Question

question 15 (3 points)
write the equation of the graph after the indicated transformation(s).
the graph of ( y = x^2 ) is shifted 4 units to the left. this graph is then vertically stretched by a factor of 6 and reflected across the ( x )-axis. finally, the graph is shifted 8 units downward.

( \bigcirc ) ( y = -6(x + 8)^2 - 4 )
( \bigcirc ) ( y = -6(x + 4)^2 - 8 )
( \bigcirc ) ( y = -6(x - 4)^2 + 8 )
( \bigcirc ) ( y = -6(x - 4)^2 - 8 )

Explanation:

Step1: Shift left 4 units

For a function \( y = f(x) \), shifting left \( h \) units gives \( y = f(x + h) \). So \( y = x^2 \) shifted left 4 units is \( y=(x + 4)^2 \).

Step2: Vertical stretch by factor 6

Vertical stretch by factor \( a \) gives \( y = a\cdot f(x) \). So now it's \( y = 6(x + 4)^2 \).

Step3: Reflect across x - axis

Reflection across x - axis changes \( y = f(x) \) to \( y=-f(x) \). So we get \( y=-6(x + 4)^2 \).

Step4: Shift downward 8 units

Shifting downward \( k \) units changes \( y = f(x) \) to \( y = f(x)-k \). So the final equation is \( y=-6(x + 4)^2-8 \).

Answer:

\( y = -6(x + 4)^2 - 8 \) (corresponding to the option \( y = -6(x + 4)^2 - 8 \))