# Graph j(x)=2x^2-5x+4 j(x)=2×2-5x+4
Find the properties of the given parabola.
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
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 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
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