Question

sketch the graph of the function.
$f(x)=0.5(x + 4)^2 + 2$

use the graphing tool to graph the function.
click to enlarge graph

Explanation:

Step1: Identify the vertex form

The function \( f(x) = 0.5(x + 4)^2 + 2 \) 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 of opening

Since \( a = 0.5>0 \), the parabola opens upward.

Step3: Find the axis of symmetry

The axis of symmetry is \( x=h=-4 \).

Step4: Plot the vertex and use the direction to sketch

Plot the vertex \( (-4,2) \). Since the parabola opens upward, we can find a few more points. For example, when \( x=-3 \), \( f(-3)=0.5(-3 + 4)^2+2=0.5(1)+2 = 2.5 \). When \( x=-5 \), \( f(-5)=0.5(-5 + 4)^2+2=0.5(1)+2 = 2.5 \). Connect these points to form the parabola.

(Note: Since the problem asks to sketch using a graphing tool, the key steps to identify the vertex, direction, and axis of symmetry help in accurately graphing the function. The actual graphing would involve plotting the vertex and other points as calculated and drawing the parabola.)

Answer:

The graph of \( f(x)=0.5(x + 4)^2 + 2 \) is a parabola with vertex at \((-4, 2)\), opening upward, and axis of symmetry \( x=-4 \). To sketch it, plot the vertex, use the direction of opening, and plot additional points (e.g., \((-3, 2.5)\), \((-5, 2.5)\)) then draw the parabola. (The actual graphing is done using the graphing tool by inputting the function and adjusting as needed based on the vertex and shape.)