# Graph q(x)=-(x-5)^2-6 q(x)=-(x-5)2-6
Find the properties of the given parabola.
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=-1
h=5
k=-6
Since the value of a is negative, the parabola opens down.
Opens Down
Find the vertex (h,k).
(5,-6)
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⋅-1
Cancel the common factor of 1 and -1.
Rewrite 1 as -1(-1).
-1(-1)4⋅-1
Move the negative in front of the fraction.
-14
-14
-14
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.
(5,-254)
(5,-254)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=5
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=-234
y=-234
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Down
Vertex: (5,-6)
Focus: (5,-254)
Axis of Symmetry: x=5
Directrix: y=-234
Direction: Opens Down
Vertex: (5,-6)
Focus: (5,-254)
Axis of Symmetry: x=5
Directrix: y=-234
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 4 in the expression.
f(4)=-(4)2+10(4)-31
Simplify the result.
Simplify each term.
Raise 4 to the power of 2.
f(4)=-1⋅16+10(4)-31
Multiply -1 by 16.
f(4)=-16+10(4)-31
Multiply 10 by 4.
f(4)=-16+40-31
f(4)=-16+40-31
Simplify by adding and subtracting.
Add -16 and 40.
f(4)=24-31
Subtract 31 from 24.
f(4)=-7
f(4)=-7
The final answer is -7.
-7
-7
The y value at x=4 is -7.
y=-7
Replace the variable x with 3 in the expression.
f(3)=-(3)2+10(3)-31
Simplify the result.
Simplify each term.
Raise 3 to the power of 2.
f(3)=-1⋅9+10(3)-31
Multiply -1 by 9.
f(3)=-9+10(3)-31
Multiply 10 by 3.
f(3)=-9+30-31
f(3)=-9+30-31
Simplify by adding and subtracting.
Add -9 and 30.
f(3)=21-31
Subtract 31 from 21.
f(3)=-10
f(3)=-10
The final answer is -10.
-10
-10
The y value at x=3 is -10.
y=-10
Replace the variable x with 6 in the expression.
f(6)=-(6)2+10(6)-31
Simplify the result.
Simplify each term.
Raise 6 to the power of 2.
f(6)=-1⋅36+10(6)-31
Multiply -1 by 36.
f(6)=-36+10(6)-31
Multiply 10 by 6.
f(6)=-36+60-31
f(6)=-36+60-31
Simplify by adding and subtracting.
Add -36 and 60.
f(6)=24-31
Subtract 31 from 24.
f(6)=-7
f(6)=-7
The final answer is -7.
-7
-7
The y value at x=6 is -7.
y=-7
Replace the variable x with 7 in the expression.
f(7)=-(7)2+10(7)-31
Simplify the result.
Simplify each term.
Raise 7 to the power of 2.
f(7)=-1⋅49+10(7)-31
Multiply -1 by 49.
f(7)=-49+10(7)-31
Multiply 10 by 7.
f(7)=-49+70-31
f(7)=-49+70-31
Simplify by adding and subtracting.
Add -49 and 70.
f(7)=21-31
Subtract 31 from 21.
f(7)=-10
f(7)=-10
The final answer is -10.
-10
-10
The y value at x=7 is -10.
y=-10
Graph the parabola using its properties and the selected points.
xy3-104-75-66-77-10
xy3-104-75-66-77-10
Graph the parabola using its properties and the selected points.
Direction: Opens Down
Vertex: (5,-6)
Focus: (5,-254)
Axis of Symmetry: x=5
Directrix: y=-234
xy3-104-75-66-77-10
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