Functions & Graphs Master Study Guide Flashcards

Four-Question Problem Solving Habit

• 1. Identify: Determine the function family involved (linear, quadratic, rational, exponential, logarithmic, or piecewise).

• 2. Target: Clarify the specific mathematical object or value requested (e.g., value, equation, intercept, vertex, domain, range, asymptote, or model).

• 3. Choose: Select the algebraic form or formula that exposes the target most directly.

• 4. Check: Verify that the sign, graph shape, domain, and approximate scale make physical and mathematical sense.

Functions, Notation, and Relations

• Definition of a Function: A rule that assigns exactly one output to each allowed input. A practical metaphor is a vending machine: pressing a single button cannot produce two different physical items simultaneously.

• Function Notation: The notation $f(x)$ represents the output of the function $f$ when given the input $x$. It does not mean $f \times x$.

  • Example: If $f(x) = 2x - 1$, then evaluating at $x = -2$ yields $f(-2) = 2(-2) - 1 = -5$

• Substitution Shortcut: Circle the target input, rewrite the function formula replacing every instance of $x$ with empty parentheses $( )$, and place the input inside every pair of parentheses. This habit prevents missing negative signs during simplification.

• Combining Function Values: To find combined expressions such as $f(-2) + g(-1)$ given $f(x) = 2x - 1$ and $g(x) = x^2 - 2x$:

  • Evaluate $f(-2) = 2(-2) - 1 = -5$

  • Evaluate $g(-1) = (-1)^2 - 2(-1) = 1 + 2 = 3$

  • Combine the outputs: $f(-2) + g(-1) = -5 + 3 = -2$

  • For product expressions like $f(0)g(4)$, evaluate each function output separately and multiply the resulting values.

• Determining if a Relation is a Function:

  • Graphical Test: Apply the vertical-line test. If any vertical line intersects the graph at more than one point, the relation is not a function of $x$.

  • Table / Ordered Pairs: A relation is not a function if a single $x$-value is paired with two or more distinct $y$-values.

  • Algebraic Test: Solve the equation for $y$. If isolating $y$ produces a dual sign $\text{something}$, $y$ is generally not a function of $x$.

  • Example: Solving $y^2 = 4x^2$ yields $y = +2|x|$ and $y = -2|x|$. Because most $x$-values yield two separate $y$-outputs, this is not a function of $x$. Conversely, $y = 5 - x^2$ assigns a single $y$ to each $x$, so it is a function.

Domain and Range Rules

• Core Definitions:

  • Domain: The set of all allowed $x$-inputs.

  • Range: The set of all possible $y$-outputs.

• Domain Mental Model: Start by assuming the domain includes all real numbers $(-\text{inf}, \text{inf})$. Remove only the inputs that break basic mathematical rules:

  • Division by zero: Denominators cannot equal $0$

  • Even roots of negative numbers: Radicands under even roots must be nonnegative ($\text{radicand} \neq 0$)

  • Logarithms of non-positive numbers: Logarithmic arguments must be strictly positive ($\text{argument} > 0$)

• Domain Rules by Expression Type:

  • Rational expression $\frac{2x - 5}{x^2 - x - 6}$:

  • Denominator restriction: $x^2 - x - 6 \neq 0$

  • Factored form: $(x - 3)(x + 2) \neq 0$

  • Excluded inputs: $x \neq 3$ and $x \neq -2$

  • Interval notation: $(-\text{inf}, -2) \text{ U } (-2, 3) \text{ U } (3, \text{inf})$

  • Square root in denominator $\frac{3x}{\text{sqrt}(5 - x)}$:

  • Radicand restriction: $5 - x > 0$ (strictly positive because it is in the denominator)

  • Inequality solution: $x < 5$

  • Interval notation: $(-\text{inf}, 5)$

  • Standard square root $\text{sqrt}(x + 7)$:

  • Radicand restriction: $x + 7 \neq 0$

  • Inequality solution: $x \neq -7$

  • Interval notation: $[-7, \text{inf})$

  • Logarithmic function $\text{log}(x - 4)$:

  • Argument restriction: $x - 4 > 0$

  • Inequality solution: $x > 4$

  • Interval notation: $(4, \text{inf})$

• Interval Notation Syntax:

  • Parentheses $( )$ indicate that an endpoint is excluded.

  • Brackets $[ ]$ indicate that an endpoint is included.

  • Infinity $\text{inf}$ and negative infinity $-\text{inf}$ always take parentheses.

  • The union symbol $\text{U}$ combines disjoint intervals.

