j(x)=2×2-5x+4

Rewrite the equation in vertex form.

Complete the square for 2×2-5x+4.

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

a=2,b=-5,c=4

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=-52(2)

Multiply 2 by 2.

d=-54

Find the value of e using the formula e=c-b24a.

Simplify each term.

Raise -5 to the power of 2.

e=4-254⋅2

Multiply 4 by 2.

e=4-258

e=4-258

To write 4 as a fraction with a common denominator, multiply by 88.

e=4⋅88-258

Combine 4 and 88.

e=4⋅88-258

Combine the numerators over the common denominator.

e=4⋅8-258

Simplify the numerator.

Multiply 4 by 8.

e=32-258

Subtract 25 from 32.

e=78

e=78

e=78

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

2(x-54)2+78

2(x-54)2+78

Set y equal to the new right side.

y=2(x-54)2+78

y=2(x-54)2+78

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

a=2

h=54

k=78

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

Opens Up

Find the vertex (h,k).

(54,78)

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⋅2

Multiply 4 by 2.

18

18

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.

(54,1)

(54,1)

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

x=54

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=34

y=34

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

Direction: Opens Up

Vertex: (54,78)

Focus: (54,1)

Axis of Symmetry: x=54

Directrix: y=34

Direction: Opens Up

Vertex: (54,78)

Focus: (54,1)

Axis of Symmetry: x=54

Directrix: y=34

Replace the variable x with 0 in the expression.

f(0)=2(0)2-5⋅0+4

Simplify the result.

Simplify each term.

Raising 0 to any positive power yields 0.

f(0)=2⋅0-5⋅0+4

Multiply 2 by 0.

f(0)=0-5⋅0+4

Multiply -5 by 0.

f(0)=0+0+4

f(0)=0+0+4

Simplify by adding zeros.

Add 0 and 0.

f(0)=0+4

Add 0 and 4.

f(0)=4

f(0)=4

The final answer is 4.

4

4

The y value at x=0 is 4.

y=4

Replace the variable x with -1 in the expression.

f(-1)=2(-1)2-5⋅-1+4

Simplify the result.

Simplify each term.

Raise -1 to the power of 2.

f(-1)=2⋅1-5⋅-1+4

Multiply 2 by 1.

f(-1)=2-5⋅-1+4

Multiply -5 by -1.

f(-1)=2+5+4

f(-1)=2+5+4

Simplify by adding numbers.

Add 2 and 5.

f(-1)=7+4

Add 7 and 4.

f(-1)=11

f(-1)=11

The final answer is 11.

11

11

The y value at x=-1 is 11.

y=11

Replace the variable x with 2 in the expression.

f(2)=2(2)2-5⋅2+4

Simplify the result.

Simplify each term.

Multiply 2 by (2)2 by adding the exponents.

Multiply 2 by (2)2.

Raise 2 to the power of 1.

f(2)=2(2)2-5⋅2+4

Use the power rule aman=am+n to combine exponents.

f(2)=21+2-5⋅2+4

f(2)=21+2-5⋅2+4

Add 1 and 2.

f(2)=23-5⋅2+4

f(2)=23-5⋅2+4

Raise 2 to the power of 3.

f(2)=8-5⋅2+4

Multiply -5 by 2.

f(2)=8-10+4

f(2)=8-10+4

Simplify by adding and subtracting.

Subtract 10 from 8.

f(2)=-2+4

Add -2 and 4.

f(2)=2

f(2)=2

The final answer is 2.

2

2

The y value at x=2 is 2.

y=2

Replace the variable x with 3 in the expression.

f(3)=2(3)2-5⋅3+4

Simplify the result.

Simplify each term.

Raise 3 to the power of 2.

f(3)=2⋅9-5⋅3+4

Multiply 2 by 9.

f(3)=18-5⋅3+4

Multiply -5 by 3.

f(3)=18-15+4

f(3)=18-15+4

Simplify by adding and subtracting.

Subtract 15 from 18.

f(3)=3+4

Add 3 and 4.

f(3)=7

f(3)=7

The final answer is 7.

7

7

The y value at x=3 is 7.

y=7

Graph the parabola using its properties and the selected points.

xy-1110454782237

xy-1110454782237

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: (54,78)

Focus: (54,1)

Axis of Symmetry: x=54

Directrix: y=34

xy-1110454782237

Graph j(x)=2x^2-5x+4