How do you describe the transformation of #f(x)=2-(x+5)^2# from a common function that occurs and sketch the graph?

Question

Serenity Jones · Accepted Answer

This way.

Consider y= #x^2# as the starting common function. Now , #x^2#------># (x+5)^2# involves a leftward (negative) direction shift of 5 units. Then, # (x+5)^2#------> #- (x+5)^2# means we have to take a mirror image about the x axis. Then, #- (x+5)^2# ------> #2- (x+5)^2# involves shift along y axis in the positive or upper direction . Hence, we get the required graph. graph{2- (x+5)^2 [-10, 10, -5, 5]}

Elias Ackerman · Answer

undefined The transformation of $f(x) = 2 - (x + 5)^2$ involves a vertical translation of 2 units upwards and a horizontal translation of 5 units to the left compared to the parent function $f(x) = -x^2$. The negative sign inside the parentheses reflects the reflection about the y-axis.

To sketch the graph:
1. Start with the graph of the parent function $f(x) = -x^2$.
2. Shift the graph 5 units to the left to account for the horizontal translation.
3. Then, shift the graph 2 units upwards to account for the vertical translation.
4. The graph will open downwards due to the negative sign associated with the quadratic term.