Question

graph the function.
$g(x) = 2(x - 2)^2 + 2$

Explanation:

Step1: Identify the vertex form

The function \( g(x) = 2(x - 2)^2 + 2 \) is in vertex form \( y = a(x - h)^2 + k \), where the vertex is \( (h, k) \). Here, \( h = 2 \) and \( k = 2 \), so the vertex should be \( (2, 2) \). But the given graph has a vertex at \( (0, 0) \), which is incorrect. To graph correctly:

• Vertex: Plot \( (2, 2) \) (since \( h = 2 \), \( k = 2 \)).
• Direction and Stretch: The coefficient \( a = 2 > 0 \), so the parabola opens upward, and it's vertically stretched by a factor of 2.
• Find other points: For \( x = 3 \), \( g(3) = 2(3 - 2)^2 + 2 = 2(1) + 2 = 4 \), so plot \( (3, 4) \). For \( x = 4 \), \( g(4) = 2(4 - 2)^2 + 2 = 2(4) + 2 = 10 \), plot \( (4, 10) \). For \( x = 1 \), \( g(1) = 2(1 - 2)^2 + 2 = 2(1) + 2 = 4 \), plot \( (1, 4) \).
• Draw the parabola: Connect the vertex \( (2, 2) \) with the other points, ensuring it opens upward and has the correct stretch.

Step2: Correct the graph

The original graph has the wrong vertex (at \( (0, 0) \) instead of \( (2, 2) \)). The correct graph should have its minimum point (vertex) at \( (2, 2) \), open upward, pass through \( (1, 4) \), \( (3, 4) \), \( (4, 10) \), etc., with a vertical stretch factor of 2.

Answer:

To graph \( g(x) = 2(x - 2)^2 + 2 \) correctly:

1. Vertex: Plot \( (2, 2) \) (since the vertex form \( y = a(x - h)^2 + k \) gives \( h = 2 \), \( k = 2 \)).
2. Direction: Opens upward ( \( a = 2 > 0 \) ).
3. Stretch: Vertically stretched by factor 2 ( \( |a| = 2 \) ).
4. Additional Points: Plot \( (1, 4) \), \( (3, 4) \), \( (4, 10) \), etc., by substituting \( x \)-values into \( g(x) \).
5. Draw the Parabola: Connect the vertex and points smoothly, opening upward.

(The given graph is incorrect; the correct vertex is \( (2, 2) \), not \( (0, 0) \).)