Given the parent function \( f(x) = |x| \) and its transformation described in the dotted line graph, we can analyze the points you've provided for both the solid line (parent function) and the dotted line (translated function).
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Parent Function Points:
- (-1, 1): \( f(-1) = |-1| = 1 \)
- (0, 0): \( f(0) = |0| = 0 \)
- (1, 1): \( f(1) = |1| = 1 \)
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Translated Function Points:
- (-1, 4): \( f(-1) = -1 + k \Rightarrow -1 + k = 4 \Rightarrow k = 5 \)
- (0, 3): \( f(0) = 0 + k \Rightarrow 0 + k = 3 \Rightarrow k = 3 \)
- (1, 4): \( f(1) = 1 + k \Rightarrow 1 + k = 4 \Rightarrow k = 3 \)
The translated function takes the form: \[ g(x) = |x| + k \] From the second point (when \( x = 0 \)), we find \( k \) = 3, meaning the function has been vertically translated upwards by \( 3 \) units.
So the translated function can be expressed as: \[ g(x) = |x| + 3 \]
Summary of the Transformation:
The transformation applied to the parent function \( f(x) = |x| \) results in moving the entire graph upward by 3 units. Thus, the translated function is \( f(x) = |x| + 3 \). This transformation affects the height of each point on the graph, increasing their y-values by 3. Hence, the final answer for the value of \( k \) in the transformation is: \[ k = 3 \]