• Range Rules for Common Graphs:

  • Nonvertical line with nonzero slope: $(-\text{inf}, \text{inf})$

  • Upward parabola with vertex $(h, k)$: $[k, \text{inf})$

  • Downward parabola with vertex $(h, k)$: $(-\text{inf}, k]$

  • Square root function shifted up by $k$ units: $[k, \text{inf})$

  • Exponential function $a \times b^x + k$: Above $k$ if $a > 0$ yielding $(k, \text{inf})$; below $k$ if $a < 0$ yielding (-\text{inf}, k)$\n- **Visual Scan Strategy**:\n - To determine domain, scan the graph left-to-right along the x-axis.\n - To determine range, scan the graph bottom-to-top along the y-axis.\n\n# Difference Quotient\n\n- **Definition**: The difference quotient measures the average rate of change of a function over a small step h:\n   rac{f(x + h) - f(x)}{h}, ext{ where } h eq 0\n- **Three-Line Algebraic Method**:\n - 1. Evaluate f(x + h)$by replacing every$x$with$(x + h).\n - 2. Subtract the complete original function f(x)$, surrounding$f(x) in parentheses.\n - 3. Simplify the numerator, factor out h$, and cancel$h with the denominator.\n- **Worked Example**: Find the difference quotient for f(x) = 3 - 2x\n - Step 1: f(x + h) = 3 - 2(x + h) = 3 - 2x - 2h\n - Step 2: f(x + h) - f(x) = (3 - 2x - 2h) - (3 - 2x) = 3 - 2x - 2h - 3 + 2x = -2h\n - Step 3: rac{-2h}{h} = -2\n- **Domain Restriction Caution**: The parameter h$cannot equal$0$because the original quotient involves division by$h$. Division by zero is undefined even if$h algebraically cancels later.\n- **Common Algebraic Error**:\n - Incorrect subtraction: f(x + h) - f(x) = 3 - 2x - 2h - 3 - 2x\n - Correct subtraction: Distribution of the negative sign across both terms of f(x)$yields$-(3 - 2x) = -3 + 2x\n- **Practice Problem**: For f(x) = x^2$, show that the difference quotient equals$2x + h\n - Expand f(x + h) = (x + h)^2 = x^2 + 2xh + h^2\n - Subtract f(x)$:$(x^2 + 2xh + h^2) - x^2 = 2xh + h^2\n - Divide by h$:$ rac{h(2x + h)}{h} = 2x + h\n\n# Graph Transformations\n\n- **Transformation Rules Summary**:\n - Vertical shift: y = f(x) + k$moves up by$k$units (down if$k < 0$). Outside operations affect$y normally.\n - Horizontal shift: y = f(x - h)$moves right by$h$units (left if written as$f(x + h)$). Inside operations affect$x oppositely.\n - Vertical stretch/shrink: y = a imes f(x)$stretches vertically if$|a| > 1$; shrinks vertically if$0 < |a| < 1. Multiplies output heights.\n - Horizontal stretch/shrink: y = f(bx)$shrinks horizontally by a factor of$ rac{1}{|b|}$if$|b| > 1. Takes the reciprocal of inside factor.\n - Reflection over x$axis:$y = -f(x)$negates all$y-outputs.\n - Reflection over y$axis:$y = f(-x)$negates all$x-inputs.\n- **Core Transformation Rule**:\n - **OUTSIDE** modifications alter the output (y) and behave intuitively.\n - **INSIDE** modifications alter the input (x) and behave inversely.\n- **Combined Transformation Example**: y = -f(x + 3) - 3\n - Shift left by 3 units\n - Reflect across the x-axis\n - Shift down by 3 units\n- **Absolute Value Transformations**: y = a|x - h| + k\n - Vertex is at (h, k).\n - Opens upward if a > 0$; opens downward if$a < 0\n - Steeper/narrower if |a| > 1$; flatter/wider if$0 < |a| < 1\n- **Quadratic Transformation Form**: y = a(x - h)^2 + k\n - Vertex is at (h, k).\n- **Cube-Root Transformation Example**: y = - ext{cube_root}(x + 1) - 2\n - Shift left by 1 unit\n - Reflect across the x-axis\n - Shift down by 2 units\n - The central inflection point moves from (0, 0)$to$(-1, -2)\n\n# Linear Functions and Applications\n\n- **Definition**: A linear function maintains a constant rate of change. The slope m$measures the change in$y$per unit increase in$x:\n  m = rac{y_2 - y_1}{x_2 - x_1} = rac{ ext{rise}}{ ext{run}}\n- **Forms of Linear Equations**:\n - Slope-intercept form: y = mx + b$(where$m$is slope and$b$is the$y-intercept)\n - Point-slope form: y - y_1 = m(x - x_1)$(used when given a point$(x_1, y_1)$and slope$m)\n - Standard form: Ax + By = C$(where$A, B, C$are integers and$A is generally positive)\n- **Slope Classifications**:\n - Positive slope: Line rises from left to right.\n - Negative slope: Line falls from left to right.\n - Zero slope: Horizontal line defined by y = c\n - Undefined slope: Vertical line defined by x = c\n- **Algebraic Conversion Example**: Convert 7x + 2y = 14 to slope-intercept form\n - 2y = -7x + 14\n - y = - rac{7}{2}x + 7\n - Slope m = - rac{7}{2}$;$y$-intercept is$(0, 7)\n- **Equation Construction Example**: Line through (-3, 2)$with slope$m = - rac{2}{3}\n - Point-slope form: y - 2 = - rac{2}{3}(x - (-3))\n - Distribute: y - 2 = - rac{2}{3}(x + 3) = - rac{2}{3}x - 2\n - Slope-intercept form: y = - rac{2}{3}x\n- **Intercept Shortcuts**:\n - To find the x$-intercept, set$y = 0$and solve for$x\n - To find the y$-intercept, set$x = 0$and solve for$y\n- **Special Two-Point Rules**:\n - If two points share the same x$-value ($x_1 = x_2$), the line is vertical ($x = x_1).