Question

create the transformed function g(x) from the basic linear function f(x) = x by shifting up 5 units, vertically dilating by a factor of 2, and vertically reflecting. basic function: f(x) transforming function history (clicked it now to invert) basic function f(x) = x edit the transforming function. transformation form: y = ±a f(x) + d choose a transformation to change the graph. reflection dilation horizontal translation vertical translation

Explanation:

Step1: Recall function transformation rules

For a function \( y = f(x) \), vertical dilation by a factor \( a \) is \( y = a f(x) \), vertical reflection is \( y=-f(x) \), and vertical shift up by \( D \) units is \( y = f(x)+D \). The base function is \( f(x)=x \). We need to vertically dilate by factor 2, vertically reflect, and shift up 5 units.

Step2: Apply vertical dilation

First, vertical dilation by factor 2: \( f_1(x)=2f(x)=2x \).

Step3: Apply vertical reflection

Then, vertical reflection (multiply by -1): \( f_2(x)= - 2f(x)=-2x \).

Step4: Apply vertical shift

Then, shift up 5 units: \( g(x)=-2x + 5 \).

Answer:

The transformed function \( g(x) \) is \( \boldsymbol{g(x)=-2x + 5} \)