6.5 Shifting and Scaling Transformations of Function Formulas

6.5 Shifting and Scaling Transformations of Function Formulas

Skill 10 of Competency 2 names exactly three families — linear, quadratic, and absolute value — and asks for the impact on the formula. Items give a described transformation and ask for the new equation, or give two equations and ask what changed.

The master rule: inside versus outside

+---------------------------------------------------------------------------+
|  INSIDE the function (affects x)  ->  OPPOSITE of what it looks like      |
|      f(x - h)   shifts RIGHT h            f(x + h)  shifts LEFT h         |
|      f(bx)      horizontal compression by factor 1/b                      |
|      f(-x)      reflection across the y-axis                              |
|                                                                           |
|  OUTSIDE the function (affects y)  ->  EXACTLY what it looks like         |
|      f(x) + k   shifts UP k               f(x) - k  shifts DOWN k         |
|      a * f(x)   vertical stretch if |a| > 1, compression if 0 < |a| < 1   |
|      -f(x)      reflection across the x-axis                              |
+---------------------------------------------------------------------------+

The counterintuitive half is the inside. f(x − 3) shifts right, not left. The reason: to get the same output the parent produced at x = 0, the new function needs x = 3, so every point moves 3 units right.

The outside behaves as expected because you are directly adding to the output value.

Quadratic transformations

The general form f(x) = a(x − h)² + k encodes all of them:

• h — horizontal shift, vertex x-coordinate
• k — vertical shift, vertex y-coordinate
• a — vertical stretch/compression and reflection

Start with f(x) = x². Shift 5 right, 3 down, and reflect across the x-axis with a vertical stretch of 2. Result: f(x) = −2(x − 5)² − 3, vertex (5, −3), opening downward, narrower than the parent.

Reading in reverse: g(x) = (1/3)(x + 2)² + 6 is the parent shifted 2 left and 6 up, and vertically compressed by a factor of 1/3, so it is wider than y = x². Its vertex is (−2, 6) and its line of symmetry is x = −2.

Converting from standard form requires completing the square, which is why section 6.2 matters here. f(x) = x² + 6x + 11 becomes (x² + 6x + 9) + 2 = (x + 3)² + 2, revealing a shift 3 left and 2 up with vertex (−3, 2).

Absolute value transformations

The form f(x) = a|x − h| + k works identically, with the same vertex (h, k) and line of symmetry x = h.

f(x) = |x| transformed to f(x) = −(1/2)|x + 4| + 1 Vertex: (−4, 1). Reflected downward because a is negative. Vertically compressed by 1/2, so the branches have slopes −1/2 and 1/2 — a wider V than the parent.

The parallel between the quadratic and absolute value forms is deliberate and heavily tested: the same (h, k) reading applies to both, which is why the blueprint groups them.

Linear transformations

For a line, the two parameters have familiar names:

• Vertical shift changes b and leaves m alone. From y = 4x − 1, shifting up 6 gives y = 4x + 5 — a parallel line. This is exactly why parallel lines share a slope: they are vertical translations of each other.
• Vertical stretch by a multiplies the whole expression: a · (mx + b) = amx + ab, changing both slope and intercept. Stretching y = 4x − 1 by 3 gives y = 12x − 3.
• Horizontal shift on a line produces a vertical shift in disguise. From f(x) = 4x − 1, f(x − 2) = 4(x − 2) − 1 = 4x − 9. The graph moved right 2, but because the line is straight, that is indistinguishable from moving down 8. This is a genuine conceptual item: only for linear functions are horizontal and vertical shifts interchangeable.
• Reflection across the x-axis negates everything: −f(x) = −4x + 1, flipping the slope's sign and the intercept's sign.

Order matters when combining

When both a stretch and a vertical shift are applied, the order changes the result.

Take f(x) = x². Stretch by 3, then shift up 5: 3x² + 5. Shift up 5, then stretch by 3: 3(x² + 5) = 3x² + 15.

Different functions. In the standard form a(x − h)² + k, the convention is stretch first, then shift, which is why k is added after the multiplication.

Reading transformations from a description

A parabola has the same shape as y = x² but opens downward with vertex (−1, 7). Same shape means |a| = 1; opening downward means a = −1. So y = −(x + 1)² + 7.

An absolute value graph has vertex (3, −2) and passes through (5, 2). Start with y = a|x − 3| − 2. Substitute (5, 2): 2 = a|5 − 3| − 2 → 4 = 2a → a = 2. So y = 2|x − 3| − 2. Verify at x = 5: 2(2) − 2 = 2 ✓

That second technique — using the vertex for h and k, then one additional point to solve for a — handles most "write the equation" items across both families.