Question

sketch the graph of the function.\\(f(x) = 2(x - 4)^2 + 2\\)\\(\dots\\)\
use the graphing tool to graph the function.\
click to enlarge graph

Explanation:

Step1: Identify the vertex form

The function \( f(x) = 2(x - 4)^2 + 2 \) is in vertex form \( f(x)=a(x - h)^2 + k \), where \( (h,k) \) is the vertex. Here, \( h = 4 \), \( k = 2 \), so the vertex is \( (4, 2) \).

Step2: Determine the direction and width

Since \( a = 2>0 \), the parabola opens upward. The value of \( |a| = 2 \) means it is narrower than the standard parabola \( y = x^2 \).

Step3: Find additional points (optional for sketching)

• When \( x = 3 \), \( f(3)=2(3 - 4)^2 + 2 = 2(1)+2 = 4 \), so the point is \( (3, 4) \).
• When \( x = 5 \), \( f(5)=2(5 - 4)^2 + 2 = 2(1)+2 = 4 \), so the point is \( (5, 4) \).
• When \( x = 2 \), \( f(2)=2(2 - 4)^2 + 2 = 2(4)+2 = 10 \), so the point is \( (2, 10) \).
• When \( x = 6 \), \( f(6)=2(6 - 4)^2 + 2 = 2(4)+2 = 10 \), so the point is \( (6, 10) \).

To sketch the graph, plot the vertex \( (4, 2) \), then use the symmetry (axis of symmetry \( x = 4 \)) and the additional points to draw the upward - opening parabola that is narrower than \( y=x^2 \).

Answer:

The graph is a parabola with vertex at \((4, 2)\), opening upward, and narrower than the standard parabola \(y = x^2\). Key points include \((3, 4)\), \((5, 4)\), \((2, 10)\), \((6, 10)\) (among others) which can be used to sketch the graph.