\n - If two points share the same y$-value ($y_1 = y_2$), the line is horizontal ($y = y_1).\n\n# Linear Regression and Modeling\n\n- **Interpretation of Parameters**: In y = mx + b\n - m carries output units per input unit (rate of change).\n - b$represents the baseline predicted output when input$x = 0\n- **Model Interpretation Example**: High school dropout rate model r = -0.308t + 13.9$, where$t is years since 1990\n - Slope -0.308$: The dropout rate decreases by$0.308 percentage points per year.\n - Intercept 13.9$: The predicted dropout rate in 1990 ($t = 0$) was$13.9\n- **Building Linear Models from Two Points**: A bulldozer is valued at ext{USD } 224,000$at$t = 0$and$ ext{USD } 100,000$at$t = 8\n - Calculate rate of depreciation: m = rac{100000 - 224000}{8 - 0} = rac{-124000}{8} = -15500 ext{ dollars/year}\n - Formulate model equation: V(t) = -15500t + 224000\n- **Year Conversion Rule**: When x represents years since a base year (e.g., 1990), input the offset value rather than the raw year number:\n - For the year 2024 with base year 1990: x = 2024 - 1990 = 34\n - Never substitute 2024 directly into the equation.\n- **TI-84 Linear Regression Steps**:\n - Enter data: Press `STAT` > `1:Edit`, enter x$-values into `L1` and$y-values into `L2`.\n - Compute model: Press `STAT` > `CALC` > `4:LinReg(ax+b)`.\n - Store regression equation directly to `Y1`: After selecting LinReg, press `VARS` > `Y-VARS` > `1:Function` > `1:Y1`.\n- **Extrapolation Warning**: Linear models are valid only within or near the empirical data range. Extrapolating far outside observed bounds produces unreliable predictions.\n\n# Quadratic Functions\n\n- **Definition**: A polynomial function of degree 2 that graphs as a parabola.\n- **Three Primary Forms**:\n - Standard form: y = ax^2 + bx + c$(exposes$y$-intercept at$(0, c); standard setup for quadratic formula)\n - Vertex form: y = a(x - h)^2 + k$(exposes vertex at$(h, k)$, axis of symmetry$x = h$, maximum/minimum value$k, and range)\n - Factored form: y = a(x - r_1)(x - r_2)$(exposes$x$-intercepts at$r_1$and$r_2)\n- **Form Selection Strategy**:\n - Select Vertex form when maximum, minimum, or vertex is requested.\n - Select Factored form when real roots or zeros are requested.\n - Select Standard form when applying the quadratic formula or finding the y-intercept.\n- **Parabola Orientation and Width**:\n - If a > 0, parabola opens upward, creating a minimum.\n - If a < 0, parabola opens downward, creating a maximum.\n - Larger values of |a|$make the graph narrower/steeper; values$0 < |a| < 1 make it wider/flatter.\n- **Vertex Formulas from Standard Form**:\n - x$-coordinate of vertex:$h = - rac{b}{2a}\n - y$-coordinate of vertex:$k = f(h)\n - Axis of symmetry: x = h\n- **Finding Intercepts**:\n - y$-intercept: Set$x = 0$to get$(0, c).\n - x$-intercepts: Set$y = 0 and solve using factoring, square roots, completing the square, or the quadratic formula:\n    x = rac{-b ext{ +/- } ext{sqrt}(b^2 - 4ac)}{2a}\n- **Discriminant Analysis**: D = b^2 - 4ac\n - If D > 0$: Two distinct real$x-intercepts.\n - If D = 0$: Exactly one repeated real$x-intercept.\n - If D < 0$: Zero real$x-intercepts (complex roots).\n\n# Completing the Square\n\n- **The Half-and-Square Rule**: For an expression x^2 + bx$, add$ rac{b}{2}^2 to form a perfect square trinomial.\n- **Worked Example 1**: Transform f(x) = -x^2 + 4x to vertex form\n - Factor out -1$from variable terms:$-(x^2 - 4x)\n - Half of -4$is$-2$; square it to get$4\n - Insert 4$inside parentheses and balance outside:$-(x^2 - 4x + 4) + 4\n - Simplified vertex form: f(x) = -(x - 2)^2 + 4\n - Properties: Vertex (2, 4)$, maximum value$4$, range$(- ext{inf}, 4]\n- **Algebraic Balance Explanation**: Adding 4$inside parentheses multiplied by an outer$-1$equals subtracting$4$overall. Adding$4 outside balances the expression back to original value.\n- **Worked Example 2**: Transform f(x) = 4x^2 + 4x - 3 to vertex form\n - Factor 4$from variable terms:$4(x^2 + x) - 3\n - Half of 1$is$ rac{1}{2}$; square it to get$ rac{1}{4}\n - Complete square: 4[(x + rac{1}{2})^2 - rac{1}{4}] - 3 = 4(x + rac{1}{2})^2 - 1 - 3\n - Simplified vertex form: f(x) = 4(x + rac{1}{2})^2 - 4\n - Properties: Vertex (- rac{1}{2}, -4)$, minimum value$-4$, range$[-4, ext{inf})\n- **Method Comparison**: Use h = - rac{b}{2a}$and$k = f(h) if only the vertex coordinates are required. Use Completing the Square when vertex form or explicit sequence of transformations is required.