Transformations of graphs; composite transformations practice questions.

The graph of y = f(x) is transformed to y = f(2x). State how the graph is affected.

• Horizontal stretch, scale factor 2
• Horizontal stretch, scale factor 1/2
• Vertical stretch, scale factor 2
• Vertical stretch, scale factor 1/2

Answer: Horizontal stretch, scale factor 1/2

Replacing x with 2x compresses the graph horizontally toward the y-axis by a factor of 1/2, since each point's x-coordinate is halved.

The graph of y = 3f(x) is obtained from y = f(x) by a:

• Horizontal stretch, scale factor 3
• Horizontal stretch, scale factor 1/3
• Vertical stretch, scale factor 3
• Vertical stretch, scale factor 1/3

Answer: Vertical stretch, scale factor 3

Multiplying the whole function by 3 stretches the graph vertically by scale factor 3; each y-value is tripled.

The point (4, 6) lies on y = f(x). Determine the image of this point on the graph of y = f(x) − 5.

• (4, 1)
• (4, 11)
• (−1, 6)
• (9, 6)

Answer: (4, 1)

Subtracting 5 outside the function is a vertical translation down by 5, so the y-coordinate changes from 6 to 1 while x stays 4.

The point (2, 7) lies on y = f(x). Determine the image of this point on y = f(x + 3).

• (5, 7)
• (−1, 7)
• (2, 10)
• (2, 4)

Answer: (−1, 7)

Replacing x with x + 3 translates the graph 3 units left, so x becomes 2 − 3 = −1 while y is unchanged.

The graph of y = f(x) is reflected in the x-axis and then translated up by 4. State the resulting equation.

• y = −f(x) + 4
• y = f(−x) + 4
• y = −f(x + 4)
• y = −(f(x) + 4)

Answer: y = −f(x) + 4

Reflection in the x-axis negates the output giving −f(x); translating up 4 adds 4, giving y = −f(x) + 4.

Given f(x) = x², the graph of y = f(x − 2) + 3 has its vertex at:

• (−2, 3)
• (2, 3)
• (2, −3)
• (−2, −3)

Answer: (2, 3)

y = (x − 2)² + 3 is a translation 2 right and 3 up of y = x², moving the vertex from (0, 0) to (2, 3).