Master Study Guide: Functions, Graphs & Transformations

Introduction and Core Concepts of Functions

• Definition of a Function: A rule where every allowed input has exactly one output.

• Machine Analogy: Put an input $x$ into the machine, the function rule acts on $x$, and exactly one output $y$-value comes out.

• Key Test for Functionality:

  • One input $x$ cannot produce two different output $y$-values.

  • Different input $x$-values are permitted to produce the exact same output $y$-value.

• Vertical Line Test:

  • A visual method to determine if a graph represents $y$ as a function of $x$.

  • If any vertical line intersects a graph at more than one point, the graph does not represent $y$ as a function of $x$.

• Illustrative Function Examples:

  • Function Example: $y = x^2$ is a function. Evaluating at $x = 3$ yields exactly one output, $y = 9$.

  • Non-Function Example: $x = y^2$ is not $y$ as a function of $x$. Evaluating at $x = 4$ produces two outputs, $y = 2$ and $y = -2$.

Function Notation and Evaluation

• Function Notation $f(x)$:

  • Naming Output: $f(x)$ is another way of naming the output $y$.

  • Operational Meaning: It means the function $f$ evaluated at $x$, and does NOT mean $f$ times $x$.

• Substitution Rule: Whatever expression or value appears inside the parentheses replaces $x$ everywhere in the function rule.

• Basic Evaluation Example:

  • Given: $f(x) = 2x + 5$

  • Evaluating $f(3)$ requires replacing every $x$ with $3$: $f(3) = 2(3) + 5 = 11$

• Algebraic Substitution Examples:

  • Given: $f(x) = x^2 - 4x + 1$

  • Substituting $a$ yields: $f(a) = a^2 - 4a + 1$

  • Substituting $x+h$ yields: $f(x+h) = (x+h)^2 - 4(x+h) + 1$

Domain, Range, and Radical Behavior

• Domain: The set of every possible input $x$.

• Range: The set of every possible output $y$.

• General Domain Finding Strategy:

  • Start by assuming the domain includes all real numbers $(-\neq, \neq)$.

  • Inspect the function expression for potential algebraic restrictions:

  • Polynomials: Expressions like $x^2 + 3x - 1$ have no restrictions and allow all real numbers for $x$.

  • Fractions / Rational Functions: The denominator cannot equal zero ($0$).

  • Even Square Roots: Even-root radicands cannot contain negative real numbers; the interior must be greater than or equal to zero ($\neq 0$).

• Even Root Domain Calculation Example:

  • For $√{x+7}$, set the interior $x + 7 ≥ 0$.

  • Solving for $x$ gives $x ≥ -7$.

• Odd Roots vs. Even Roots Behavior:

  • Odd roots (such as cube roots and fifth roots) can contain negative numbers inside the radicand.

  • Example Equation 1: $x^4 + y^5 = 23$ simplifies to $y = √[5]{23 - x^4}$. This produces exactly one real $y$ for every real $x$, so it is a valid function.

  • Example Equation 2: $x^4 + y^2 = 23$ simplifies to $y = ±√{23 - x^4}$. The $±$ creates two distinct outputs for many inputs, so it is not $y$ as a function of $x$.

Interval Notation

• Overview: Compact notation for describing sets of real numbers.

• Inclusion and Exclusion Symbols:

  • Parentheses $(\,)$ exclude an endpoint.

  • Brackets $[\,]$ include an endpoint.

  • Infinity $∞$ and negative infinity $-∞$ always use parentheses.

• Interval Examples:

  • Inequality $x ≥ -7$ becomes $[-7, ∞)$.

  • Inequality $-2 < x ≤ 5$ becomes $(-2, 5]$.

  • All real numbers becomes $(-∞, ∞)$.

Linear Functions

• Definition: A linear function graphs as a straight line and can be written in slope-intercept form $y = mx + b$, where:

  • $m$ represents the slope.

  • $b$ represents the y-intercept.

• Linear Conversion Example:

  • Given equation: $x = 6y + 7$

  • Solve for $y$:     $x - 7 = 6y$     $y = \frac{1}{6}x - \frac{7}{6}$

  • Slope $m = \frac{1}{6}$ and y-intercept $b = -\frac{7}{6}$.

• Identifying Nonlinear Functions: Expressions containing $x^2$, $\sqrt{x}$, $|x|$, $xy$, or other non-linear powers are generally not linear functions.

Core Parent Functions

• Quadratic Parent Function: $y = x^2$

  • U-shaped parabola with vertex at $(0,0)$.

• Absolute Value Parent Function: $y = |x|$

  • V-shaped graph with vertex at $(0,0)$.

• Square-Root Parent Function: $y = \sqrt{x}$

  • Graph starts at endpoint $(0,0)$ and curves to the right.

• Linear Parent Function: $y = x$

  • Straight line passing through $(0,0)$ with slope $1$.

General Transformation Formula and Rules

• Master Transformation Formula: $y = a f(x - h) + k$

  • $a$: Controls vertical stretch, vertical compression, and vertical reflection.

  • $h$: Controls horizontal translations.