\n\n# Quadratic Optimization and Applications\n\n- **Application Workflow**:\n - 1. Express physical constraints in terms of variables.\n - 2. Eliminate variables to build a single-variable quadratic objective function.\n - 3. Identify the vertex to find maximum or minimum values.\n- **Fencing Optimization Example**: Construction of two side-by-side rectangular pens using 840 ext{ ft} of available fencing\n - Constraint formulation: 3 vertical lengths (x$) and 2 combined horizontal lengths ($2y)\n - Equation: 3x + 4y = 840\n - Isolate variable: y = rac{840 - 3x}{4} = 210 - 0.75x\n - Objective function (Total Area): A(x) = 2x imes y = 2x( rac{840 - 3x}{4}) = 420x - 1.5x^2\n - Vertex calculation: Since a = -1.5$, parabola opens down, yielding maximum area at$x = - rac{b}{2a} = - rac{420}{2(-1.5)} = 140 ext{ ft}\n - Solve corresponding dimension: y = rac{840 - 3(140)}{4} = 105 ext{ ft}\n - Dimensions and Maximum Area: Total dimensions are 140 ext{ ft}$by$210 ext{ ft}$($2y = 210\n - Maximum total area: A(140) = 29,400 ext{ ft}^2\n- **Physical Domain Boundaries**: Dimensions must remain positive (x > 0$and$y > 0).\n - Setting rac{840 - 3x}{4} > 0$gives$x < 280\n - Physical domain restriction: 0 < x < 280\n- **Unit Reasoning**: Area calculations multiply two linear dimensions ( ext{ft} imes ext{ft} = ext{ft}^2). The measurement unit is squared, not the function itself.\n\n# Polynomial and Rational Functions\n\n- **Polynomial Classification**:\n - Terms contain variables with nonnegative whole-number exponents.\n - Degree is the highest exponent present after full simplification.\n - Degree 1: Linear; Degree 2: Quadratic; Degree 0: Constant.\n - Simplification Example: y = 8x + 2(10 - 4x) = 8x + 20 - 8x = 20. This is a constant function of degree 0, not a linear function.\n- **Rational Functions Definition**: Formed by the quotient of two polynomials rac{P(x)}{Q(x)}$, where$Q(x) eq 0. Every polynomial is a rational function (with denominator 1), but not all rational functions are polynomials.\n- **Vertical Asymptotes and Holes Procedure**:\n - 1. Fully factor numerator P(x)$and denominator$Q(x).\n - 2. Cancel common factors.\n - 3. Denominator factors that cancel completely produce **holes** (removable discontinuities).\n - 4. Uncanceled denominator factors set to zero produce **vertical asymptotes**.\n- **Worked Example**: rac{x^2 + 3x}{x^2 + 2x}\n - Factored form: rac{x(x + 3)}{x(x + 2)}\n - Cancel common factor x$:$ rac{x + 3}{x + 2}\n - Canceled factor x = 0$creates a hole at$x = 0\n - Remaining denominator zero x + 2 = 0$yields vertical asymptote at$x = -2\n- **Horizontal Asymptotes Rules** (Compare degree of numerator N$vs denominator$D):\n - If N < D$: Horizontal asymptote is$y = 0\n - If N = D$: Horizontal asymptote is$y = rac{ ext{leading coefficient of numerator}}{ ext{leading coefficient of denominator}}\n - If N > D: No horizontal asymptote (may exhibit slant/oblique asymptote)\n- **Horizontal Asymptote Examples**:\n - rac{3x^2 + 2x - 1}{4x^2 - 5x + 3}$: Degrees are equal ($2 = 2$), so asymptote is$y = rac{3}{4}\n - rac{x^2 + 4}{100x + 1}$: Numerator degree higher ($2 > 1), so no horizontal asymptote exists.\n\n# Exponential Functions and Financial Models\n\n- **General Exponential Form**: y = a imes b^x$, where$b > 0$and$b eq 1. The variable is located in the exponent.\n - Base behavior: Growth if b > 1$; Decay if$0 < b < 1\n- **Growth and Decay Formulas**:\n - Discrete Compound Interest: A = P(1 + rac{r}{n})^{nt}\n - Continuous Growth/Decay: A = P e^{rt}\n - Parameters: P = ext{principal/starting amount}$,$r = ext{annual percentage rate (as decimal)}$,$n = ext{compoundings per year}$,$t = ext{time in years}\n- **Percentage Decimal Conversions**: Shift decimal point 2 places left (1.25 ext{%} = 0.0125\n- **Compounding Frequency Values**:\n - Quarterly: n = 4\n - Monthly: n = 12\n - Daily: n = 365\n- **Doubling and Tripling Time Formulas**:\n - Continuous model doubling time: t = rac{ ext{ln}(2)}{r}\n - Continuous model tripling time: t = rac{ ext{ln}(3)}{r}\n - Discrete model doubling/tripling solution: Solve t = rac{ ext{ln}(A/P)}{n imes ext{ln}(1 + r/n)}\n- **Repeated Doubling Model**: A biological cell doubles every rac{1}{2}$day ($0.5 ext{ days}\n - Number of doubling periods in t$days:$ rac{t}{0.5} = 2t\n - Model function: N(t) = 1 imes 2^{2t} = 2^{2t}\n- **Solving Exponential Equations**:\n - Matching bases: If 5^{2x - 3} = 5^4$, set exponents equal:$2x - 3 = 4 ightarrow 2x = 7 ightarrow x = 3.5\n - Mismatched bases: Isolate exponential term and take natural log ext{ln} of both sides.\n\n# Logarithmic Functions and Equations\n\n- **Definition**: A logarithm answers the question: *What exponent turns the base into this input?