SOLUTION: Mario is going to fly through this scene, following the path of the quadratic function 𝑓(𝑥). Based on the table and graph, you will write equations for this quadratic functi

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: Mario is going to fly through this scene, following the path of the quadratic function 𝑓(𝑥). Based on the table and graph, you will write equations for this quadratic functi      Log On


   

Question 1194535: Mario is going to fly through this scene, following the path of the quadratic function 𝑓(𝑥). Based on the
table and graph, you will write equations for this quadratic function in each of three forms: standard
form, vertex form, and factored form. You may start with any form you choose, but you need to write
equivalent equations in all three forms.
equivalent equations in all three forms.
𝑥 𝑓(𝑥)
0 3
1 1.25
2 0
3 -0.75
4 -1
5 -0.75
6 0
7 1.25
8 3
1.
Found 4 solutions by greenestamps, MathLover1, math_tutor2020, MathTherapy:
Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


The two zeros are given as x=2 and x=6.

You could start with that, knowing that the equation is of the form

f%28x%29=a%28x-2%29%28x-6%29

Then determine the value of a using any of the given points. e.g.,

f%280%29=a%28-2%29%28-6%29=12a=3 --> a=1%2F4

Then one form of the equation is

f%28x%29=%281%2F4%29%28x-2%29%28x-6%29

Reformat it into any form you want.

Alternatively, with the zeros at x=2 and x=6, you know the axis of symmetry is x=4; and since f(4)=-1, you know the vertex is (4,-1). Then you could start with vertex form,

f%28x%29=a%28x-h%29%5E2%2Bk
f%28x%29=a%28x-4%29%5E2-1

Again determine the value of a by using any of the given data points (except (4,-1))

f%280%29=a%28-4%29%5E2-1+=+16a-1+=+3
16a=4
a=1%2F4

And so

f%28x%29=%281%2F4%29%28x-4%29%5E2-1

Again reformat that in any way you need to.

Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
A quadratic function:
f%28x%29+=+ax%5E2+%2B+bx+%2B+c+

given:
x| f%28x%29
0| 3
1| 1.25
2 |0
3| -0.75
4+|-1
5 |-0.75
6 |0
7+|1.25
8| 3
f%28x%29+=+ax%5E2+%2B+bx+%2B+c+......use given points to set up a system of three equations
0| 3=x| f%28x%29
3+=+a%2A0%5E2+%2B+b%2A0+%2B+c+
c=3...........eq.1
2| 0=x| f%28x%29
0+=+a%2A2%5E2+%2B+b%2A2+%2B+3+
0+=+4a+%2B+2b+%2B+3+
-2b-3+=+4a++
a=-b%2F2-3%2F4+........eq.2

6|+0=x| f%28x%29
0+=+a%2A6%5E2+%2B+b%2A6+%2B+3+
0+=+36a+%2B+6b+%2B+3+.......simplify
0+=+12a+%2B+2b+%2B+1+
-2b-1+=+12a++
a=-b%2F6-1%2F12+........eq.3
from eq.2 and eq.3 we have
-b%2F2-3%2F4=-b%2F6-1%2F12++......solve for b
1%2F12-3%2F4=b%2F2-b%2F6++
-2%2F3=b%2F3+
b=-2+
go to
a=-b%2F2-3%2F4+........eq.2, substitute b
a=-%28-2%29%2F2-3%2F4+
a=1%2F4+

your equation is:

f%28x%29+=+%281%2F4%29x%5E2+-2x+%2B+3+-> standard form
f%28x%29+=%281%2F4%29+%28x+-+2%29+%28x+-+6%29-> factored form
f%28x%29+=%28+1%2F4%29+%28x+-+4%29%5E2+-+1-> vertex form


Answer by math_tutor2020(3816) About Me  (Show Source):
You can put this solution on YOUR website!

Here is the graph confirming what the other tutors have written is correct.
https://www.desmos.com/calculator/i8vtpiw08v
Desmos is a free graphing app. GeoGebra is another handy tool I use all the time.

Side note: There's something a bit strange about the given data points. I would think that Mario would jump in a path similar to a bridge arch; in other words, the parabola your teacher gave you should be flipped upside-down. I'm not sure if there's a major typo going on or not. It's possible I'm missing something, and if so, then I apologize.

Answer by MathTherapy(10551) About Me  (Show Source):
You can put this solution on YOUR website!
Mario is going to fly through this scene, following the path of the quadratic function 𝑓(𝑥). Based on the
table and graph, you will write equations for this quadratic function in each of three forms: standard
form, vertex form, and factored form. You may start with any form you choose, but you need to write
equivalent equations in all three forms.
equivalent equations in all three forms.
𝑥 𝑓(𝑥)
0 3
1 1.25
2 0
3 -0.75
4 -1
5 -0.75
6 0
7 1.25
8 3
1.
For this problem, you don't have to go through all that one of the other persons went through to get the VERTEX form of this parabola. It's OVERKILL!

Notice that the y-values on either side of point (4, - 1) repeat (up and down)? This means that (4, - 1) is the VERTEX (h, k) of this parabola. 

Vertex form of a PARABOLA: matrix%281%2C3%2C+y%2C+%22=%22%2C+a%28x+-+h%29%5E2+%2B+k%29
                           matrix%281%2C3%2C+3%2C+%22=%22%2C+a%280+-+4%29%5E2+-+1%29 ------- Substituting (4, - 1) and (0, 3) for (h, k) and (x, y), respectively
                            3 = 16a - 1
                        3 + 1 = 16a
                            4 = 16a
                           matrix%281%2C5%2C+4%2F16%2C+or%2C+1%2F4%2C+%22=%22%2C+a%29 

To determine the VERTEX form of this equation, we get:
                           matrix%281%2C3%2C+y%2C+%22=%22%2C+%281%2F4%29%28x+-+4%29%5E2+-+1%29 ------- Substituting 1%2F4 and (4, - 1) for "a", and (h, k), respectively

(2, 0) and (6, 0) are the 2 zeroes, and "a" has already been determined. Knowing this, you can now find the other 2 forms!