Comprehensive Notes on Function Transformations, Parent Functions, Piecewise Functions, and Function Composition

General Study Advice and Graphing Fundamentals

• Exam and Homework Preparation Strategy:

  • Even when homework assignments are completed online, always maintain written notes on paper.

  • Write down each problem statement fully, solve or sketch the graph on paper step-by-step, and then enter the final response into the online portal.

  • Understanding paper-based graphical methods is essential for mastery on paper-based exams.

• Linear Equations Review:

  • An equation is classified as linear when both variables ($x$ and $y$) are raised to the first power (e.g., $y = x$ or $y = mx + b$).

  • Graphing a Line using Slope: For a linear function with an ordered pair at $(0, -1)$ and a slope of $4$ ($\frac{4}{1}$), start at $(0, -1)$, move $4$ units up and $1$ unit to the right to locate the next point $(1, 3)$, and connect the points with a straight line.

• Table of Values Method for Graphing:

  • If the general shape of any parent function or transformed graph is forgotten, select arbitrary $x$-values across the domain, substitute them into the function to find corresponding $y$-values, plot the resulting ordered pairs $(x, y)$, and connect them to reveal the graph's curve.

Basic Tool Field (Parent) Functions

• Constant Function:

  • Equation: $y = c$

  • Graph: A horizontal line passing through $(0, c)$.

• Identity Function:

  • Equation: $y = x$

  • Graph: A straight line passing through the origin $(0, 0)$ with a slope of $1$, bisecting the first and third quadrants.

• Absolute Value Function:

  • Equation: $y = |x|$

  • Graph: A V-shaped graph composed of two linear rays joining at a central point called the vertex $(0, 0)$.

  • Behavior: The left branch has a negative slope ($y = -x$ for $x < 0$) and the right branch has a positive slope ($y = x$ for $x \neq 0$).

  • Range: $[0, \frac{\infty}{1})$ (or [0, \text{\infty})), because absolute value outputs can never be negative.

• Quadratic Function:

  • Equation: $y = x^2$

  • Graph: A U-shaped curve called a parabola opening upward with its vertex at $(0, 0)$.

• Cubic Function:

  • Equation: $y = x^3$

  • Graph: An S-shaped curve passing through the origin $(0, 0)$, extending into the first and third quadrants.

• Square Root Function:

  • Equation: $y = \sqrt{x}$

  • Graph: A curve representing the upper half of a sideways parabola, starting at $(0, 0)$ and extending into quadrant I.

  • Domain and Range: $[0, \infty)$.

• Cube Root Function:

  • Equation: $y = \sqrt[3]{x}$

  • Graph: An origin-symmetric curve extending infinitely in both positive and negative directions across all real numbers.

• Reciprocal Function:

  • Equation: $y = \frac{1}{x}$

  • Domain Restriction: $x \neq 0$ (division by zero is undefined).

  • Vertical Asymptote: The line $x = 0$ (coinciding with the y-axis). The graph approaches this line infinitely close but never touches or crosses it.

  • Horizontal Asymptote: The line $y = 0$ (coinciding with the x-axis), because $1$ divided by any real number yields a non-zero value.

• Reciprocal Square Function:

  • Equation: $y = \frac{1}{x^2}$

  • Domain Restriction: $x \neq 0$

  • Behavior: Similar asymptotes to the reciprocal function at $x = 0$ and $y = 0$, but because $x^2 > 0$ for all non-zero $x$, both branches lie entirely above the x-axis in quadrants I and II.

Transformations of Functions

• General Transformation Formula Notation:     $y = a \cdot f(b(x + c)) + d$

• Vertical Stretch and Compress ($a$):

  • $a$ is multiplied outside the parent function $f$

  • Vertical Stretch: Occurs when $|a| > 1$. The graph is elongated vertically, making it narrower. All $y$-coordinates of the parent function are multiplied by $a$

  • Vertical Compression (Shrink): Occurs when $0 < |a| < 1$. The graph is flattened vertically, making it wider

• Vertical Reflection ($a$):

  • Positive $a$ ($a > 0$): Graph opens/remains in its standard vertical orientation

  • Negative $a$ ($a < 0$): Graph reflects vertically across the x-axis (flips upside down)