*\n  y = ext{log}b(x) ext{ is equivalent to } b^y = x\n- **Conversion Rules**:\n - The log base remains the exponential base.\n - The output of the logarithm becomes the exponent.\n - The logarithm argument becomes the exponential result.\n- **Conversion Examples**:\n - ext{log}_3(x) = 2 ightarrow 3^2 = x ightarrow x = 9\n - ext{log}_x(36) = 2 ightarrow x^2 = 36 ightarrow x = 6$(Base must be positive and$x eq 1\n - ext{log}_2(16) = x ightarrow 2^x = 16 ightarrow x = 4\n - ext{ln}(M) = N ightarrow e^N = M\n - ext{log}(u) = v ightarrow 10^v = u\n- **Base Types**:\n - Common Logarithm ext{log}(x): Base 10\n - Natural Logarithm ext{ln}(x)$: Base$e\n - Both require strictly positive inputs (x > 0\n- **Laws of Logarithms**:\n - Product Law: ext{log}_b(MN) = ext{log}_b(M) + ext{log}_b(N)\n - Quotient Law: ext{log}_b( rac{M}{N}) = ext{log}_b(M) - ext{log}_b(N)\n - Power Law: ext{log}_b(M^p) = p imes ext{log}_b(M)\n- **Domain Restrictions and Verification**: Always establish that arguments are positive ( ext{argument} > 0) prior to combining logs. Check all final solutions in the original unsimplified equation to catch extraneous solutions.\n\n# Solving Logarithmic and Exponential Equations\n\n- **One Logarithm per Side**: ext{log}(x + 5) = ext{log}(2x - 3)\n - Equate arguments: x + 5 = 2x - 3 ightarrow x = 8\n - Check domain: 8 + 5 = 13 > 0$and$2(8) - 3 = 13 > 0. Valid solution.\n- **Combining Logarithms**: ext{ln}(2x - 2) - ext{ln}(x - 1) = ext{ln}(x)\n - Apply Quotient Law: ext{ln}( rac{2x - 2}{x - 1}) = ext{ln}(x)\n - Simplify left argument: rac{2(x - 1)}{x - 1} = 2$(valid for$x > 1\n - Equate: ext{ln}(2) = ext{ln}(x) ightarrow x = 2\n- **Solving Exponential Equations using Natural Log**: Solve 0.01 = e^{-0.05x}\n - Take natural log of both sides: ext{ln}(0.01) = ext{ln}(e^{-0.05x})\n - Simplify exponent: ext{ln}(0.01) = -0.05x\n - Solve: x = rac{ ext{ln}(0.01)}{-0.05} ext{ approx } 92.10\n- **Fractional/Variable Base Example**: Solve ext{log}{1/3}(9) = x\n - Convert: ( rac{1}{3})^x = 9\n - Rewrite using base 3: (3^{-1})^x = 3^2 ightarrow 3^{-x} = 3^2\n - Solve exponent: -x = 2 ightarrow x = -2\n- **Forbidden Base Caution**: Base b = 1$is forbidden because$1^y = 1$for all$y. It cannot generate non-zero positive outputs or define a unique inverse function.\n\n# Piecewise Functions and Tiered Applications\n\n- **Definition**: A piecewise function applies different sub-rules across distinct input intervals.\n- **Evaluation Steps**:\n - 1. Identify which interval contains the given input x\n - 2. Apply only the rule corresponding to that specific interval.\n - 3. Ignore all inactive rules.\n- **Graphing Guidelines**:\n - Graph each rule exclusively over its specified domain interval.\n - Use closed circles for eq$or$ eq inequalities.\n - Use open circles for <$or$> inequalities.\n- **Tiered Electricity Pricing Mental Model**: Cumulative tiered rates apply lower rates to initial usage brackets and higher rates only to incremental usage exceeding thresholds. Do not apply higher tier rates to total consumption.\n- **Continuity Check at Boundaries**: Calculate left-hand and right-hand outputs at boundary points. If both pieces match at the boundary and the included point is defined, the pieces connect continuously.\n- **Translation Habit**: Underline interval boundary phrases such as "first 20," "next 180," or "usage over 1000" before constructing algebraic expressions.\n\n# Model Selection and Regression Shortcuts\n\n- **Selecting Regression Models by Trend Shape**:\n - Straight trend line: Linear model y = ax + b (`LinReg`)\n - Single turn parabola: Quadratic model y = ax^2 + bx + c (`QuadReg`)\n - Two turns / S-curve segment: Cubic model (`CubicReg`)\n - Constant percentage growth/decay: Exponential model y = a imes b^x (`ExpReg`)\n - Rapid initial change leveling off: Logarithmic model y = a + b imes ext{ln}(x) (`LnReg`)\n- **Market Equilibrium Calculation**:\n - Set Demand function equal to Supply function: D(x) = S(x)\n - Solve for x to determine equilibrium quantity.\n - Substitute quantity x back into either model to compute equilibrium price.\n- **Model Sanity Checks**:\n - Check units, domain limits, signs, and order of magnitude.\n - A high coefficient of determination r^2 indicates good statistical fit but does not guarantee real-world mathematical validity.\n- **Data-Year Shift Shortcut**: Transform calendar years into smaller numbers (x = ext{year} - ext{base year}) to simplify inputs and prevent numerical instability.\n- **Interpolation vs Extrapolation**:\n - **Interpolation**: Predicting values *within* known data bounds (reliable).\n - **Extrapolation**: Predicting values *outside* known data bounds (high risk).