The graph of f (in blue) is translated a whole number of units horizontally and vertically to obtain the graph of k (in red). The function f is defined by f(x)= x squared Write down the expression for k(x) .

$\bf \qquad \qquad \qquad \qquad \textit{function transformations} \\ \quad \\\\ \begin{array}{rllll} % left side templates f(x)=&{{ A}}({{ B}}x+{{ C}})+{{ D}} \\ \quad \\ y=&{{ A}}({{ B}}x+{{ C}})+{{ D}} \\ \quad \\ f(x)=&{{ A}}\sqrt{{{ B}}x+{{ C}}}+{{ D}} \\ \quad \\ f(x)=&{{ A}}(\mathbb{R})^{{{ B}}x+{{ C}}}+{{ D}} \\ \quad \\ f(x)=&{{ A}} sin\left({{ B }}x+{{ C}} \right)+{{ D}} \end{array}$

$\bf \begin{array}{llll} % right side info \bullet \textit{ stretches or shrinks horizontally by } {{ A}}\cdot {{ B}}\\\\ \bullet \textit{ flips it upside-down if }{{ A}}\textit{ is negative} \\\\\end{array}$
$\bf \begin{array}{llll} \bullet \textit{ horizontal shift by }\frac{{{ C}}}{{{ B}}}\\ \qquad if\ \frac{{{ C}}}{{{ B}}}\textit{ is negative, to the right}\\\\ \qquad if\ \frac{{{ C}}}{{{ B}}}\textit{ is positive, to the left}\\\\ \bullet \textit{ vertical shift by }{{ D}}\\ \qquad if\ {{ D}}\textit{ is negative, downwards}\\\\ \qquad if\ {{ D}}\textit{ is positive, upwards}\\\\ \bullet \textit{ period of }\frac{2\pi }{{{ B}}} \end{array}$

now, notice k(x) in the picture, is down by 4 units, and to the left by 3 units, thus $\bf \begin{array}{llll} k(x)=&\sqrt{1x+3}&-4\\ &\ \ \uparrow \qquad \uparrow &\uparrow \\ &B\qquad C&D \end{array}$

smecloudbird17 smecloudbird17 · Answer 2 · 2022-06-17T01:40:48-04:00

Using translation of a graph, the expression for k(x) is

$k(x) = \sqrt{x+3} - 4$

What is translation of a graph?

A translation of a graph is its rigid movement, vertically or horizontally.

If the graph of y = f(x) is translated a units horizontally and b units vertically, then the equation of the translated graph is y − b = (x − a).

$y = \sqrt{x}\\ y +4 = \sqrt{x+3}\\ y = \sqrt{x+3} - 4\\ k(x) = \sqrt{x+3} - 4$

Learn more about translation of a graph here

https://brainly.com/question/11805053

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