• Horizontal Stretch and Compress ($b$):

  • $b$ is multiplied inside the function argument directly next to $x$

  • Horizontal Compression (Shrink): Occurs when $b > 1$

  • Horizontal Stretch: Occurs when $0 < b < 1$

• Horizontal Reflection ($b$):

  • Negative $b$ ($b < 0$): Graph reflects horizontally across the y-axis (flips left-to-right)

• Vertical Shift ($d$):

  • $d$ is added or subtracted outside the main function

  • $+d$: Shifts the entire graph UP by $d$ units

  • $-d$: Shifts the entire graph DOWN by $d$ units

• Horizontal Shift ($c$):

  • $c$ is added or subtracted inside the function argument directly to $x$

  • Opposite Sign Rule: The shift direction is opposite to the sign of $c$ (the value that brings the argument back to zero)

  • $+c$ (e.g., $f(x + c)$): Shifts the graph LEFT by $c$ units

  • $-c$ (e.g., $f(x - c)$): Shifts the graph RIGHT by $c$ units

Transformation Examples and Graphing Applications

• Identification Examples:

  • $f(x) = (x + 18)^2$ compared to $f(x) = x^2$: Shifted 18 units to the left because $+18$ is inside the function.

  • $f(x) = x^2 + 18$ compared to $f(x) = x^2$: Shifted 18 units up because $+18$ is outside the function.

  • $f(x) = 18\sqrt{x}$ compared to $f(x) = \sqrt{x}$: Vertical stretch by a factor of $18$ because $18 > 1$ is outside the square root.

  • $f(x) = \sqrt{18x}$ compared to $f(x) = \sqrt{x}$: Horizontal compression (shrinking) by a factor of $18$ because $18 > 1$ is inside the square root.

• Absolute Value Transformations:

  • $y = -|x|$: Reflects the standard absolute value graph vertically over the x-axis (V-shape opening downward).

  • $y = |x + 4| + 3$:

    • Shift 4 units left and 3 units up.

    • Parent vertex $(0, 0)$ moves to $(-4, 3)$.

    • Parent point $(1, 1)$ transforms to $(-3, 4)$ via $x_{\text{new}} = 1 - 4 = -3$ and $y_{\text{new}} = 1 + 3 = 4$

• Graph Matching Equations:

  • Absolute value vertex located at $(1, -3)$: Equation is $y = |x - 1| - 3$

  • Absolute value vertex shifted to $(-3, 0)$: Equation is $y = |x + 3|$

• Advanced Parabola Transformation Example: $y = -2(x - 1)^2 + 3$

  • Parent function: $f(x) = x^2$

  • Shift right by $1$ unit ($-1$ inside) and shift up by $3$ units ($+3$ outside).

  • New vertex: $(1, 3)$.