\n\n# TI-84 Plus Keystroke Guide\n\n- **Enter Graph Function**: Press `Y=`, enter expression using `X,T,theta,n` key.\n- **Reset Standard Window**: Press `ZOOM` > `6:ZStandard`.\n- **Find Zero / x-intercept**: Press `2nd` > `TRACE (CALC)` > `2:zero`. Set Left Bound, Right Bound, and Guess.\n- **Find Minimum Point**: Press `2nd` > `TRACE` > `3:minimum`. Set bounds and guess.\n- **Find Maximum Point**: Press `2nd` > `TRACE` > `4:maximum`. Set bounds and guess.\n- **Find Intersection**: Press `2nd` > `TRACE` > `5:intersect`. Select first curve, second curve, and guess.\n- **Generate Table**: Press `2nd` > `WINDOW (TBLSET)`, define `TblStart` and `DeltaTbl`, then press `2nd` > `GRAPH`.\n- **Fraction Expressions**: Enclose fractions in parentheses `(-2)/(3)` or press `ALPHA` > `Y=` to access the `n/d` template.\n- **Decimal to Fraction**: Press `MATH` > `1:Frac`.\n- **Natural Log / e^x$**: Press `LN`; for$e^x press `2nd` > `LN`.\n- **Common Log / 10^x$**: Press `LOG`; for$10^x press `2nd` > `LOG`.\n- **Execute Regression**: Input values in `STAT` > `1:Edit`. Press `STAT` > `CALC`, choose model, and calculate.\n- **Calculator Usage Principle**: Use the calculator as an analytical checker. Estimate results manually prior to calculator entry.\n\n# Master Formula Sheet\n\n- **Linear Slope**: m = rac{y_2 - y_1}{x_2 - x_1}\n- **Point-Slope Form**: y - y_1 = m(x - x_1)\n- **Slope-Intercept Form**: y = mx + b\n- **Quadratic Vertex Coordinates**: h = - rac{b}{2a}$,$k = f(h)\n- **Quadratic Formula**: x = rac{-b ext{ +/- } ext{sqrt}(b^2 - 4ac)}{2a}\n- **Vertex Form**: y = a(x - h)^2 + k\n- **Difference Quotient**: rac{f(x+h) - f(x)}{h}$, where$h eq 0\n- **Compound Interest (Discrete)**: A = P(1 + rac{r}{n})^{nt}\n- **Compound Growth (Continuous)**: A = P e^{rt}\n- **Doubling Time (Continuous)**: t = rac{ ext{ln}(2)}{r}\n- **Logarithm Conversion**: y = ext{log}_b(x) ext{ iff } b^y = x\n- **Horizontal Asymptotes**:\n - Numerator degree < Denominator degree ightarrow y = 0\n - Numerator degree = Denominator degree ightarrow y = rac{ ext{leading coef numerator}}{ ext{leading coef denominator}}\n- **Domain Exclusions**:\n - Denominator eq 0\n - Even root radicand eq 0\n - Logarithm argument > 0\n\n# Five Common Errors to Avoid\n\n- **1. Sign Dropping**: Dropping negative signs during substitution steps.\n- **2. Inside Transformation Misinterpretation**: Shifting inside transformations in the intuitive direction instead of the opposite direction.\n- **3. Raw Year Insertion**: Substituting actual calendar years directly into models instead of calculating elapsed time from base year (t = ext{year} - ext{base year}).\n- **4. Domain Loss after Cancellation**: Failing to list domain exclusions for cancelled denominator terms when analyzing rational graphs.\n- **5. Premature Rounding**: Rounding intermediate steps early. Maintain full calculator precision until the final reported value.\n\n# Mixed Practice Problems\n\n- 1. Given f(x) = 2x - 1$and$g(x) = x^2 - 2x$, calculate$f(-2) + g(-1).\n- 2. Is x^2 = y^2$a function of$x? Explain using outputs or the vertical-line test.\n- 3. Find the domain of rac{2x - 5}{x^2 - x - 6}.\n- 4. Describe the transformations required to convert y = f(x)$to$y = -f(x + 3) - 3\n- 5. Write the equation of the line passing through (-3, 2)$with slope$- rac{2}{3}$in$y = mx + b form.\n- 6. Find the intercepts and slope of 2x - 3y = 18\n- 7. Rewrite -x^2 + 4x in vertex form and state its vertex and maximum/minimum value.\n- 8. For G(x) = 0.3x^2 + 1.2x - 6.9, determine the vertex and range.\n- 9. Find all vertical asymptotes and holes for rac{x^2 + 3x}{x^2 + 2x}.\n- 10. Find the horizontal asymptote of rac{3x^2 + 2x - 1}{4x^2 - 5x + 3}.\n- 11. Convert ext{log}_3(x) = 2$to exponential form and solve for$x\n- 12. Solve ext{log}(x + 5) = ext{log}(2x - 3) and verify log domains.\n- 13. Solve 0.01 = e^{-0.05x} rounded to two decimal places.\n- 14. Write a model equation for a population starting at 1 cell that doubles every half-day.\n- 15. Given linear model r = -0.308t + 13.9$, where$t$represents years since 1990, predict$r in 2024.\n- 16. Explain why x = 0$cannot represent the year 1950 in a logarithmic model$y = a + b imes ext{ln}(x).\n- 17. Simplify y = 8x + 2(10 - 4x) and classify the resulting function family.\n- 18. Find the simplified difference quotient for f(x) = 3 - 2x\n\n# Answer Key and Detailed Explanations\n\n- 1. **Answer**: -2\n - Reasoning: f(-2) = 2(-2) - 1 = -5$;$g(-1) = (-1)^2 - 2(-1) = 3$;$-5 + 3 = -2\n- 2. **Answer**: No\n - Reasoning: Solving gives y = ext{sqrt}(x^2) = ext{+/-}|x|$. Single $x$ inputs yield two distinct outputs; vertical lines intersect the graph twice.

