Question

what does a graph look like that is y=x^2+4x+4 with a repeated root of x=-2

Answer 1

y = x^2 + 4x + 4 = (x + 2)^2.

Key features
- Factorization shows a repeated root at x = −2 (multiplicity 2).
- The graph is a parabola opening upward.
- Vertex: (−2, 0). This is also the x-intercept; the parabola just touches the x-axis there (tangent), it does not cross.
- Axis of symmetry: x = −2.
- Minimum value: y = 0 at x = −2.
- y-intercept: y(0) = 4.

A few points:
- x = −4 → y = 4
- x = −3 → y = 1
- x = −2 → y = 0 (vertex)
- x = −1 → y = 1
- x = 0 → y = 4

Simple ASCII sketch (vertical scale exaggerated):

y
^
| *
| * *
| * *
| * *
|---*----*----*----> x
-4 -2 0

The "*" at x = −2 touches the x-axis (vertex) and the arms rise symmetrically away from it.