Question

move at least one of the 5 guide points below to complete the graph of $y = (x + 2)^2 + 1$. moving the red points changes the vertical stretch or compression. moving the blue point shifts the function left/right/up/down. click the buttons below to start over or reflect over the x-axis. reset reflect over x-axis

Explanation:

Step1: Identify vertex of target function

The vertex form of a parabola is $y=a(x-h)^2+k$, where $(h,k)$ is the vertex. For $y=(x+2)^2+1$, rewrite as $y=(x-(-2))^2+1$, so vertex $(h,k)=(-2,1)$. Move the blue vertex point from $(0,0)$ to $(-2,1)$.

Step2: Find points 1 unit left/right of vertex

For $x=-2-1=-3$:
$y=(-3+2)^2+1=(-1)^2+1=1+1=2$
For $x=-2+1=-1$:
$y=(-1+2)^2+1=(1)^2+1=1+1=2$
Move the lower red points from $(-1,1)$ and $(1,1)$ to $(-3,2)$ and $(-1,2)$.

Step3: Find points 2 units left/right of vertex

For $x=-2-2=-4$:
$y=(-4+2)^2+1=(-2)^2+1=4+1=5$
For $x=-2+2=0$:
$y=(0+2)^2+1=(2)^2+1=4+1=5$
Move the upper red points from $(-2,4)$ and $(2,4)$ to $(-4,5)$ and $(0,5)$.

Answer:

1. Blue vertex point: $(-2, 1)$
2. Lower red points: $(-3, 2)$ and $(-1, 2)$
3. Upper red points: $(-4, 5)$ and $(0, 5)$

This completes the graph of $y=(x+2)^2+1$.