  • $k$: Controls vertical translations.

• Horizontal Shift Rules:

  • Operations inside the function perform the opposite of intuition.

  • $f(x - 3)$ shifts the graph RIGHT by $3$ units.

  • $f(x + 3)$ shifts the graph LEFT by $3$ units.

• Vertical Shift Rules:

  • Operations outside the function behave normally.

  • $f(x) + 4$ shifts the graph UP by $4$ units.

  • $f(x) - 4$ shifts the graph DOWN by $4$ units.

• Reflection Rules:

  • $-f(x)$ reflects the graph vertically across the x-axis.

  • $f(-x)$ reflects the graph horizontally across the y-axis.

Vertical Stretch and Compression

• Parameter Evaluation: In $y = a f(x)$, evaluate $|a|$ relative to $1$:

  • If $|a| > 1$, the graph is vertically stretched, becoming steeper and narrower.

  • If $0 < |a| < 1$, the graph is vertically compressed, becoming flatter and wider.

• Job Separation Principle:

  • Magnitude $|a|$ controls vertical stretch/compression.

  • Sign of $a$ controls reflection across the x-axis.

• Example Analysis $y = -3x^2$:

  • Magnitude $|a| = 3$ stretches the parabola vertically by a factor of $3$.

  • Negative sign reflects the graph across the x-axis, making the parabola open downward.

Quadratics and Parabolas

• Parent Function: $y = x^2$

• Vertex Form: $y = a(x - h)^2 + k$

  • Vertex location is $(h, k)$.

• Opening Direction Rules:

  • If $a > 0$, the parabola opens upward.

  • If $a < 0$, the parabola opens downward.

  • Reasoning: A squared number is non-negative. A positive multiplier $a$ places the arms above the vertex; a negative multiplier $a$ reflects them below the vertex.

• Transformation Example $y = 2(x - 3)^2 - 4$:

  • Vertex is $(3, -4)$.

  • Horizontal shift: Right $3$ units.

  • Vertical shift: Down $4$ units.

  • Opening direction: Opens upward ($a = 2 > 0$).

  • Vertical stretch: Stretched vertically by a factor of 2$.\n\n\n# Sideways Parabolas\n\n- Standard Form: When y$is squared instead of$x$, the parabola opens sideways:$x = a(y - k)^2 + h\n\n- Key Coordinates: The vertex is (h, k).\n\n- Detailed Example Analysis x = y^2 + 8:\n - Vertex is (8, 0).\n - Opening direction: Opens to the right.\n - Evaluating Points:\n - Substituting y = 1$yields$x = (1)^2 + 8 = 9$, giving point$(9, 1).\n - Substituting y = -1$yields$x = (-1)^2 + 8 = 9$, giving point$(9, -1).\n - Function Status: Fails the vertical line test because input x = 9$yields two distinct outputs ($y = 1$and$y = -1).\n\n\n# Absolute-Value Transformations\n\n- Parent Graph: y = |x|$forms a V shape with vertex$(0,0).\n\n- Transformation Form: y = a|x - h| + k\n - Vertex is at (h, k).\n\n- Example Analysis y = -2|x + 3| + 1:\n - Vertex is (-3, 1).\n - Shifts: Left 3$units, up$1 unit.\n - Reflection: Reflected across the x-axis.\n - Stretch: Vertically stretched by a factor of 2$.

• Graphing Procedure:

  1. Plot the vertex $(h, k)$ first.

  2. Use $|a|$ to determine how quickly each arm rises or falls for every $1$ unit moved left or right.

Square-Root Transformations

• Parent Graph: $y = \sqrt{x}$ begins at $(0,0)$ and extends right.

• Transformation Form: $y = a\sqrt{x - h} + k$

  • Starting endpoint is $(h, k)$.

• Example Analysis $y = \sqrt{x + 7}$:

  • Begins at endpoint $(-7, 0)$ because $x + 7 = 0$ when $x = -7$.

  • Domain is $[-7, \infty)$.

• Useful Reference Points: Select perfect square inputs for evaluating parent function points:

  • $x = 0 \rightarrow y = 0$

  • $x = 1 \rightarrow y = 1$

  • $x = 4 \rightarrow y = 2$

  • $x = 9 \rightarrow y = 3$

The Difference Quotient

• Formula: $\frac{f(x+h) - f(x)}{h}$ where $h \ne 0$

• Purpose: Measures the average rate of change and serves as the foundation for derivatives in calculus.

• Step-by-Step Evaluation for $f(x) = x^2$:

  1. Evaluate $f(x+h)$: $f(x+h) = (x+h)^2 = x^2 + 2xh + h^2$

  2. Subtract $f(x)$: $(x^2 + 2xh + h^2) - x^2 = 2xh + h^2$

  3. Divide by $h$: $\frac{2xh + h^2}{h} = \frac{h(2x + h)}{h} = 2x + h$

• Simplification Rules:

  • An expression like $\frac{6h}{h}$ simplifies to $6$ because $\frac{h}{h} = 1$ when $h \ne 0$

• Common Error Warning: A primary mistake is forgetting to distribute subtraction across every term in $f(x)$.

