Graph y=-x^2-6x+34

Math
y=-x2-6x+34
Find the properties of the given parabola.
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Rewrite the equation in vertex form.
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Complete the square for -x2-6x+34.
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Use the form ax2+bx+c, to find the values of a, b, and c.
a=-1,b=-6,c=34
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(-1)
Simplify the right side.
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Cancel the common factor of 6 and 2.
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Factor 2 out of 6.
d=-2⋅32⋅-1
Move the negative one from the denominator of 3-1.
d=-(-1⋅3)
d=-(-1⋅3)
Multiply.
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Multiply -1 by 3.
d=3
Multiply -1 by -3.
d=3
d=3
d=3
Find the value of e using the formula e=c-b24a.
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Simplify each term.
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Raise -6 to the power of 2.
e=34-364⋅-1
Multiply 4 by -1.
e=34-36-4
Divide 36 by -4.
e=34+9
Multiply -1 by -9.
e=34+9
e=34+9
Add 34 and 9.
e=43
e=43
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
-(x+3)2+43
-(x+3)2+43
Set y equal to the new right side.
y=-(x+3)2+43
y=-(x+3)2+43
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=-1
h=-3
k=43
Since the value of a is negative, the parabola opens down.
Opens Down
Find the vertex (h,k).
(-3,43)
Find p, the distance from the vertex to the focus.
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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⋅-1
Cancel the common factor of 1 and -1.
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Rewrite 1 as -1(-1).
-1(-1)4⋅-1
Move the negative in front of the fraction.
-14
-14
-14
Find the focus.
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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.
(-3,1714)
(-3,1714)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=-3
Find the directrix.
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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=1734
y=1734
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Down
Vertex: (-3,43)
Focus: (-3,1714)
Axis of Symmetry: x=-3
Directrix: y=1734
Direction: Opens Down
Vertex: (-3,43)
Focus: (-3,1714)
Axis of Symmetry: x=-3
Directrix: y=1734
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.
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Replace the variable x with -4 in the expression.
f(-4)=-(-4)2-6⋅-4+34
Simplify the result.
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Simplify each term.
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Raise -4 to the power of 2.
f(-4)=-1⋅16-6⋅-4+34
Multiply -1 by 16.
f(-4)=-16-6⋅-4+34
Multiply -6 by -4.
f(-4)=-16+24+34
f(-4)=-16+24+34
Simplify by adding numbers.
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Add -16 and 24.
f(-4)=8+34
Add 8 and 34.
f(-4)=42
f(-4)=42
The final answer is 42.
42
42
The y value at x=-4 is 42.
y=42
Replace the variable x with -5 in the expression.
f(-5)=-(-5)2-6⋅-5+34
Simplify the result.
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Simplify each term.
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Raise -5 to the power of 2.
f(-5)=-1⋅25-6⋅-5+34
Multiply -1 by 25.
f(-5)=-25-6⋅-5+34
Multiply -6 by -5.
f(-5)=-25+30+34
f(-5)=-25+30+34
Simplify by adding numbers.
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Add -25 and 30.
f(-5)=5+34
Add 5 and 34.
f(-5)=39
f(-5)=39
The final answer is 39.
39
39
The y value at x=-5 is 39.
y=39
Replace the variable x with -2 in the expression.
f(-2)=-(-2)2-6⋅-2+34
Simplify the result.
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Simplify each term.
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Raise -2 to the power of 2.
f(-2)=-1⋅4-6⋅-2+34
Multiply -1 by 4.
f(-2)=-4-6⋅-2+34
Multiply -6 by -2.
f(-2)=-4+12+34
f(-2)=-4+12+34
Simplify by adding numbers.
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Add -4 and 12.
f(-2)=8+34
Add 8 and 34.
f(-2)=42
f(-2)=42
The final answer is 42.
42
42
The y value at x=-2 is 42.
y=42
Replace the variable x with -1 in the expression.
f(-1)=-(-1)2-6⋅-1+34
Simplify the result.
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Simplify each term.
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Multiply -1 by (-1)2 by adding the exponents.
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Multiply -1 by (-1)2.
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Raise -1 to the power of 1.
f(-1)=(-1)(-1)2-6⋅-1+34
Use the power rule aman=am+n to combine exponents.
f(-1)=(-1)1+2-6⋅-1+34
f(-1)=(-1)1+2-6⋅-1+34
Add 1 and 2.
f(-1)=(-1)3-6⋅-1+34
f(-1)=(-1)3-6⋅-1+34
Raise -1 to the power of 3.
f(-1)=-1-6⋅-1+34
Multiply -6 by -1.
f(-1)=-1+6+34
f(-1)=-1+6+34
Simplify by adding numbers.
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Add -1 and 6.
f(-1)=5+34
Add 5 and 34.
f(-1)=39
f(-1)=39
The final answer is 39.
39
39
The y value at x=-1 is 39.
y=39
Graph the parabola using its properties and the selected points.
xy-539-442-343-242-139
xy-539-442-343-242-139
Graph the parabola using its properties and the selected points.
Direction: Opens Down
Vertex: (-3,43)
Focus: (-3,1714)
Axis of Symmetry: x=-3
Directrix: y=1734
xy-539-442-343-242-139
Graph y=-x^2-6x+34

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