6.3 Function Transformations

6.3 Function Transformations

Quick Answer: The master transformation model g(x) = a · f(b(x - h)) + k systematically alters parent function graphs. Parameters a and k govern vertical changes (a stretches if $|a| > 1$, compresses if $0 < |a| < 1$, reflects across the x-axis if $a < 0$; k shifts vertically up or down). Parameters b and h govern horizontal changes (b compresses by factor $1/|b|$ if $|b| > 1$, stretches if $0 < |b| < 1$, reflects across the y-axis if $b < 0$; h shifts horizontally right $h$ or left $-h$). In coordinate mapping: $(x, y) \to (\frac{x}{b} + h, \; ay + k)$.

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The Master Transformation Architecture & Vertical Operations (AII-F.BF.3b)

Every function in Algebra II—including polynomials, radicals, exponentials, logarithms, and trigonometric functions—derives from a basic parent function $y = f(x)$. Applying algebraic constants to the inputs and outputs transforms the parent graph via reflections, dilations (stretches/compressions), and translations.

The general transformation form is represented as:

$$g(x) = a \cdot f(b(x - h)) + k$$

Vertical Transformations: Operating Outside the Function

Operations performed outside the parent function evaluate after $f$ has processed the input, directly modifying the output $y$-coordinates in an intuitive, direct manner:

1. Vertical Translation ($k$):

   • $g(x) = f(x) + k$ shifts the graph vertically upward by $k$ units ($k > 0$).
   • $g(x) = f(x) - k$ shifts the graph vertically downward by $k$ units ($k > 0$).
   • Coordinate effect: $(x, y) \to (x, y + k)$.

2. Vertical Dilation ($a$):

   • If $|a| > 1$, the graph undergoes a vertical stretch by a factor of $|a|$ (points move farther away from the $x$-axis).
   • If $0 < |a| < 1$, the graph undergoes a vertical compression (shrink) by a factor of $|a|$ (points move closer to the $x$-axis).
   • Coordinate effect: $(x, y) \to (x, a \cdot y)$.

3. Vertical Reflection (Across the $x$-Axis):

   • $g(x) = -f(x)$ negates all output values, flipping the graph vertically across the $x$-axis.
   • Coordinate effect: $(x, y) \to (x, -y)$.

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Horizontal Transformations & The Factoring Rule

Operations performed inside the function's argument modify the input values $x$ before $f$ processes them. Horizontal transformations behave in a counter-intuitive, inverse manner because the input $x$ must adjust to produce the original function argument:

1. Horizontal Translation ($h$):

   • $g(x) = f(x - h)$ shifts the graph horizontally right by $h$ units ($h > 0$).
   • $g(x) = f(x + h) = f(x - (-h))$ shifts the graph horizontally left by $h$ units ($h > 0$).
   • Coordinate effect: $(x, y) \to (x + h, y)$.

2. Horizontal Dilation ($b$):

   • If $|b| > 1$, the graph undergoes a horizontal compression by a factor of $\frac{1}{|b|}$ (points move closer to the $y$-axis).
   • If $0 < |b| < 1$, the graph undergoes a horizontal stretch by a factor of $\frac{1}{|b|}$ (points move farther away from the $y$-axis).
   • Coordinate effect: $(x, y) \to (\frac{x}{b}, y)$.

3. Horizontal Reflection (Across the $y$-Axis):

   • $g(x) = f(-x)$ negates the input values, reflecting the graph across the $y$-axis.
   • Coordinate effect: $(x, y) \to (-x, y)$.

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Coordinate Mapping & Transformation Sequencing

When a function undergoes multiple combined transformations, the order of operations dictates how coordinates transform:

• Vertical Transformations (Outside): Follow standard algebraic order of operations—apply multiplications first (vertical stretch/compression and $x$-axis reflection via $a$), followed by additions/subtractions (vertical translation via $k$).
• Horizontal Transformations (Inside): In factored form $b(x - h)$, inputs scale by $\frac{1}{b}$ and translate by $+h$.
• The Master Coordinate Mapping Formula: Any reference point $(x, y)$ on the parent curve maps to the transformed point: $$(x, y) \longrightarrow \left(\frac{x}{b} + h, \;\; a \cdot y + k\right)$$

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Even and Odd Functions: Algebraic and Graphical Symmetry

A function can exhibit specific rotational or reflectional symmetry across its entire domain:

Even Functions: Line Symmetry Across the $y$-Axis

• Algebraic Condition: A function $f$ is even if and only if for every $x$ in its domain: $$f(-x) = f(x)$$
• Graphical Symmetry: The graph is symmetric with respect to the $y$-axis (the line $x = 0$). Folding the graph along the vertical $y$-axis produces perfect alignment.
• Polynomial Rule: A polynomial function is even if all variable terms contain even exponents (including a non-zero constant term $c = c \cdot x^0$, since 0 is an even integer). Examples: $f(x) = x^4 - 3x^2 + 7$, $f(x) = \cos(x)$, $f(x) = |x|$.

