Question

use technology to find points and then graph the function $y = (x + 3)^2 + 2$, following the instructions below.
done plotting points
plot at least five points that fit on the axes below. click a point to delete it.

Explanation:

Step1: Identify the vertex

The function \( y=(x + 3)^2+2 \) is in vertex form \( y=a(x - h)^2+k \), where the vertex is \( (h,k) \). Here, \( h=- 3 \) and \( k = 2 \), so the vertex is \( (-3,2) \).

Step2: Choose x - values

Let's choose \( x=-5,-4,-3,-2,-1 \) (symmetric around the vertex \( x = - 3 \)).

Step3: Calculate y - values for each x

• For \( x=-5 \): \( y=(-5 + 3)^2+2=(-2)^2+2=4 + 2=6 \), so the point is \( (-5,6) \).
• For \( x=-4 \): \( y=(-4 + 3)^2+2=(-1)^2+2=1 + 2=3 \), so the point is \( (-4,3) \).
• For \( x=-3 \): \( y=(-3 + 3)^2+2=0^2+2=2 \), so the point is \( (-3,2) \) (vertex).
• For \( x=-2 \): \( y=(-2 + 3)^2+2=(1)^2+2=1 + 2=3 \), so the point is \( (-2,3) \).
• For \( x=-1 \): \( y=(-1 + 3)^2+2=(2)^2+2=4 + 2=6 \), so the point is \( (-1,6) \).

These five points \( (-5,6),(-4,3),(-3,2),(-2,3),(-1,6) \) can be plotted on the graph.

Answer:

The five points to plot are \((-5, 6)\), \((-4, 3)\), \((-3, 2)\), \((-2, 3)\), \((-1, 6)\) (and the graph is a parabola opening upwards with vertex at \((-3,2)\) passing through these points).