given the vertex of (-2, 1) and the intercept of (0,5) please show how to reconstruct the equation.

Question

How do you figure out how to do a problem like this

Answer 1

The standard equation of a parabola is y=ax² + bx + c. In this problem, you are given the coordinates of the vertex (-2,1) and the y-intercept (0,5). The point of this question is to find out the values of "a", "b" and "c". 

If you plug in the values of the y-intercept (given) into the standard equation...

1. y=ax² + bx + c
2. (5)=a(0)² + b(0) + c.
3. c=5

 

For the vertex values, you must rearrange the standard equation a little bit. 

y=ax² + bx + c

y= a(x-h)² + k with the vertex being (h,k)

Now, plugging in the vertex values... 

1. y= a( x- (-2))² + 1
2. y= a (x² + 4x + 4) + 1
3. y= ax² + 4ax + 4a + 1

Recalling the standard equation of the parabola, you can match up the components of the equation according to the powers of x: 

ax^{2 =} ax² --> a=a (No new value is learned)

4a + 1 = c

1. 4a + 1 = (5)
2. a= 1

4ax= bx

1. 4a= b
2. 4(1) = b
3. b=4

You know have all the values. a= 1, b= 4 and c=5

If you plug these values into the standard parabolic equation, y=ax² + bx + c, and simplify, you successfully reconstruct the equation. 

y= (1)x² + (4)x + (5)

y= x² + 4x + 5

Hope this helps :) 

 *NOTE: Thank you, Natalia D. for catching my error. My response has now been edited to the correct version.*

 

 

Answer 2

y = ax² + bx + c
"c" is the y-intercept
y = a(x - h)² + k
(h,k) are coordinate of the vertex
x-coordinate of vertex: -b / (2a)
~~~~~~~~~~~~
c = 5
Let's use (-2, -1) :
1. a(-2)² + b(-2) + 5 = 1
4a - 2b = - 4
2. -b / (2a) = -2 ---> b = 4a
Now, let's solve system of equation:
4a - 2b = - 4
b = 4a
4a - 2(4a) = - 4
a = 1
b = 4
c = 5
y = x² + 4x + 5   <---- final answer

Answer 3

To start out, the equation of a parabola is y = (x-a)² + b where a and b are constants. The point (a,b) is the vertex of the parabola. So in your case, if you take -2 and 1, the equation would now look like y = (x+2)² + 1. The intercept given can be checked by entering in 0 for x, so y = (0+2)² + 1 = 5 as desired. I hope this helps!