# Graph f(x)=-3x^2-4x f(x)=-3×2-4x
Find the properties of the given parabola.
Rewrite the equation in vertex form.
Complete the square for -3×2-4x.
Use the form ax2+bx+c, to find the values of a, b, and c.
a=-3,b=-4,c=0
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=-42(-3)
Simplify the right side.
Cancel the common factor of 4 and 2.
Factor 2 out of 4.
d=-2⋅22⋅-3
Cancel the common factors.
Factor 2 out of 2⋅-3.
d=-2⋅22(-3)
Cancel the common factor.
d=-2⋅22⋅-3
Rewrite the expression.
d=-2-3
d=-2-3
d=-2-3
Move the negative in front of the fraction.
d=23
Multiply –23.
Multiply -1 by -1.
d=1(23)
Multiply 23 by 1.
d=23
d=23
d=23
Find the value of e using the formula e=c-b24a.
Simplify each term.
Cancel the common factor of (-4)2 and 4.
Rewrite -4 as -1(4).
e=0-(-1⋅4)24(-3)
Apply the product rule to -1(4).
e=0-(-1)2⋅424(-3)
Raise -1 to the power of 2.
e=0-1⋅424(-3)
Multiply 42 by 1.
e=0-424(-3)
Factor 4 out of 42.
e=0-4⋅44⋅-3
Cancel the common factors.
Factor 4 out of 4⋅-3.
e=0-4⋅44(-3)
Cancel the common factor.
e=0-4⋅44⋅-3
Rewrite the expression.
e=0-4-3
e=0-4-3
e=0-4-3
Move the negative in front of the fraction.
e=0+43
Multiply –43.
Multiply -1 by -1.
e=0+1(43)
Multiply 43 by 1.
e=0+43
e=0+43
e=0+43
Add 0 and 43.
e=43
e=43
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
-3(x+23)2+43
-3(x+23)2+43
Set y equal to the new right side.
y=-3(x+23)2+43
y=-3(x+23)2+43
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=-3
h=-23
k=43
Since the value of a is negative, the parabola opens down.
Opens Down
Find the vertex (h,k).
(-23,43)
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.
(-23,54)
(-23,54)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=-23
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=1712
y=1712
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Down
Vertex: (-23,43)
Focus: (-23,54)
Axis of Symmetry: x=-23
Directrix: y=1712
Direction: Opens Down
Vertex: (-23,43)
Focus: (-23,54)
Axis of Symmetry: x=-23
Directrix: y=1712
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)2-4⋅-2
Simplify the result.
Simplify each term.
Raise -2 to the power of 2.
f(-2)=-3⋅4-4⋅-2
Multiply -3 by 4.
f(-2)=-12-4⋅-2
Multiply -4 by -2.
f(-2)=-12+8
f(-2)=-12+8
Add -12 and 8.
f(-2)=-4
The final answer is -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)2-4⋅-3
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)2-4⋅-3
Use the power rule aman=am+n to combine exponents.
f(-3)=(-3)1+2-4⋅-3
f(-3)=(-3)1+2-4⋅-3
Add 1 and 2.
f(-3)=(-3)3-4⋅-3
f(-3)=(-3)3-4⋅-3
Raise -3 to the power of 3.
f(-3)=-27-4⋅-3
Multiply -4 by -3.
f(-3)=-27+12
f(-3)=-27+12
Add -27 and 12.
f(-3)=-15
The final answer is -15.
-15
-15
The y value at x=-3 is -15.
y=-15
Replace the variable x with 0 in the expression.
f(0)=-3(0)2-4⋅0
Simplify the result.
Simplify each term.
Raising 0 to any positive power yields 0.
f(0)=-3⋅0-4⋅0
Multiply -3 by 0.
f(0)=0-4⋅0
Multiply -4 by 0.
f(0)=0+0
f(0)=0+0
Add 0 and 0.
f(0)=0
The final answer is 0.
0
0
The y value at x=0 is 0.
y=0
Replace the variable x with 1 in the expression.
f(1)=-3(1)2-4⋅1
Simplify the result.
Simplify each term.
One to any power is one.
f(1)=-3⋅1-4⋅1
Multiply -3 by 1.
f(1)=-3-4⋅1
Multiply -4 by 1.
f(1)=-3-4
f(1)=-3-4
Subtract 4 from -3.
f(1)=-7
The final answer is -7.
-7
-7
The y value at x=1 is -7.
y=-7
Graph the parabola using its properties and the selected points.
xy-3-15-2-4-2343001-7
xy-3-15-2-4-2343001-7
Graph the parabola using its properties and the selected points.
Direction: Opens Down
Vertex: (-23,43)
Focus: (-23,54)
Axis of Symmetry: x=-23
Directrix: y=1712
xy-3-15-2-4-2343001-7
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