# Graph y=1/2x^2+2x-3 y=12×2+2x-3
Combine 12 and x2.
y=x22+2x-3
Find the properties of the given parabola.
Rewrite the equation in vertex form.
Complete the square for x22+2x-3.
Use the form ax2+bx+c, to find the values of a, b, and c.
a=12,b=2,c=-3
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=22(12)
Simplify the right side.
Cancel the common factor of 2.
Cancel the common factor.
d=22(12)
Rewrite the expression.
d=112
d=112
Multiply the numerator by the reciprocal of the denominator.
d=1⋅2
Multiply 2 by 1.
d=2
d=2
Find the value of e using the formula e=c-b24a.
Simplify each term.
Raise 2 to the power of 2.
e=-3-44(12)
Combine 4 and 12.
e=-3-442
Divide 4 by 2.
e=-3-42
Divide 4 by 2.
e=-3-1⋅2
Multiply -1 by 2.
e=-3-2
e=-3-2
Subtract 2 from -3.
e=-5
e=-5
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
12⋅(x+2)2-5
12⋅(x+2)2-5
Set y equal to the new right side.
y=12⋅(x+2)2-5
y=12⋅(x+2)2-5
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=12
h=-2
k=-5
Since the value of a is positive, the parabola opens up.
Opens Up
Find the vertex (h,k).
(-2,-5)
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⋅12
Simplify.
Combine 4 and 12.
142
Divide 4 by 2.
12
12
12
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.
(-2,-92)
(-2,-92)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=-2
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=-112
y=-112
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Up
Vertex: (-2,-5)
Focus: (-2,-92)
Axis of Symmetry: x=-2
Directrix: y=-112
Direction: Opens Up
Vertex: (-2,-5)
Focus: (-2,-92)
Axis of Symmetry: x=-2
Directrix: y=-112
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)22+2(-4)-3
Simplify the result.
Simplify each term.
Raise -4 to the power of 2.
f(-4)=162+2(-4)-3
Divide 16 by 2.
f(-4)=8+2(-4)-3
Multiply 2 by -4.
f(-4)=8-8-3
f(-4)=8-8-3
Simplify by subtracting numbers.
Subtract 8 from 8.
f(-4)=0-3
Subtract 3 from 0.
f(-4)=-3
f(-4)=-3
The final answer is -3.
-3
-3
The y value at x=-4 is -3.
y=-3
Replace the variable x with -3 in the expression.
f(-3)=(-3)22+2(-3)-3
Simplify the result.
Find the common denominator.
Write 2(-3) as a fraction with denominator 1.
f(-3)=(-3)22+2(-3)1-3
Multiply 2(-3)1 by 22.
f(-3)=(-3)22+2(-3)1⋅22-3
Multiply 2(-3)1 and 22.
f(-3)=(-3)22+2(-3)⋅22-3
Write -3 as a fraction with denominator 1.
f(-3)=(-3)22+2(-3)⋅22+-31
Multiply -31 by 22.
f(-3)=(-3)22+2(-3)⋅22+-31⋅22
Multiply -31 and 22.
f(-3)=(-3)22+2(-3)⋅22+-3⋅22
f(-3)=(-3)22+2(-3)⋅22+-3⋅22
Combine fractions.
Combine fractions with similar denominators.
f(-3)=(-3)2+2(-3)⋅2-3⋅22
Multiply.
Multiply 2 by -3.
f(-3)=(-3)2-6⋅2-3⋅22
Multiply -6 by 2.
f(-3)=(-3)2-12-3⋅22
Multiply -3 by 2.
f(-3)=(-3)2-12-62
f(-3)=(-3)2-12-62
f(-3)=(-3)2-12-62
Simplify the numerator.
Raise -3 to the power of 2.
f(-3)=9-12-62
Subtract 12 from 9.
f(-3)=-3-62
Subtract 6 from -3.
f(-3)=-92
f(-3)=-92
Move the negative in front of the fraction.
f(-3)=-92
The final answer is -92.
-92
-92
The y value at x=-3 is -92.
y=-92
Replace the variable x with 0 in the expression.
f(0)=(0)22+2(0)-3
Simplify the result.
Simplify each term.
Raising 0 to any positive power yields 0.
f(0)=02+2(0)-3
Divide 0 by 2.
f(0)=0+2(0)-3
Multiply 2 by 0.
f(0)=0+0-3
f(0)=0+0-3
Simplify by adding zeros.
Add 0 and 0.
f(0)=0-3
Subtract 3 from 0.
f(0)=-3
f(0)=-3
The final answer is -3.
-3
-3
The y value at x=0 is -3.
y=-3
Replace the variable x with -1 in the expression.
f(-1)=(-1)22+2(-1)-3
Simplify the result.
Simplify each term.
Raise -1 to the power of 2.
f(-1)=12+2(-1)-3
Multiply 2 by -1.
f(-1)=12-2-3
f(-1)=12-2-3
Find the common denominator.
Write -2 as a fraction with denominator 1.
f(-1)=12+-21-3
Multiply -21 by 22.
f(-1)=12+-21⋅22-3
Multiply -21 and 22.
f(-1)=12+-2⋅22-3
Write -3 as a fraction with denominator 1.
f(-1)=12+-2⋅22+-31
Multiply -31 by 22.
f(-1)=12+-2⋅22+-31⋅22
Multiply -31 and 22.
f(-1)=12+-2⋅22+-3⋅22
f(-1)=12+-2⋅22+-3⋅22
Combine fractions.
Combine fractions with similar denominators.
f(-1)=1-2⋅2-3⋅22
Multiply.
Multiply -2 by 2.
f(-1)=1-4-3⋅22
Multiply -3 by 2.
f(-1)=1-4-62
f(-1)=1-4-62
f(-1)=1-4-62
Simplify the numerator.
Subtract 4 from 1.
f(-1)=-3-62
Subtract 6 from -3.
f(-1)=-92
f(-1)=-92
Move the negative in front of the fraction.
f(-1)=-92
The final answer is -92.
-92
-92
The y value at x=-1 is -92.
y=-92
Graph the parabola using its properties and the selected points.
xy-4-3-3-92-2-5-1-920-3
xy-4-3-3-92-2-5-1-920-3
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex: (-2,-5)
Focus: (-2,-92)
Axis of Symmetry: x=-2
Directrix: y=-112
xy-4-3-3-92-2-5-1-920-3
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