Question

tell what transformations each variable controls in the formula:
$f(x)=asqrt{x - h}+k$

a:

if a is negative then the graph

h:

k:

Explanation:

Step1: Analyze variable \( a \)

In the function \( f(x)=a\sqrt{x - h}+k \), the coefficient \( a \) controls the vertical stretch or compression of the graph. If \( |a|>1 \), it's a vertical stretch; if \( 0<|a|<1 \), it's a vertical compression. Also, the sign of \( a \) affects reflection. If \( a \) is negative, the graph reflects over the \( x \)-axis.

Step2: Analyze variable \( h \)

The term \( x - h \) in the square root means that \( h \) controls the horizontal shift of the graph. If \( h>0 \), the graph shifts \( h \) units to the right; if \( h<0 \), it shifts \( |h| \) units to the left (since \( x - h=x+|h| \) when \( h<0 \)).

Step3: Analyze variable \( k \)

The constant \( k \) added at the end controls the vertical shift of the graph. If \( k>0 \), the graph shifts \( k \) units up; if \( k<0 \), it shifts \( |k| \) units down.

Answer:

• \( a \): Controls vertical stretch/compression and reflection over the \( x \)-axis (if \( a<0 \), reflects over \( x \)-axis; \( |a|>1 \) stretches, \( 0<|a|<1 \) compresses vertically).
• If \( a \) is negative then the graph: Reflects over the \( x \)-axis.
• \( h \): Controls horizontal shift (shifts \( h \) units right if \( h>0 \), \( |h| \) units left if \( h<0 \)).
• \( k \): Controls vertical shift (shifts \( k \) units up if \( k>0 \), \( |k| \) units down if \( k<0 \)).