48⋅1-192
Multiply -48 by 1.
f(1)=16-48-192
f(1)=16-48-192
Simplify by subtracting numbers.
Subtract 48 from 16.
f(1)=-32-192
Subtract 192 from -32.
f(1)=-224
f(1)=-224
-224
-224
The y value at x=1 is -224.
y=-224
Graph the parabola using its properties and the selected points.
xy1-22432-228
xy1-22432-228
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex: (32,-228)
Focus: (32,-1459164)
Axis of Symmetry: x=32
Directrix: y=-1459364
xy1-22432-228
Graph 16(t^2-3t-12)     