Transformations of graphs practice questions.

Compared with y = f(x), the graph of y = f(x) + 3 is translated

• 3 units up
• 3 units down
• 3 units right
• 3 units left

Answer: 3 units up

Adding a constant outside the function shifts the graph vertically upward by 3.

The graph of y = 3f(x) is obtained from the graph of y = f(x) by which single transformation?

• A horizontal stretch, scale factor 3
• A vertical stretch, scale factor 3
• A vertical stretch, scale factor 1/3
• A translation of 3 units upwards

Answer: A vertical stretch, scale factor 3

Multiplying the output f(x) by 3 triples every y-coordinate, which is a vertical stretch with scale factor 3; the x-coordinates are unchanged.

The graph of y = f(x + 2) − 5 is obtained from the graph of y = f(x) by a translation with vector

• (−2, −5)
• (2, −5)
• (−2, 5)
• (2, 5)

Answer: (−2, −5)

Replacing x by x + 2 shifts the graph 2 units left, and subtracting 5 shifts it 5 units down, giving the vector (−2, −5).

The graph of y = x² is transformed to give y = (x − 4)² + 1. State the coordinates of the vertex of the new graph.

• (−4, 1)
• (4, −1)
• (4, 1)
• (−4, −1)

Answer: (4, 1)

The transformation translates the vertex (0, 0) by 4 units right and 1 unit up, so the new vertex is (4, 1); the sign inside the bracket works opposite to intuition.

The point P(6, −2) lies on the graph of y = f(x). Determine the coordinates of the image of P on the graph of y = 2f(x/3).

• (2, −4)
• (18, −1)
• (2, −1)
• (18, −4)

Answer: (18, −4)

f(x/3) is a horizontal stretch with scale factor 3 (x: 6 → 18) and the factor 2 is a vertical stretch (y: −2 → −4), giving (18, −4).

Which sequence of transformations maps the graph of y = x² onto the graph of y = 2(x + 3)²?

• Translation 3 units right, then vertical stretch scale factor 2
• Translation 3 units left, then vertical stretch scale factor 2
• Vertical stretch scale factor 1/2, then translation 3 units right
• Translation 3 units left, then horizontal stretch scale factor 2

Answer: Translation 3 units left, then vertical stretch scale factor 2

Replacing x by x + 3 translates the graph 3 units left to give (x + 3)², and multiplying by 2 is a vertical stretch with scale factor 2.