Question

question
select all the correct answers.
annie transforms the graph of ( y = 2x ) into the graph of ( y = 2x - 4 ). which statements correctly describe this transformation?

• it is the graph of ( y = 2x ) translated 4 units to the right.
• it is the graph of ( y = 2x ) translated 4 units downward.
• it is the graph of ( y = 2x ) translated 4 units upward.
• it is the graph of ( y = 2x ) translated 2 units to the right.
• it is the graph of ( y = 2x ) compressed horizontally 4 units to the left.
• it is the graph of ( y = 2x ) stretched vertically 2 units upward.

Explanation:

Step1: Recall Vertical Translation Rules

For a function \( y = f(x) \), the transformation \( y = f(x)+k \) (or \( y = f(x)-|k| \) when \( k<0 \)) represents a vertical translation. If \( k>0 \), it's a translation upward by \( k \) units; if \( k<0 \), it's a translation downward by \( |k| \) units. Also, for horizontal translation, \( y = f(x - h) \) is a translation to the right by \( h \) units, and \( y = f(x + h) \) is a translation to the left by \( h \) units.

Step2: Analyze the Given Functions

The original function is \( y = 2x \), and the transformed function is \( y = 2x - 4 \). We can also rewrite the transformed function in terms of horizontal translation: solve \( 2x - 4=2(x - 2) \), so \( y = 2(x - 2) \).

Step3: Evaluate Each Statement

• Statement 1: Translated 4 units right

For horizontal translation, the transformed function would need to be \( y = 2(x - 4) = 2x - 8 \), which is not our case. So this is incorrect.

• Statement 2: Translated 4 units downward

Using the vertical translation rule, \( y = 2x-4 \) is \( y = 2x+(-4) \), so it's a vertical translation downward by 4 units. This is correct.

• Statement 3: Translated 4 units upward

A vertical translation upward would be \( y = 2x + 4 \), not \( -4 \). So this is incorrect.

• Statement 4: Translated 2 units right

From \( y = 2(x - 2) \), this is a horizontal translation to the right by 2 units (since \( h = 2 \) in \( y = f(x - h) \)). This is correct.

• Statement 5: Compressed horizontally 4 units left

Horizontal compression/stretch involves multiplying \( x \) by a factor, not subtracting a constant inside. The form here is a translation, not a compression. So this is incorrect.

• Statement 6: Stretched vertically 2 units upward

Vertical stretch involves multiplying the function by a factor greater than 1, e.g., \( y = 2\cdot2x \) for a vertical stretch. Here, the change is a translation, not a stretch. So this is incorrect.

Answer:

B. It is the graph of \( y = 2x \) translated 4 units downward,
D. It is the graph of \( y = 2x \) translated 2 units to the right