Question

Find the horizontal intercepts of the quadratic by rewriting the quadratic into transformation form.
f(x) = -4x^2+10x-5

Answer 1

@jtryon, are you here?

Answer 2

yes, I am here

Answer 3

You are given
f(x) = -4x^2 + 10x - 5

and you want to write the quadratic in transformational form, correct?

Answer 4

correct, now I know that one way to find the x-intercepts is to use the quadratic formula but I saw in my textbook that you can also find the x-int. once you write the quadratic in transformational form but I am not understanding that part

Answer 5

The standard form of a quadratic equation is \(y = a(x - h)^2 + k\) and the transformational form is \(\dfrac{1}{a}(y - k) = (x - h)^2\) correct?

Answer 6

correct

Answer 7

Which means that if we convert the given quadratic to standard form, it should be relatively easy to convert it to transformational form, right?

Answer 8

no, wait

Answer 9

the standard form is defined as ax^2+bx+c and the transformational form is a(x-h)^2+k

Answer 10

yes, converting it to transformational form was relatively easy, but with the vertex that I found at 5/4, it was then difficult to plug that back in and come up with the x-intercepts

Answer 11

The example that they have given use a whole integer vertex of -1 so that is easy to plug in

Answer 12

What did you get for the transformational form of f(x) = -4x^2+10x-5 ?

Answer 13

f(x) = -4(x+1)^2+1.25

Answer 14

I think that is where I may have made and error

Answer 15

@Hero

Answer 16

Would you mind showing your steps please?

Answer 17

Sure
-4(5/4)^2+10(5/4) - 5
-6.25 + 12.5 - 5 = 1.25

Answer 18

I am not certain about the (x+1) in the transformational form as I still have only a slight understanding of this form

Answer 19

If I got the idea of transformational form, I''m thinking it would be easier to find the x-intercepts than having to use the quadratic formula

Answer 20

I got something different for the transformational form

Answer 21

@jtryon

Answer 22

What did you get? I probably made a mistake somewhere

Answer 23

The first step I did was solve for my h which is 1.25 and then my k which is also 1.25

Answer 24

There are a specific set of steps that you perform to convert from standard form to vertex form.

Answer 25

The transformational form that they are describing here is also known as the vertex form of a quadratic equation.

Answer 26

First, @jtryon, since y = f(x), we can write the given quadratic equation as

\(y = -4x^2 + 10x - 5\)

Answer 27

Yes, that is what I like to do instead of have it as f(x) or h(r)

Answer 28

It helps me to call it y

Answer 29

Next what we can do is multiply (or divide) both sides by -1 to get
\(-y = 4x^2 - 10x + 5\)

Do you agree?

Answer 30

Yes

Answer 31

Why do we want to change it to -y? I understand the process of multiplying both sides by -1

Answer 32

You'll see...

Afterwards, we can factor 4 from the quadratic expression to get
\(-y = 4(x^2 - \dfrac{10}{4}x + \dfrac{5}{4})\)

agreed?

Answer 33

Ok, continue

Answer 34

Now we divide both sides by 4 to get

\(-\dfrac{y}{4} = x^2 - \dfrac{10}{4}x + \dfrac{5}{4}\)

Answer 35

Since \(\dfrac{10}{4} = \dfrac{5}{2}\) we can re-write the previous step as:

\(-\dfrac{y}{4} = x^2 -\dfrac{5}{2}x + \dfrac{5}{4}\)

Agreed?

Answer 36

That makes sense

Answer 37

Now we will subtract \(\dfrac{5}{4}\) from both sides to get

\(-\dfrac{y}{4} - \dfrac{5}{4} = x^2 - \dfrac{5}{2}x\)

Which leads us to the whole point of doing all that manipulation. Notice that we have the form \(x^2 + bx\) on the right side. Do you know what this means?

Answer 38

That is the standard form?

Answer 39

Not quite. But if we add \(\left(\frac{b}{2}\right)^2\) to both sides, we'll end up with a complete square on the right side. Does the concept of "completing the square" sound familiar to you?

Answer 40

Yes, it is familiar to me. I have not used that in awhile.

Answer 41

How do we get the x-intercepts when we have changed from standard form to transformational form?

Answer 42

Once we finish converting to transformational/vertex form, we'll be able to find these x-intercepts you refer to.

Answer 43

The interesting thing is that converting to tranformational form is the complete long way to find x-intercepts.

Answer 44

There is a series of steps you have to complete before you convert to transformational form so I am wondering if it is faster to use the quadratic formula

Answer 45

It is relatively easy to identify a,b,c from the terms in the quadratic

Answer 46

Oh, so it is a long way of finding the x-intercepts

Answer 47

Basically, if your goal was to find the x-intercepts, converting from general form to transformational form is about the most time consuming possible method to use. Especially considering this quadratic.

Answer 48

They made it seem really simple in the textbook using the following example:
y = 2x^2+4x-4
0 = 2x^2 + 4x - 4
h = b/2a = 4/2(2) = -1
k = f(-1) = 2(-1)^2+4(-1)-4=6

0 = 2(x+1)^2-6
6 = 2(x+1)^2
3 = (x+1)^2
x+1=+- sqrt(3)
x = -1+- sqrt(3)

Answer 49

It wasn't that simple when I attempted to use it with the vertex of (1.25, 1.25)

Answer 50

Scan the page from your textbook and upload it here so I can see exactly how they typed out the steps.

Answer 51

@Hero

Answer 52

But yeah, they are right. The method to find (h,k) is much faster than completing the square.

Answer 53

Because (h,k) is the vertex of the parabola. So actually, it is silly to complete the square.

Answer 54

Basically the vertex (transformational) form of a quadratic is

\(y = (x - h)^2 + k\)

\(h = -\dfrac{b}{2a}\)

\(k = c - \dfrac{b^2}{4a}\)

You are given
\(f(x) = -4x^2 + 10x - 5\)

And a = -4, b = 10, c = -5

First solve for h:
\(h = -\dfrac{10}{2(-4)} = \dfrac{10}{8} = \dfrac{5}{4}\)

Then solve for k:
\(k = -5 - \dfrac{10^2}{4(-4)} = -5 + \dfrac{100}{16} = -5 + \dfrac{25}{4} = -\dfrac{20}{4} + \dfrac{25}{4} = \dfrac{5}{4}\)

Therefore the transformational form of f(x) = -4x^2 + 10x - 5 is
\(f(x) = -4\left(x - \dfrac{5}{4}\right)^2 + \dfrac{5}{4}\)

Check:
http://www.wolframalpha.com/input/?i=-4x%5E2+%2B+10x+-+5+%3D+-4%28x+-+5%2F4%29%5E2+%2B+5%2F4