Question

1. $y = 2(x + 1)^2 + 1$

Explanation:

Step1: Identify the vertex form

The equation \( y = 2(x + 1)^2 + 1 \) is in vertex form \( y = a(x - h)^2 + k \), where the vertex is \( (h, k) \). Here, \( h=-1 \) (since \( x+1=x - (-1) \)) and \( k = 1 \), so the vertex is \( (-1, 1) \).

Step2: Determine the direction and stretch

The coefficient \( a = 2 \), which is positive, so the parabola opens upward. The value \( |a| = 2>1 \), so it is a vertical stretch by a factor of 2.

Step3: Find key points

• Vertex: \( (-1, 1) \)
• For \( x = 0 \): \( y=2(0 + 1)^2+1=2 + 1=3 \), so the point is \( (0, 3) \)
• For \( x=-2 \): \( y=2(-2 + 1)^2+1=2(1)+1 = 3 \), so the point is \( (-2, 3) \)
• For \( x = 1 \): \( y=2(1 + 1)^2+1=2(4)+1=9 \), so the point is \( (1, 9) \)
• For \( x=-3 \): \( y=2(-3 + 1)^2+1=2(4)+1 = 9 \), so the point is \( (-3, 9) \)

Step4: Plot the points and draw the parabola

Plot the vertex \( (-1, 1) \), and the other points \( (0, 3) \), \( (-2, 3) \), \( (1, 9) \), \( (-3, 9) \) on the grid. Then draw a smooth upward - opening parabola passing through these points.

Answer:

To graph \( y = 2(x + 1)^2+1 \):

1. Vertex: Plot the vertex \( (-1, 1) \).
2. Direction/Stretch: Opens upward, vertically stretched by factor 2.
3. Key Points: Plot \( (0, 3) \), \( (-2, 3) \), \( (1, 9) \), \( (-3, 9) \) (and others as needed) and draw the parabola through them.