Question

What is the equation, in standard form, of a parabola that contains the following points?

(-2, -20), (0, -4), (4, -20)

Please show all steps ! Will give medal to best answer.

Answer 1

@Coolsector @Mr.ClayLordMath @ehuman

Answer 2

find the slope 1st
y2 - y1
------- = slope
x2 - x1

Answer 3

pick 2 of the ordered pairs, remember that they represent (x,y), plug in the values and find slope

Answer 4

from the slope and one ordered pair you can construct the equation because you will have the x, the y and the slope (m)

equation of a line is
y=mx+b

Answer 5

b is the y intercept, or what y value is at x=0

Answer 6

The slope is 8 I think, @ehuman? So the equation of the line would be -20=8(-2)-20?

Answer 7

@ehuman I dont think thats a right answer since parabola means a quadratic equation not leaner as you find it .

Answer 8

Do you know how to do it the right way @ikram002p?

Answer 9

\(\bf (-2, -20)\qquad (0, -4)\qquad (4, -20)\\ \quad \\
\textit{a parabola equation is a quadratic equation, thus}\qquad y=ax^2+bx+c
\)

so let us plug in those 3 values given for "x" and "y" in the \(\bf y=ax^2+bx+c\)
form, so we'll end up with a system of equations of 3 variables

gimme a sec

Answer 10

\(\begin{array}{llll}
(-2, -20)\qquad (0, -4)\qquad (4, -20)\qquad ax^2+bx+c\\ \quad \\
(-2, -20)\implies -20 = a(-2)^2+b(-2)+c\implies &\bf4a-2b+c=-20\\ \quad \\
(0, -4)\implies -4= a(0)^2+b(0)+c\implies &\bf c = -4\\ \quad \\
(4, -20)\implies -20= a(4)^2+b(4)+c\implies &\bf 16a+4b+c=-20
\end{array}\)

Answer 11

so based on the 2nd equation, we know c = -4

so we can use that in the 1st and 3rd one and use them like

4a -2b = -24
16a +4b = -24

and from there, all you'd need to do, is find either by elimination or substitution
we have "c", already, so just need the "a" and "b" coefficients
once you find them, plug them back in at the form \(\bf ax^2+bx+c\)

Answer 12

Would a be -5.5?

Answer 13

@jdoe0001

Answer 14

@ikram002p

Answer 15

c is not -5.5 it is -4
as can be seen from the point (0,-4)
y = ax^2 +bx + c
when x = 0 y=-4
y(0) = c
c = -4

Answer 16

I know c is -4.. I asked if A was -5.5

Answer 17

follow jdoe0001 steps to find a and b

Answer 18

I did follow his steps, I wanted to check the values I got..

Answer 19

4a -2b = -24
16a +4b = -24

if we multiply the first eq by 2 then add it to the second:
24a = -72

Answer 20

B=3.14??

Answer 21

a is not -5.5
look at what i wrote

Answer 22

a=-3

Answer 23

good..
now find b using one of the equations

Answer 24

(i mean plug a=-3 into one of the equations and then you will get b)

Answer 25

\[y=2x^2+12x-4\]
try this

Answer 26

b=1.71?

Answer 27

Whats that equation for @ikram002p ?

Answer 28

no, b should be 6.
show me your steps

Answer 29

mmmm sry
\[y=2x^2-12x-4\]

Answer 30

16(-3)+4b+4=-20
-48+4b+4=-20
-48=-28b
1.71=b

Answer 31

a=2
b=-12
c=-4

Answer 32

16(-3)+4b+4=-20
-48+4b+4=-20
-48=-28b <- mistake is here why 28b
1.71=b

Answer 33

16(-3)+4b+4=-20
-48+4b+4=-20
4b = 24
b = 6

Answer 34

I subtracted 4b from 4b and -24

Answer 35

Oh I see now. Now I plug it all in to the y=ax^2+bx+c ?

Answer 36

@Coolsector

Answer 37

\[y=ax^2+bx+c\]
\[-4=a(0)+b(0)+c \rightarrow c=-4\]
\[-20=a(-2)^2+b(-2)-4\rightarrow 4a+2b=16\]
\[-20=a(4)^2+b(4)-4\rightarrow 16a+4b=-16\]
a=6 &b =-4
the equation would be
\[y=6x^2-4x-4\]

Answer 38

a=-6 ..

Answer 39

im sorry but those 2 eq
4a -2b = -24
16a +4b = -24
was incorrect.

the correct ones are:
4a-2b=-16
16a+4b=-16

now multiply the first by 2 and add them:

24a = -48
a=-2

now plug a=-2 into the first for example and get:
-8-2b=-16
-2b =-8
b =4

so you have

y=-2x^2+4x-4

as you can see it pass through all the points:
(-2,-20):
y(-2) = -8-8-4 = -20

(0,-4)
obvious

and (4,-20):
y(4) = -32+16-4=-20

again, final answer:
y=-2x^2+4x-4

Answer 40

hm so much for my miss there \(\bf \begin{array}{llll}
(-2, -20)\qquad (0, -4)\qquad (4, -20)\qquad ax^2+bx+c\\ \quad \\
(-2, -20)\implies -20 = a(-2)^2+b(-2)+c\implies &\bf4a-2b+c=-20\\ \quad \\
(0, -4)\implies -4= a(0)^2+b(0)+c\implies &\bf c = -4\\ \quad \\
(4, -20)\implies -20= a(4)^2+b(4)+c\implies &\bf 16a+4b+c=-20
\end{array}\\ \quad \\
4a-2b-4=-20\\ \quad \\
16a+4b-4=-20\\ \quad \\
\textit{should be as corrected by Coolsector}\\ \quad \\
4a-2b=-16\\ \quad \\
16a+4b=-16\\ \quad \\\)