Translating Word Problems into Functions

• Translation Procedure: First identify unknown quantities, then establish algebraic relationships using geometric or physical constraints.

• Example Problem Walkthrough:

  • Scenario: A rectangle has a perimeter of $420\,m$.

  • Step 1: Write the perimeter equation:     $P = 2L + 2W = 420$

  • Step 2: Simplify by dividing by $2$:     $L + W = 210$

  • Step 3: Express width in terms of length:     $W = 210 - L$

  • Step 4: Express area as a function of length $L$:     $A = L \times W$     $A(L) = L(210 - L) = 210L - L^2$

  • Step 5: Units verification: Area uses square meters ($m^2$) because meters times meters equals $m^2$.

  • Step 6: Determine physical domain restrictions:

  • Both physical dimensions must be strictly positive ($L > 0$ and $210 - L > 0$).

  • Solving $210 - L > 0$ gives $L < 210$

  • Physical Domain: $0 < L < 210$

Reliable Graphing Strategy

• Step 1: Identify the parent function.

• Step 2: Rewrite the equation into transformation form if possible.

• Step 3: Locate the primary vertex, endpoint, or intercept.

• Step 4: Apply horizontal and vertical shifts.

• Step 5: Apply reflection across axes and vertical stretch/compression.

• Step 6: Plot a few easy, key points.

• Step 7: Verify domain, range, opening direction, and test against the vertical line test.

Common Traps and Pitfalls

• Notation Error: $f(x)$ represents function evaluation, not multiplication of $f$ and x$.\n\n- Direction Confusion: Inside horizontal shifts move in the opposite direction of the sign (x + 7$means shift LEFT by$7).\n\n- Radical Fallacy: A square root alone does not make an expression fail to be a function. y = \sqrt{x}$IS a valid function, whereas$y = \pm\sqrt{x} is NOT a function.\n\n- Parameter Isolation: A negative sign on a$means reflection across the x-axis, while magnitude$|a| determines vertical stretch/compression.\n\n- Domain Exclusions: Even-root radicands must be \ge 0$, and rational denominators cannot equal$0$.

• Variable Power Orientation: $x^2$ forms a vertical parabola; $y^2$ forms a sideways parabola.

Practice Questions and Detailed Answers

• Question 1: Is $y = x^2 + 4$ a function? Explain why.

  • Answer 1: Yes. Every input $x$ produces exactly one output y$.\n\n- Question 2: Is x = y^2 + 8$a function of$x? Explain with the vertical line test.\n - Answer 2: No. Many x$-values correspond to both a positive and negative$y; a vertical line intersects the graph twice.\n\n- Question 3: Find the domain of f(x) = \sqrt{x-5} in interval notation.\n - Answer 3: Set x - 5 \ge 0 \rightarrow x \ge 5$. Domain is$[5, \infty).\n\n- Question 4: Describe every transformation from y = x^2$to$y = -3(x+2)^2 + 5.\n - Answer 4: Shifted left 2$units, shifted up$5$units, vertically stretched by a factor of$3, and reflected across the x-axis.\n\n- Question 5: Give the vertex and opening direction of y = 2(x-4)^2 - 7$.

  • Answer 5: Vertex is $(4, -7)$; opens upward because a = 2 > 0$.\n\n- Question 6: Give the vertex and direction of x = (y-3)^2 - 2$.

  • Answer 6: Vertex is $(-2, 3)$; opens to the right.

• Question 7: Describe $y = \frac{1}{2}|x-6| + 1$ relative to y = |x|$.\n - Answer 7: Shifted right 6$units, shifted up$1$unit, and vertically compressed by a factor of$\frac{1}{2}.\n\n- Question 8: If f(x) = 3x^2 - 2x$, find$f(2).\n - Answer 8: 3(2^2) - 2(2) = 3(4) - 4 = 12 - 4 = 8.\n\n- Question 9: If f(x) = x^2$, simplify$\frac{f(x+h) - f(x)}{h}.\n - Answer 9: \frac{(x+h)^2 - x^2}{h} = \frac{2xh + h^2}{h} = 2x + h\n\n- Question 10: A rectangle has perimeter 100$. Write area as a function of$L and give its physical domain.\n - Answer 10: Width W = 50 - L$, area function$A(L) = L(50 - L) = 50L - L^2$, physical domain$0 < L < 50$.

Mastery Check and Recommended Study Strategy

• Mastery Criteria: Mastery is achieved when looking at any given equation allows immediate determination of:

  • Parent function

  • Function status (whether it is a valid function)

  • Domain and range

  • All applied transformations

  • Vertex or starting endpoint coordinates

  • Opening or extension direction

  • Key graph coordinates

  • Justification for why each property is true

• Recommended Study Strategy:

  • Do not memorize left/right/up/down rules as isolated shortcuts.

  • Start from the parent function graph and analyze what the equation mathematically does to input $x$ and output $$y$.

  • Transforming isolated rules into underlying reasoning allows application to unfamiliar and complex problems.