Question

1. a graph of a function f(x) = x undergoes a sequence of transformations to form a new function g(x) = -2f(x + 6) - 8. which statement is true about the original function?

□ the graph of f(x) is shifted up 6 units.
□ the graph of f(x) is shifted right 8 units.
□ the graph of f(x) is reflected over the x-axis and stretched vertically.
□ the graph of f(x) is reflected over the y-axis and stretched horizontally.

Explanation:

Step1: Recall Function Transformation Rules

For a function \( y = a f(b(x - h)) + k \):

• Vertical stretch/compression: \( |a| > 1 \) is stretch, \( 0 < |a| < 1 \) is compression; reflection over x - axis if \( a < 0 \).
• Horizontal stretch/compression: \( |b| > 1 \) is compression, \( 0 < |b| < 1 \) is stretch; reflection over y - axis if \( b < 0 \).
• Horizontal shift: \( h \) units right if \( h>0 \), left if \( h < 0 \).
• Vertical shift: \( k \) units up if \( k>0 \), down if \( k < 0 \).

Given \( g(x)=-2f(x + 6)-8 \), compare with \( y = a f(b(x - h)) + k \). Here, \( a=-2 \), \( b = 1 \), \( h=-6 \), \( k=-8 \).

Step2: Analyze Each Option

• Option 1: Shift up 6 units? The vertical shift is \( k=-8 \) (down 8), horizontal shift is \( h = - 6 \) (left 6). So this is wrong.
• Option 2: Shift right 8 units? Horizontal shift is left 6, vertical shift down 8. Wrong.
• Option 3: Reflect over x - axis (\( a=-2<0 \)) and vertical stretch (\( |a| = 2>1 \)). This matches the transformation from \( f(x) \) to \( g(x) \) (since \( a=-2 \) gives reflection over x - axis and vertical stretch).
• Option 4: Reflect over y - axis? \( b = 1>0 \), no reflection over y - axis. Horizontal stretch? \( |b| = 1 \), no stretch. Wrong.

Answer:

The graph of \( f(x) \) is reflected over the x - axis and stretched vertically. (The option: "The graph of \( f(x) \) is reflected over the x - axis and stretched vertically.")