Question

in the equation f(x)=5(x+3)²-10 what does transformation does the 5 cause?
vertical stretch by 5
vertical shift by 5

Explanation:

Step1: Recall function transformation rules

For a function of the form \( y = a f(x - h) + k \), the coefficient \( a \) affects vertical stretch/compression. If \( |a|>1 \), it's a vertical stretch by a factor of \( |a| \); if \( 0<|a|<1 \), it's a vertical compression. A vertical shift is caused by \( k \), and horizontal shift by \( h \).

Step2: Analyze the given function

The function is \( f(x)=5(x + 3)^2-10 \). Comparing with \( y = a f(x - h)+k \), here \( a = 5 \), \( h=- 3 \), \( k=-10 \). The \( 5 \) is the coefficient of the squared term, so it affects the vertical stretch/compression. Since \( |5|>1 \), it's a vertical stretch by a factor of \( 5 \).

Answer:

Vertical Stretch by 5