Question

the graph of (f) is translated a whole number of units horizontally and vertically to obtain the graph of (k).

the function (f) is defined by (f(x) = -\frac{1}{2}x^2).

find the expression for (k(x)).

(k(x) = \\)

Explanation:

Identify the vertex of the original function

The original function is given by:

$$f(x) = -\frac{1}{2}x^2$$

The vertex of this parabola is at the origin:

$$(0, 0)$$

Identify the vertex of the translated function

Looking at the graph, the translated parabola \(k(x)\) has its peak (vertex) located at:

$$(-3, -2)$$

Determine the horizontal and vertical shifts

The vertex shifted from \((0, 0)\) to \((-3, -2)\).
This represents:

• A horizontal shift of \(3\) units to the left.
• A vertical shift of \(2\) units down.

Write the transformed function formula

Using the transformation rules:

$$k(x) = f(x - h) + k$$

Substituting \(h = -3\) and \(k = -2\):

$$k(x) = -\frac{1}{2}(x + 3)^2 - 2$$

Answer:

The graph of \(f\) is translated a whole number of units horizontally and vertically to obtain the graph of \(k\). The function \(f\) is defined by \(f(x) = -\frac{1}{2}x^2\). Find the expression for \(k(x)\).

\(k(x) =\) <blank>\(-\frac{1}{2}(x+3)^2-2\)</blank>