Question

question
drag the tiles to the correct boxes to complete the pairs.
match each transformation with its description.
tiles
3f(x) f(x + 3) f(3x) f(x) + 3
pairs
shifts f(x) 3 units upward
shifts f(x) 3 units left
compression f(x) by a factor of \\(\frac{1}{3}\\) toward the y - axis
stretches f(x) by a factor of 3 away from the x - axis

Explanation:

Step1: Recall function transformation rules

• Vertical shift: \( f(x) + k \) shifts \( f(x) \) up \( k \) units (if \( k>0 \)) or down \( |k| \) units (if \( k<0 \)).
• Horizontal shift: \( f(x + h) \) shifts \( f(x) \) left \( h \) units (if \( h>0 \)) or right \( |h| \) units (if \( h<0 \)).
• Horizontal compression/stretch: \( f(bx) \) compresses \( f(x) \) horizontally by a factor of \( \frac{1}{b} \) (if \( b>1 \)) or stretches by a factor of \( \frac{1}{b} \) (if \( 0 < b < 1 \)).
• Vertical stretch/compression: \( a f(x) \) stretches \( f(x) \) vertically by a factor of \( |a| \) (if \( |a|>1 \)) or compresses by a factor of \( |a| \) (if \( 0 < |a| < 1 \)).

Step2: Match each transformation

1. Shifts \( f(x) \) 3 units upward: This is a vertical shift up. Using the vertical shift rule, the transformation is \( f(x)+3 \).
2. Shifts \( f(x) \) 3 units left: This is a horizontal shift left. Using the horizontal shift rule, the transformation is \( f(x + 3) \).
3. Compresses \( f(x) \) by a factor of \( \frac{1}{3} \) (toward the \( y \)-axis): This is a horizontal compression. For horizontal compression by a factor of \( \frac{1}{3} \), we use \( f(3x) \) (since \( b = 3 \), and \( \frac{1}{b}=\frac{1}{3} \)).
4. Stretches \( f(x) \) by a factor of 3 (away from the \( x \)-axis): This is a vertical stretch. Using the vertical stretch rule, the transformation is \( 3f(x) \).

Answer:

• Shifts \( f(x) \) 3 units upward: \( f(x)+3 \)
• Shifts \( f(x) \) 3 units left: \( f(x + 3) \)
• Compresses \( f(x) \) by a factor of \( \frac{1}{3} \) (toward the \( y \)-axis): \( f(3x) \)
• Stretches \( f(x) \) by a factor of 3 (away from the \( x \)-axis): \( 3f(x) \)