# Graph f(x)=x^2+7x+9 f(x)=x2+7x+9
Find the properties of the given parabola.
Rewrite the equation in vertex form.
Complete the square for x2+7x+9.
Use the form ax2+bx+c, to find the values of a, b, and c.
a=1,b=7,c=9
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=72(1)
Multiply 2 by 1.
d=72
Find the value of e using the formula e=c-b24a.
Simplify each term.
Raise 7 to the power of 2.
e=9-494⋅1
Multiply 4 by 1.
e=9-494
e=9-494
To write 9 as a fraction with a common denominator, multiply by 44.
e=9⋅44-494
Combine 9 and 44.
e=9⋅44-494
Combine the numerators over the common denominator.
e=9⋅4-494
Simplify the numerator.
Multiply 9 by 4.
e=36-494
Subtract 49 from 36.
e=-134
e=-134
Move the negative in front of the fraction.
e=-134
e=-134
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
(x+72)2-134
(x+72)2-134
Set y equal to the new right side.
y=(x+72)2-134
y=(x+72)2-134
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=1
h=-72
k=-134
Since the value of a is positive, the parabola opens up.
Opens Up
Find the vertex (h,k).
(-72,-134)
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.
Cancel the common factor.
14⋅1
Rewrite the expression.
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.
(-72,-3)
(-72,-3)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=-72
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=-72
y=-72
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Up
Vertex: (-72,-134)
Focus: (-72,-3)
Axis of Symmetry: x=-72
Directrix: y=-72
Direction: Opens Up
Vertex: (-72,-134)
Focus: (-72,-3)
Axis of Symmetry: x=-72
Directrix: y=-72
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 -5 in the expression.
f(-5)=(-5)2+7(-5)+9
Simplify the result.
Simplify each term.
Raise -5 to the power of 2.
f(-5)=25+7(-5)+9
Multiply 7 by -5.
f(-5)=25-35+9
f(-5)=25-35+9
Subtract 35 from 25.
f(-5)=-10+9
f(-5)=-1
f(-5)=-1
-1
-1
The y value at x=-5 is -1.
y=-1
Replace the variable x with -6 in the expression.
f(-6)=(-6)2+7(-6)+9
Simplify the result.
Simplify each term.
Raise -6 to the power of 2.
f(-6)=36+7(-6)+9
Multiply 7 by -6.
f(-6)=36-42+9
f(-6)=36-42+9
Subtract 42 from 36.
f(-6)=-6+9
f(-6)=3
f(-6)=3
3
3
The y value at x=-6 is 3.
y=3
Replace the variable x with -3 in the expression.
f(-3)=(-3)2+7(-3)+9
Simplify the result.
Simplify each term.
Raise -3 to the power of 2.
f(-3)=9+7(-3)+9
Multiply 7 by -3.
f(-3)=9-21+9
f(-3)=9-21+9
Subtract 21 from 9.
f(-3)=-12+9
f(-3)=-3
f(-3)=-3
-3
-3
The y value at x=-3 is -3.
y=-3
Replace the variable x with -2 in the expression.
f(-2)=(-2)2+7(-2)+9
Simplify the result.
Simplify each term.
Raise -2 to the power of 2.
f(-2)=4+7(-2)+9
Multiply 7 by -2.
f(-2)=4-14+9
f(-2)=4-14+9
Subtract 14 from 4.
f(-2)=-10+9
f(-2)=-1
f(-2)=-1
-1
-1
The y value at x=-2 is -1.
y=-1
Graph the parabola using its properties and the selected points.
xy-63-5-1-72-134-3-3-2-1
xy-63-5-1-72-134-3-3-2-1
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex: (-72,-134)
Focus: (-72,-3)
Axis of Symmetry: x=-72
Directrix: y=-72
xy-63-5-1-72-134-3-3-2-1
Graph f(x)=x^2+7x+9     