The graph of [tex]$f$[/tex] is translated a whole number of units horizontally and vertically to obtain the graph of [tex]$k$[/tex]. The function [tex]$f$[/tex] is defined by [tex]$f(x)-|x|$[/tex]. Write down the expression for [tex]$k(x)$[/tex].
[tex]$k(x)=\square$[/tex]
Answer (2)
The function f ( x ) = ∣ x ∣ is translated horizontally by h units, resulting in ∣ x − h ∣ .
The translated function is then shifted vertically by v units, resulting in ∣ x − h ∣ + v .
The final expression for k ( x ) is the combination of these translations.
Therefore, k ( x ) = ∣ x − h ∣ + v .
Explanation
Understanding the Problem We are given the function f ( x ) = ∣ x ∣ and told that its graph is translated horizontally and vertically to obtain the graph of k ( x ) . Our goal is to find the expression for k ( x ) .
Defining Translations Let h be the number of units the graph of f is translated horizontally and v be the number of units the graph of f is translated vertically.
Horizontal Translation A horizontal translation by h units transforms f ( x ) to f ( x − h ) . In our case, since f ( x ) = ∣ x ∣ , this becomes ∣ x − h ∣ .
Vertical Translation A vertical translation by v units transforms f ( x − h ) to f ( x − h ) + v . Therefore, k ( x ) = ∣ x − h ∣ + v .
Final Expression Thus, the expression for k ( x ) is k ( x ) = ∣ x − h ∣ + v , where h and v are integers representing the horizontal and vertical translations, respectively.
Examples
Imagine you are designing a simple navigation app. The absolute value function helps determine the distance to a location regardless of direction. If you shift the graph horizontally and vertically, you're essentially changing the starting point or adjusting the scale. Understanding these transformations is crucial for accurately mapping locations and distances in your app.
To find the expression for k ( x ) after translating the function f ( x ) = ∣ x ∣ , we first apply a horizontal shift of h and then a vertical shift of v . The resulting expression is k ( x ) = ∣ x − h ∣ + v . Hence, both translations are accounted for in this final form.
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