• 3. Answer: $(-\text{inf}, -2) \text{ U } (-2, 3) \text{ U } (3, \text{inf})$

  • Reasoning: Factor denominator $(x - 3)(x + 2) \neq 0$, yielding exclusions $x \neq 3$ and $x \neq -2$

• 4. Answer: Shift left 3 units, reflect across the $x$-axis, shift down 3 units.

  • Reasoning: Inside $+3$ moves left 3; outside negative reflects vertically; outside $-3$ moves down 3.

• 5. Answer: $y = -\frac{2}{3}x$

  • Reasoning: Point-slope gives $y - 2 = -\frac{2}{3}(x + 3) \rightarrow y - 2 = -\frac{2}{3}x - 2 \rightarrow y = -\frac{2}{3}x$

• 6. Answer: $x$-intercept $(9, 0)$; $y$-intercept $(0, -6)$; slope $m = \frac{2}{3}$

  • Reasoning: Set $y = 0 \rightarrow 2x = 18 \rightarrow x = 9$; set $x = 0 \rightarrow -3y = 18 \rightarrow y = -6$; rearrange to $y = \frac{2}{3}x - 6$

• 7. Answer: Vertex form $-(x - 2)^2 + 4$; Vertex $(2, 4)$; Maximum value $4$

  • Reasoning: $-(x^2 - 4x + 4) + 4 = -(x - 2)^2 + 4$

• 8. Answer: Vertex $(-2, -8.1)$; Range $[-8.1, \text{inf})$

  • Reasoning: $h = -\frac{1.2}{2(0.3)} = -2$; $k = G(-2) = 0.3(4) + 1.2(-2) - 6.9 = 1.2 - 2.4 - 6.9 = -8.1$. Parabola opens upward.

