Question

the graph of f is translated a whole number of units horizontally and vertically to obtain
the function f is defined by f(x) = \sqrt{x}.
write down the expression for k(x).

Explanation:

Step1: Identify horizontal translation

The original function \( f(x) = \sqrt{x} \) has its vertex at \( (0,0) \). The graph of \( k(x) \) has its vertex at \( (-1, 3) \). So, the horizontal translation is 1 unit to the left (since we go from \( x = 0 \) to \( x = -1 \), which is a shift of \( -1 \) in the x - direction, or equivalently, replacing \( x \) with \( x+1 \) for a left shift).

Step2: Identify vertical translation

The vertical translation is 3 units up (since we go from \( y = 0 \) to \( y = 3 \)). For a function \( y = f(x) \), a vertical shift of \( a \) units up is given by \( y=f(x)+a \).

Step3: Write the expression for \( k(x) \)

First, apply the horizontal shift: replacing \( x \) with \( x + 1 \) in \( f(x)=\sqrt{x} \), we get \( \sqrt{x + 1} \). Then apply the vertical shift of 3 units up: \( k(x)=\sqrt{x + 1}+3 \).

Answer:

\( k(x)=\sqrt{x + 1}+3 \)