How do you find the equation (or the vertical stretch) of a parabola?

Question

I've never been taught, and my math teacher is totally useless. I can determine the horizontal and vertical translations and if there's a reflection on the x-axis, but I can't figure out a way to determine the vertical stretch of a parabola.

Example:
A parabola with a vertex of (-1,-4) intercepts the x-axis at (-3,0) and (1,0).
Using this function:
f(x) = RxVS(x-HT)^2+VT
I need to come up with the equation of the graph. So far I have this:
f(x) = VS(x+1)-4

How would I get the vertical stretch?

Answer 1

You put one of the intercepts into the equation and solve for VS.

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See this similar problem from the below link:
Example 1: Given the vertex (4, -1) and the point (2, 3), what is the equation of the quadratic function.

y = a (x - h)² + k
3 = a (2 - 4)² + (-1)
4 = a (-2)²
4 = a (4)
1 = a

y = 1 (x - 4)² - 1

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for yours
F(x) = VS×(x+1)² - 4
choose one of the two:
-3 = VS×(0+1)² - 4
or
4 - 3 = VS×(0+1)² = VS

so,
┏━━━━━━┓
┃✓ ∴ VS = 1 ┃
┗━━━━━━┛

and

┏━━━━━━━━┓
┃✓F(x) = (x+1)² - 4┃
┗━━━━━━━━┛


(test it with any of the three points given.)

Answer 2

The equation should be y+4 = 2p (x+1)^2
Plug x=1 and y =0 into the above equation getting:
0+4 = 2p(1+1)^2
4 = 8p --> 2p = 1
So equation is y+4 = (x+1)^2
The coefficient of x determines the horizontal stretch. If the coefficient of x^2 is >1 the parabola gets more narrow. If the coefficient of x is less than 1 the parabola becomes wider (has a horizontal stretch)
So y= x^2 is standard parabola with vertex at (0,0)
y = 5x^2 would be a very narrow parabola
y = x^2/5 would be a very wide parabola.

I do not know what you mean by vertical stretch.

Answer 3

How To Find Vertical Stretch