Question

Determine quadratic function given vertex and a point: Vertex: (1,1) and point: (2,4) Please explain Thanks!

Answer 1

Go back to the form: \[y=ax ^{2}+bx+c\]
Now, the x-coord of the vertex = -b/2a, so
\[1=\frac{ -b }{ 2a }\]
Then:
\[\frac{ 1 }{ 1 }=\frac{ -b }{ 2a }\]
cross multiply and we get b = -2a

Answer 2

Now we can replace b with 2a:
\[y=ax ^{2}+2ax+c\]
Plug in the first point replacing x with 1 and y with 1 and you get an equation with just 2 unknowns: a and c
Plug in the second point replacing the x and y again and you get a second equation with just 2 unknowns: a and c
Now you can solve this system of equations as you have 2 unknowns and 2 equations.

Answer 3

Do you know how to solve a system of equations?

Answer 4

no

Answer 5

sorry, i didn't get a message when you answered
Did you get:
3a + c = 1
8a + c = 4
?

Answer 6

To solve a system, we want to be able to add 2 equations and have one of the variables "disappear".
We could multiply that first equation by (-1).
We get:
-3a - c = -1
8a + c = 4
Notice that the signs in the first equation all changed because we mult by -1
Now add the equations downward. The c's should disappear.

Answer 7

thank you

Answer 8

I made an error.\[-\frac{ b }{ 2a } = 1\]
So b = -2a ... sorry

Answer 9

ok

Answer 10

\[y=ax ^{2}-2ax+c\]

Answer 11

when we plug in (1,1), we get: -a + c = 1
When we plug in (2, 4), we get: 4 = 4a - 4a + c
So: 4 = c
Now you can use that to find a since: -a + c = 1 and c = 4,
-a + 4 = 1 so a = ?

Answer 12

also, b = -2a, and now that we know 'a' we can use it to find 'b'.

Answer 13

would it be -2a?

Answer 14

-a + 4 = 1 a=3

Answer 15

so now what did you get for 'a' and 'b'?

Answer 16

sorry, 'c' and 'b'

Answer 17

a=3,b=-2a

Answer 18

c=4, b=2a?

Answer 19

b = -2a = -2(3) = _?

Answer 20

-6

Answer 21

remember I made that error and b = - 2a, not 2a

Answer 22

Good
then you go back to \[y=ax ^{2}+bx+c \] and put 3 in for 'a', -6 in for b and 4 in for c. That gives you this specific quadratic equation
A lot of work on that one!

Answer 23

I know thank you so much I need to work on my math skills more

Answer 24

if you plug in the given points to your resulting equation, you will find they both work. If you find the x-coordinate of the vertex, you will find it is 1

Answer 25

good luck
I am a math teacher / tutor at a community college but i have developed nerve issues so I had to cut back and this is fun for me

Answer 26

What college is that? I should attend