Odd Functions: Point Symmetry About the Origin

• Algebraic Condition: A function $f$ is odd if and only if for every $x$ in its domain: $$f(-x) = -f(x)$$
• Graphical Symmetry: The graph possesses $180^\circ$ rotational symmetry about the origin $(0, 0)$. Rotating the graph upside down leaves it unchanged. If $(x, y)$ lies on the curve, then $(-x, -y)$ must also lie on the curve.
• Polynomial Rule: A polynomial function is odd if all variable terms contain odd exponents and the constant term is strictly zero. Examples: $f(x) = x^5 - 4x^3 + 2x$, $f(x) = \sin(x)$, $f(x) = \frac{1}{x}$.

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Worked Examples

Worked Problem 1: Coordinate Mapping Under Multiple Transformations

Problem: The graph of $y = f(x)$ contains anchor points $(-4, 0)$, $(0, 6)$, and $(2, -2)$. Determine the coordinates of these points on the graph of $g(x) = -2f(x - 3) + 4$ and describe the complete transformation sequence.

• Step 1: Identify transformation parameters. From $g(x) = -2f(x - 3) + 4$:

  • $a = -2$: Vertical stretch by factor of 2 and reflection across the $x$-axis.
  • $b = 1$: No horizontal dilation.
  • $h = 3$: Horizontal shift right 3 units.
  • $k = 4$: Vertical shift up 4 units.

• Step 2: Construct the coordinate mapping rule.

  $$(x, y) \longrightarrow (x + 3, \; -2y + 4)$$

• Step 3: Map each anchor point.

  • Point 1: $(-4, 0) \longrightarrow (-4 + 3, \; -2(0) + 4) = (-1, 4)$
  • Point 2: $(0, 6) \longrightarrow (0 + 3, \; -2(6) + 4) = (3, -12 + 4) = (3, -8)$
  • Point 3: $(2, -2) \longrightarrow (2 + 3, \; -2(-2) + 4) = (5, 4 + 4) = (5, 8)$

Worked Problem 2: Factoring Horizontal Arguments in Radical Functions

Problem: Describe the transformations that transform the parent function $f(x) = \sqrt{x}$ into $g(x) = \sqrt{4x - 12} + 1$.

• Step 1: Factor the coefficient of $x$ inside the radical.

  $$g(x) = \sqrt{4(x - 3)} + 1$$

• Step 2: Extract transformation parameters.

  • Inside: $b = 4$, indicating a horizontal compression by a factor of $\frac{1}{4}$.
  • Inside: $h = 3$, indicating a horizontal shift right 3 units.
  • Outside: $k = 1$, indicating a vertical shift up 1 unit.

• Step 3: Alternative algebraic verification via radical properties.

  $$g(x) = \sqrt{4(x - 3)} + 1 = \sqrt{4} \cdot \sqrt{x - 3} + 1 = 2\sqrt{x - 3} + 1$$

  This confirms that a horizontal compression by a factor of $\frac{1}{4}$ on a square root function is algebraically equivalent to a vertical stretch by a factor of $2$, followed by a shift right 3 units and up 1 unit.

Worked Problem 3: Algebraic Proofs for Even, Odd, and Neither Functions

Problem: Algebraically prove whether each function is even, odd, or neither:

1. $f(x) = x^4 - 6x^2 + 8$
2. $g(x) = 2x^3 - 7x$
3. $h(x) = x^3 + 4x^2 - 5$

• Proof 1 for $f(x)$: Substitute $(-x)$ for $x$:

  $$f(-x) = (-x)^4 - 6(-x)^2 + 8 = x^4 - 6x^2 + 8$$

  Because $f(-x) = f(x)$, $f(x)$ is an even function.

• Proof 2 for $g(x)$: Substitute $(-x)$ for $x$:

  $$g(-x) = 2(-x)^3 - 7(-x) = -2x^3 + 7x$$

  Factor out $-1$:

  $$g(-x) = -(2x^3 - 7x) = -g(x)$$

  Because $g(-x) = -g(x)$, $g(x)$ is an odd function.

• Proof 3 for $h(x)$: Substitute $(-x)$ for $x$:

  $$h(-x) = (-x)^3 + 4(-x)^2 - 5 = -x^3 + 4x^2 - 5$$

  Compute $-h(x)$:

  $$-h(x) = -(x^3 + 4x^2 - 5) = -x^3 - 4x^2 + 5$$

  Notice that $h(-x) \neq h(x)$ (since $-x^3 \neq x^3$) and $h(-x) \neq -h(x)$ (since $+4x^2 \neq -4x^2$). Therefore, $h(x)$ is neither even nor odd.

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Common Regents Pitfalls & Exam Strategies

• Pitfall 1: Failing to factor inside horizontal expressions. Looking at $f(3x - 12)$ and claiming a horizontal shift of 12 units right is an immediate error. Factor first: $f(3(x - 4))$, revealing a shift of 4 units right.
• Pitfall 2: Confusing reflection axes. $-f(x)$ negates outputs ($y$-values), reflecting across the $x$-axis. $f(-x)$ negates inputs ($x$-values), reflecting across the $y$-axis.
• Pitfall 3: Assuming a constant term preserves odd symmetry. In $g(x) = x^3 - 5x + 4$, the presence of the non-zero constant $+4$ breaks odd symmetry because $g(-x) = -x^3 + 5x + 4 \neq -(x^3 - 5x + 4)$.