Question

algebra 2a semester online practice
complete this assessment to review what youve learned. it will not count toward your grade.
which function would vertically stretch the function ( k(x) = -3sqrt{2x - 1} + 6 ) by a factor of 7? (1 point)
( circ ) ( g(x) = -10sqrt{2x - 1} + 6 )
( circ ) ( m(x) = 4sqrt{2x - 1} + 6 )
( circ ) ( p(x) = -21sqrt{2x - 1} + 6 )
( circ ) ( k(x) = -3sqrt{14x - 1} + 6 )

Explanation:

Step1: Recall Vertical Stretch Rule

For a function \( y = a \cdot f(x) \), a vertical stretch by factor \( c \) (where \( c>1 \)) means multiplying the entire function \( f(x) \) by \( c \), so the new function is \( y = c \cdot f(x) \).

Step2: Apply the Rule to \( k(x) \)

Given \( k(x) = -3\sqrt{2x - 1}+6 \), to vertically stretch by factor 7, we multiply the entire function (excluding the vertical shift part? No, the vertical stretch affects the non - shift part. Wait, the general form for vertical stretch of \( y = f(x) \) is \( y = c\cdot f(x) \) where \( c \) is the stretch factor. So here, \( f(x)=-3\sqrt{2x - 1}+6 \)? No, wait, actually, the function \( k(x) \) can be thought of as \( k(x)=a\sqrt{2x - 1}+b \), where \( a=-3 \) and \( b = 6 \). A vertical stretch by factor 7 means we multiply the coefficient of the square - root term (the part that is not the vertical shift) by 7. So the new coefficient of \( \sqrt{2x - 1} \) will be \( -3\times7=-21 \).

So the new function \( p(x) \) after vertical stretch by factor 7 is \( p(x)=7\times(-3\sqrt{2x - 1})+6=-21\sqrt{2x - 1}+6 \).

Answer:

\( p(x)=-21\sqrt{2x - 1}+6 \) (the option corresponding to this function, which is the third option: \( p(x)=-21\sqrt{2x - 1}+6 \))