  • Negative sign (-$action) reflects parabola downward across x-axis.\n * Coefficient 2$stretches graph vertically by a factor of$2 (making it narrower).\n\n# Piecewise Functions\n\n* **Definition**: A piecewise function is a function defined by multiple sub-functions, each applying to a distinct interval of the domain (x-values).\n\n* **Endpoint Conventions**:\n * **Filled (Solid) Circle**: Used when an endpoint is included in the domain interval (indicated by \le$or$\ge).\n * **Open Circle**: Used when an endpoint is excluded from the domain interval (indicated by strict inequalities <$or$>).\n\n* **Comprehensive Piecewise Example**:\n         f(x) = \begin{cases} -x - 6 & \text{if } x \le -4 \ 2 & \text{if } -4 < x \le 3 \ 2x - 4 & \text{if } x > 3 \end{cases}     \n\n * **Piece 1 (y = -x - 6$for$x \le -4)**:\n * Evaluate endpoint at x = -4$:$y = -(-4) - 6 = -2 \rightarrow (-4, -2)$. Plot with a **solid circle** because inequality is$\le.\n * Evaluate secondary point at x = -5$:$y = -(-5) - 6 = -1 \rightarrow (-5, -1).\n * Graph shape: A ray starting at (-4, -2) extending upward to the left.\n\n * **Piece 2 (y = 2$for$-4 < x \le 3)**:\n * Horizontal line segment at constant height y = 2\n * Left endpoint at x = -4$:$(-4, 2)$plotted with an **open circle** because inequality is strictly$> -4\n * Right endpoint at x = 3$:$(3, 2)$plotted with a **solid circle** because inequality is$\le 3\n\n * **Piece 3 (y = 2x - 4$for$x > 3)**:\n * Evaluate boundary at x = 3$:$y = 2(3) - 4 = 2 \rightarrow (3, 2). Plot with an **open circle** (overlaps the solid endpoint from Piece 2).\n * Evaluate secondary point at x = 4$:$y = 2(4) - 4 = 4 \rightarrow (4, 4).\n * Graph shape: A ray extending upward to the right starting from (3, 2).\n\n# Composition of Functions\n\n* **Definition & Notation**:\n * Composition notation: (f \circ g)(x) = f(g(x))\n * The output of the inside function g(x)$becomes the input domain for the outside function$f(x).\n\n* **Real-World Application (Successive Discounts)**:\n * Scenario: A pair of \$50$jeans is discounted by$50\%$, with an additional$10\% discount applied at checkout when using a store credit card.\n * Step 1 (First function g(x)$): Apply$50\%$discount to$\$50$:$g(50) = 50 \times (1 - 0.50) = \$25\n * Step 2 (Second function f(x)$): Apply$10\%$credit card discount to the discounted price of$\$25$:$f(25) = 25 - (25 \times 0.10) = \$22.50\n * Composite Result: f(g(50)) = \$22.50$(Note: This is not equal to a flat$60\%$discount, which would yield$\$20).\n\n* **Numerical Evaluation Example**:\n * Given f(x) = 4x - 3$and$g(x) = 4x^2 + 8x + 7$, evaluate$f(g(-3)):\n * Step 1: Evaluate inside function g(-3):\n        g(-3) = 4(-3)^2 + 8(-3) + 7 = 4(9) - 24 + 7 = 36 - 24 + 7 = 19\n * Step 2: Substitute 19$into outside function$f(x):\n        f(19) = 4(19) - 3 = 76 - 3 = 73\n * Final Answer: f(g(-3)) = 73\n\n* **Algebraic Composition Example 1 (f(g(x)))**:\n * Given f(x) = \frac{1}{x - 3}$and$g(x) = \frac{1}{x} - 3:\n * Substitute g(x)$into$f(x):\n        f(g(x)) = \frac{1}{\left(\frac{1}{x} - 3\right) - 3} = \frac{1}{\frac{1}{x} - 6}\n * Find a common denominator for the denominator expression:\n        \frac{1}{x} - 6 = \frac{1 - 6x}{x}\n * Invert and multiply to simplify the complex fraction:\n        f(g(x)) = \frac{x}{1 - 6x}\n\n* **Algebraic Composition Example 2 (g(f(x)))**:\n * Given f(x) = \frac{1}{x - 3}$and$g(x) = \frac{1}{x} - 3:\n * Substitute f(x)$into$g(x):\n        g(f(x)) = \frac{1}{\left(\frac{1}{x - 3}\right)} - 3\n * Simplify the complex fraction:\n        \frac{1}{\frac{1}{x - 3}} = x - 3\n * Subtract 3:\n        g(f(x)) = (x - 3) - 3 = x - 6\n\n# Questions & Discussion\n\n* **Question**: Can formula sheets or specific transformation notes be brought into exams?\n * **Answer**: Yes, notes summarizing general transformation formulas (such as the roles of a$,$b$,$c$, and$d) are permitted for exam reference.\n\n* **Question**: Is the constant d in function transformations always equivalent to the y-intercept?\n * **Answer**: No. For linear functions, d$corresponds to the y-intercept. However, for non-linear functions (such as quadratic or absolute value functions),$d$represents vertical shifting, and the y-intercept must be found separately by evaluating$f(0).\n\n* **Question**: Does multiplying a function by a constant factor greater than 1 make the graph wider?\n * **Answer**: No. A coefficient |a| > 1$causes vertical stretching, making the graph narrower. A graph becomes wider when the coefficient lies between$0$and$1$(e.g.,$a = \frac{1}{2}$$).