y=x2-4x+10

Rewrite the equation in vertex form.

Complete the square for x2-4x+10.

Use the form ax2+bx+c, to find the values of a, b, and c.

a=1,b=-4,c=10

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(1)

Simplify the right side.

Cancel the common factor of 4 and 2.

Factor 2 out of 4.

d=-2⋅22⋅1

Cancel the common factors.

Factor 2 out of 2⋅1.

d=-2⋅22(1)

Cancel the common factor.

d=-2⋅22⋅1

Rewrite the expression.

d=-21

Divide 2 by 1.

d=-1⋅2

d=-1⋅2

d=-1⋅2

Multiply -1 by 2.

d=-2

d=-2

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=10-(-1⋅4)24(1)

Apply the product rule to -1(4).

e=10-(-1)2⋅424(1)

Raise -1 to the power of 2.

e=10-1⋅424(1)

Multiply 42 by 1.

e=10-424(1)

Factor 4 out of 42.

e=10-4⋅44⋅1

Cancel the common factors.

Factor 4 out of 4⋅1.

e=10-4⋅44(1)

Cancel the common factor.

e=10-4⋅44⋅1

Rewrite the expression.

e=10-41

Divide 4 by 1.

e=10-1⋅4

e=10-1⋅4

e=10-1⋅4

Multiply -1 by 4.

e=10-4

e=10-4

Subtract 4 from 10.

e=6

e=6

Substitute the values of a, d, and e into the vertex form a(x+d)2+e.

(x-2)2+6

(x-2)2+6

Set y equal to the new right side.

y=(x-2)2+6

y=(x-2)2+6

Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.

a=1

h=2

k=6

Since the value of a is positive, the parabola opens up.

Opens Up

Find the vertex (h,k).

(2,6)

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.

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.

(2,254)

(2,254)

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=234

y=234

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Up

Vertex: (2,6)

Focus: (2,254)

Axis of Symmetry: x=2

Directrix: y=234

Direction: Opens Up

Vertex: (2,6)

Focus: (2,254)

Axis of Symmetry: x=2

Directrix: y=234

Replace the variable x with 1 in the expression.

f(1)=(1)2-4⋅1+10

Simplify the result.

Simplify each term.

One to any power is one.

f(1)=1-4⋅1+10

Multiply -4 by 1.

f(1)=1-4+10

f(1)=1-4+10

Simplify by adding and subtracting.

Subtract 4 from 1.

f(1)=-3+10

Add -3 and 10.

f(1)=7

f(1)=7

The final answer is 7.

7

7

The y value at x=1 is 7.

y=7

Replace the variable x with 0 in the expression.

f(0)=(0)2-4⋅0+10

Simplify the result.

Simplify each term.

Raising 0 to any positive power yields 0.

f(0)=0-4⋅0+10

Multiply -4 by 0.

f(0)=0+0+10

f(0)=0+0+10

Simplify by adding zeros.

Add 0 and 0.

f(0)=0+10

Add 0 and 10.

f(0)=10

f(0)=10

The final answer is 10.

10

10

The y value at x=0 is 10.

y=10

Replace the variable x with 3 in the expression.

f(3)=(3)2-4⋅3+10

Simplify the result.

Simplify each term.

Raise 3 to the power of 2.

f(3)=9-4⋅3+10

Multiply -4 by 3.

f(3)=9-12+10

f(3)=9-12+10

Simplify by adding and subtracting.

Subtract 12 from 9.

f(3)=-3+10

Add -3 and 10.

f(3)=7

f(3)=7

The final answer is 7.

7

7

The y value at x=3 is 7.

y=7

Replace the variable x with 4 in the expression.

f(4)=(4)2-4⋅4+10

Simplify the result.

Simplify each term.

Raise 4 to the power of 2.

f(4)=16-4⋅4+10

Multiply -4 by 4.

f(4)=16-16+10

f(4)=16-16+10

Simplify by subtracting numbers.

Subtract 16 from 16.

f(4)=0+10

Add 0 and 10.

f(4)=10

f(4)=10

The final answer is 10.

10

10

The y value at x=4 is 10.

y=10

Graph the parabola using its properties and the selected points.

xy010172637410

xy010172637410

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: (2,6)

Focus: (2,254)

Axis of Symmetry: x=2

Directrix: y=234

xy010172637410

Graph y=x^2-4x+10