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Quiz: Transformations

Test your understanding of translations, reflections, stretches, and composite transformations with these review questions.

1. The graph of \(y = f(x) + 5\) is obtained from \(y = f(x)\) by:

  1. Shifting left by \(5\)
  2. Shifting right by \(5\)
  3. Shifting up by \(5\)
  4. Shifting down by \(5\)
Show Answer

The correct answer is C. Adding a constant \(b\) outside the function shifts the graph vertically. Since \(5 > 0\), the shift is upward by \(5\) units. Every point \((x, y)\) on \(f\) becomes \((x, y + 5)\) on the new graph. Horizontal shifts occur when the change is made to the input inside the function, not outside.

Concept Tested: Vertical Translation

2. The graph of \(y = f(x - 3)\) is obtained from \(y = f(x)\) by:

  1. Shifting right by \(3\)
  2. Shifting left by \(3\)
  3. Shifting down by \(3\)
  4. Shifting up by \(3\)
Show Answer

The correct answer is A. Horizontal translations work opposite to the sign inside the function: \(f(x - 3)\) shifts the graph right by \(3\) (not left). This counterintuitive behavior is the most common transformation mistake. To get the same \(y\) value, the new \(x\) must be \(3\) larger than the original, so every point moves right.

Concept Tested: Horizontal Translation

3. What transformation maps \(y = x^2\) to \(y = -x^2\)?

  1. Reflection in the \(y\)-axis
  2. Vertical translation down
  3. Horizontal stretch
  4. Reflection in the \(x\)-axis
Show Answer

The correct answer is D. The transformation \(y = -f(x)\) negates every output, which reflects the graph across the \(x\)-axis. The upward parabola becomes a downward parabola. Note that for \(y = x^2\), reflection in the \(y\)-axis (\(y = (-x)^2 = x^2\)) would have no visible effect because the function is even — so option A is the wrong kind of reflection here.

Concept Tested: Reflection in X-Axis

4. The point \((3, 4)\) is on the graph of \(y = f(x)\). Where is the corresponding point on \(y = f(x) + 2\)?

  1. \((5, 4)\)
  2. \((3, 6)\)
  3. \((3, 2)\)
  4. \((1, 4)\)
Show Answer

The correct answer is B. Adding \(2\) outside the function shifts the graph vertically up by \(2\). The \(x\)-coordinate stays the same, but the \(y\)-coordinate increases by \(2\): \((3, 4) \to (3, 4 + 2) = (3, 6)\). Option A incorrectly shifts horizontally, and option C subtracts instead of adding.

Concept Tested: Transformation of Points

5. Which equation represents the graph of \(y = x^2\) after a vertical stretch by factor \(3\)?

  1. \(y = x^2 + 3\)
  2. \(y = (3x)^2\)
  3. \(y = 3x^2\)
  4. \(y = (x + 3)^2\)
Show Answer

The correct answer is C. A vertical stretch by factor \(p\) multiplies the entire function by \(p\): \(y = p \cdot f(x) = 3x^2\). Every \(y\)-value triples. Option A is a vertical translation, option B is a horizontal compression (note: \((3x)^2 = 9x^2\) not \(3x^2\)), and option D is a horizontal translation.

Concept Tested: Vertical Stretch

6. The graph of \(y = f(2x)\) compared to \(y = f(x)\) is:

  1. Stretched horizontally by factor \(2\)
  2. Compressed horizontally by factor \(\frac{1}{2}\)
  3. Stretched vertically by factor \(2\)
  4. Shifted right by \(2\)
Show Answer

The correct answer is B. Replacing \(x\) with \(2x\) causes a horizontal compression by factor \(\frac{1}{2}\) (the graph becomes narrower). This is counterintuitive — multiplying by a value greater than \(1\) inside the function compresses rather than stretches. Every \(x\)-coordinate is halved: \((x, y) \to (x/2, y)\). This is the horizontal analog of the "opposite direction" rule.

Concept Tested: Horizontal Stretch

7. Starting from \(y = x^2\), what transformations produce \(y = (x - 3)^2 + 4\)?

  1. Right \(3\), up \(4\)
  2. Left \(3\), up \(4\)
  3. Right \(3\), down \(4\)
  4. Left \(3\), down \(4\)
Show Answer

The correct answer is A. The term \((x - 3)\) inside the function means a horizontal translation right by \(3\) (opposite to the sign). The \(+4\) outside means a vertical translation up by \(4\). The vertex of the parabola moves from \((0, 0)\) to \((3, 4)\). This is the vertex form of a quadratic, which makes the vertex location obvious from the equation.

Concept Tested: Composite Transformation

8. The point \((2, 5)\) is on \(y = f(x)\). Which point is on \(y = f(-x)\)?

  1. \((2, -5)\)
  2. \((-2, 5)\)
  3. \((-2, -5)\)
  4. \((5, 2)\)
Show Answer

The correct answer is B. The transformation \(y = f(-x)\) is a reflection in the \(y\)-axis, which negates the \(x\)-coordinate of every point while keeping the \(y\)-coordinate unchanged. So \((2, 5) \to (-2, 5)\). Option A describes reflection in the \(x\)-axis, option C is reflection through the origin, and option D is reflection across the line \(y = x\) (inverse).

Concept Tested: Reflection in Y-Axis

9. A scale factor of \(\frac{1}{2}\) applied vertically to \(y = f(x)\) produces:

  1. \(y = \frac{1}{2}f(x)\), the graph is half as tall
  2. \(y = 2f(x)\), the graph is twice as tall
  3. \(y = f(2x)\), the graph is half as wide
  4. \(y = f(x/2)\), the graph is twice as wide
Show Answer

The correct answer is A. A vertical scale factor of \(\frac{1}{2}\) multiplies every \(y\)-value by \(\frac{1}{2}\), compressing the graph vertically to half its original height. The equation is \(y = \frac{1}{2}f(x)\). Options C and D describe horizontal transformations, and option B describes a stretch (not a compression).

Concept Tested: Scale Factor

10. In what order should the transformations in \(y = 3f(x - 2) + 1\) be applied to the graph of \(f\)?

  1. Shift up by \(1\), then stretch by \(3\), then shift right by \(2\)
  2. Stretch by \(3\), then shift up by \(1\), then shift right by \(2\)
  3. Shift right by \(2\), then stretch vertically by \(3\), then shift up by \(1\)
  4. Shift left by \(2\), then stretch by \(3\), then shift down by \(1\)
Show Answer

The correct answer is C. Work from inside out. The inner operation \((x - 2)\) affects the input first, so the graph shifts right by \(2\). Next, the factor \(3\) multiplies the outputs, giving a vertical stretch by \(3\). Finally, \(+1\) shifts the result up by \(1\). Horizontal operations first, vertical operations second — and within each, stretches before translations (or the rule "inside-out" handles it automatically).

Concept Tested: Graph Transformation Order