• 9. Answer: Hole at $x = 0$; Vertical Asymptote at $x = -2$

  • Reasoning: Factor $\frac{x(x + 3)}{x(x + 2)}$. Cancel factor $x$ (hole at $x = 0$). Remaining zero in denominator is $x = -2$ (vertical asymptote).

• 
  

10. Answer: $y = \frac{3}{4}$

• Reasoning: Degrees of numerator and denominator are both equal to 2. Take ratio of leading coefficients $\frac{3}{4}$.

11. Answer: $3^2 = x \rightarrow x = 9$

• Reasoning: Applying base conversion triangle yields $3^2 = 9$

12. Answer: $x = 8$

• Reasoning: Equating arguments gives $x + 5 = 2x - 3 \rightarrow x = 8$. Check log arguments: $8 + 5 = 13 > 0$ and $2(8) - 3 = 13 > 0$

13. Answer: $x \text{ approx } 92.10$

• Reasoning: Take natural log: $\text{ln}(0.01) = -0.05x \rightarrow x = \frac{\text{ln}(0.01)}{-0.05} \text{ approx } 92.1034$

14. Answer: $N(t) = 2^{2t}$

• Reasoning: Number of doubling periods in $t$ days is $\frac{t}{0.5} = 2t$. Exponential model is $1 \times 2^{2t}$.

15. Answer: r = 3.428\text{%} \text{ approx } 3.43\text{%}

• Reasoning: Calculate offset years $t = 2024 - 1990 = 34$. Substitute: $r = -0.308(34) + 13.9 = -10.472 + 13.9 = 3.428$

16. Answer: Logarithm of zero is undefined.

• Reasoning: $\text{ln}(0)$ is mathematically undefined. Use shifted variables like $x = \text{year} - 1949$ to ensure positive inputs.

17. Answer: $y = 20$; Constant function

• Reasoning: Expand $y = 8x + 20 - 8x = 20$. Variable terms cancel, leaving a constant polynomial of degree 0.

18. Answer: $-2$

• Reasoning: $\frac{[3 - 2(x + h)] - [3 - 2x]}{h} = \frac{3 - 2x - 2h - 3 + 2x}{h} = \frac{-2h}{h} = -2$

Knowt-Ready Recall Flashcards

• Q: What criteria defines a relation as a function?

  • A: Every allowed input is assigned exactly one output.

• Q: What are the three primary domain restrictions?

  • A: Denominators cannot equal zero; even-root radicands must be nonnegative; logarithmic arguments must be strictly positive.

• Q: How do inside vs outside transformations differ?

  • A: Outside changes alter $y$-outputs normally; inside changes alter $x$-inputs oppositely.

• Q: How do you algebraically find an $x$-intercept?

  • A: Set $y$ or $f(x)$ equal to $0$ and solve for $x$

• Q: What does the sign of a quadratic leading coefficient $a$ indicate?

  • A: Positive $a$ opens upward (has minimum); negative $a$ opens downward (has maximum).

• Q: What information does the discriminant $b^2 - 4ac$ provide?

  • A: Indicates the number of real $x$-intercepts: positive $> 0$ yields two; zero $= 0$ yields one; negative $< 0$ yields zero.

• Q: What distinguishes a hole from a vertical asymptote in rational functions?

  • A: Cancelling factors in the denominator creates holes; remaining uncanceled denominator zeros create vertical asymptotes.

• Q: What question is solved by evaluating a logarithm?

  • A: It identifies the exponent required to raise the base to obtain the argument.

• Q: Under what condition does an exponential function represent growth?

  • A: When its base $b$ is strictly greater than 1 ($b > 1$

• Q: How is market equilibrium calculated algebraically?

  • A: Set the demand function equal to the supply function ($D(x) = S(x)$), solve for quantity $x$, and compute price.

• Q: What is the rule for completing the square on $x^2 + bx$?

  • A: Take half of the linear coefficient $b$ and square it: add $\frac{b}{2}^2$

• Q: What is the four-step workflow for solving algebra problems?

  • A: 1. Identify family, 2. Target requested value, 3. Choose formula, 4. Check reasonableness.

Final Exam Strategy

• Solve straightforward family identification problems first to build momentum.

• Write out the requested target parameter and chosen formula before entering values into the calculator.

• Maintain full floating-point precision in intermediate steps; apply rounding rules only to the final calculated answer.

• Perform sanity checks on final answers using qualitative visual checks (graph shape) and unit consistency.