Express a Function in Vertex Form

[BeccaP]

I need to express the function:

f(x)= -5x² + 20x + 2

In Vertex Form.

I have attempted to do it two different ways, but am unsure if I am doing it correctly.

Attempt 1: Perfect Square

f(x)= -5x²+ 20x + 2

= -5 ( x² - 4x ) + 2

b² = [1/2 (-4)]²
= 2²
= 4

= -5 ( x² -4x + 4 - 4) + 2

= -5 ( x²- 4x + 4) - 20 + 2

= -5 (x-2)²- 18

The vertex is (4, -18). The parabola is concave down, maximum is y= -18.

Attempt 2: Partial Factoring

f(x)= -5x²+ 20x + 2

= -5x (x-4) + 2 [1]


-5x (x-4) = 0

-5x=0 or (x-4)=0

x=0 x= 4

Insert x=0 to find y
y= -5x (x-4) + 2

=-5(0) (0-4) + 2

= 0 - 4 + 2

= -4 + 2

= -2

Insert x=4 to find y

y= -5x (x-4) + 2

= -5(4) (4-4) + 2

= -20 + 2

= -18

Thus the mid points are : (0, -2) and (4, -18)

Find Mid Value

0+4
2

= 4
2

= 2

Insert 2 into equation

y= -5x²+ 20x + 2

= -5(2)^{2 + 20(2) + 2}

= -5 (4) + 40 + 2

= -20 + 40 + 2

= 22

Thus the vertex is (2, 22)

Any guidance would be helpful.

Thank you

 

[pka]

I need to express the function:
f(x)= -5x² + 20x + 2 In Vertex Form.

.
Just do in one step.
f(x)=-5x^2+20x+2=-5(x-2)^2+22.

 

[BeccaP]

.
Just do in one step.
f(x)=-5x^2+20x+2=-5(x-2)^2+22.

So the partial factoring method is the correct way to do the problem?

 

[pka]

So the partial factoring method is the correct way to do the problem?

There no one correct way.
If you are good at completing squares, that is the quickest way.

 

[Mrspi]

I need to express the function:

f(x)= -5x² + 20x + 2

In Vertex Form.

I have attempted to do it two different ways, but am unsure if I am doing it correctly.

Attempt 1: Perfect Square

f(x)= -5x²+ 20x + 2

= -5 ( x² - 4x ) + 2

b² = [1/2 (-4)]²
= 2²
= 4

= -5 ( x² -4x + 4 - 4) + 2

= -5 ( x²- 4x + 4) - 20 + 2 ERROR HERE! -5(-4) is + 20, so you should have -5(x² - 4x + 4) + 20 + 2, or -5(x² - 4x + 4) + 22

= -5 (x-2)²- 18 Should be -5(x - 2)² + 22

The vertex is (4, -18). The parabola is concave down, maximum is y= -18. "vertex form" is y = a(x - h)² + k; the vertex is at (h, k). So the correct vertex should be (2, 22)

Attempt 2: Partial Factoring

f(x)= -5x²+ 20x + 2

= -5x (x-4) + 2 [1]


-5x (x-4) = 0

-5x=0 or (x-4)=0

x=0 x= 4

Insert x=0 to find y
y= -5x (x-4) + 2

=-5(0) (0-4) + 2

= 0 - 4 + 2

= -4 + 2

= -2

Insert x=4 to find y

y= -5x (x-4) + 2

= -5(4) (4-4) + 2

= -20 + 2

= -18

Thus the mid points are : (0, -2) and (4, -18)

Find Mid Value

0+4
2

= 4
2

= 2

Insert 2 into equation

y= -5x²+ 20x + 2

= -5(2)^{2 + 20(2) + 2}

= -5 (4) + 40 + 2

= -20 + 40 + 2

= 22

Thus the vertex is (2, 22)

Any guidance would be helpful.

Thank you

You should get the correct vertex using either method, PROVIDING that you are careful with your arithmetic. See my comments in red above.