General Mathematics 11 Study Guide - Function Families

Linear Functions

• Definition: A linear function is a straight-line function represented by the general equation $y = mx + b$.

• Core Concepts & Formulas:

  • Slope-Intercept Form: $y = mx + b$, where $m$ represents the slope and $b$ represents the y-intercept.
  • Slope Formula (Two Points): For two distinct points $(x_1, y_1)$ and $(x_2, y_2)$, the slope $m$ is calculated using:         $m = \frac{y_2 - y_1}{x_2 - x_1}$
  • Point-Slope Form: $y - y_1 = m(x - x_1)$
  • Domain and Range: Both the domain and range of any non-horizontal linear function consist of all real numbers, expressed in interval notation as $(-\infty, \infty)$.

• Comprehensive Problem Walkthrough:

  • Problem Statement: Given a line passing through the points $(-1, 4)$ and $(3, -4)$, determine its slope, derive its equation in slope-intercept form, calculate its x-intercept and y-intercept, and sketch its graph.
  • Step 1 — Calculate the Slope:
    • Assign points: $(x_1, y_1) = (-1, 4)$ and $(x_2, y_2) = (3, -4)$.
    • Substitute points into the slope formula:             $m = \frac{-4 - 4}{3 - (-1)}$
    • Simplify:             $m = \frac{-8}{4} = -2$
    • Slope: $m = -2$
  • Step 2 — Determine Equation via Point-Slope Form:
    • Substitute $m = -2$ and $(x_1, y_1) = (-1, 4)$ into $y - y_1 = m(x - x_1)$:             $y - 4 = -2(x - (-1))$
    • Simplify the algebraic expression:             $y - 4 = -2(x + 1)$$y - 4 = -2x - 2$
    • Add $4$ to both sides to isolate $y$ in slope-intercept form:             $y = -2x + 2$
    • Verification: Test using the second point $(3, -4)$:             $y = -2(3) + 2 = -6 + 2 = -4$             The result matches the given point $(3, -4)$.
  • Step 3 — Calculate the x-Intercept:
    • Set $y = 0$ in equation $y = -2x + 2$:             $0 = -2x + 2$
    • Solve for $x$:             $2x = 2$$x = 1$
    • x-intercept: $(1, 0)$
  • Step 4 — Calculate the y-Intercept:
    • Set $x = 0$ in equation $y = -2x + 2$:             $y = -2(0) + 2 = 2$
    • y-intercept: $(0, 2)$ (which directly corresponds to value $b = 2$).
  • Step 5 — Graphing the Function:
    • Plot the initial given points $(-1, 4)$ and $(3, -4)$.
    • Plot key intercepts: x-intercept $(1, 0)$ and y-intercept $(0, 2)$.
    • Draw a straight line connecting all points, placing arrows at both ends to indicate continuous extension in both directions.

Quadratic Functions

• Definition: A quadratic function forms a U-shaped or dome-shaped parabola defined in standard form as $y = ax^2 + bx + c$ or in vertex form as $y = a(x - h)^2 + k$.

• Core Concepts & Formulas:

  • Vertex Coordinates from Standard Form: The x-coordinate of the vertex $h$ is calculated by:         $h = -\frac{b}{2a}$         The y-coordinate of the vertex $k$ is found by evaluating the function at $h$:         $k = f(h)$
  • Axis of Symmetry: The vertical line passing directly through the vertex, defined as $x = h$
  • Direction of Opening & Extrema:
    • If $a > 0$, the parabola opens upward, making the vertex an absolute minimum point.
    • If $a < 0$, the parabola opens downward, making the vertex an absolute maximum point.
  • Domain and Range:
    • Domain: Always all real numbers, $(-\infty, \infty)$.
    • Range: If $a > 0$, the range is $y \ge k$ or $[k, \infty)$. If $a < 0$, the range is $y \le k$ or $(-\infty, k]$.

• Comprehensive Problem Walkthrough:

  • Problem Statement: Given the quadratic function $y = -x^2 + 4x + 5$, find its vertex, axis of symmetry, x-intercepts, y-intercept, and sketch its graph.
  • Step 1 — Calculate the Vertex (Completing the Square):
    • Identify standard form coefficients: $a = -1$, $b = 4$, $c = 5$
    • Group x-terms and factor out $a = -1$:             $y = -(x^2 - 4x) + 5$
    • Determine the value to complete the square:             Take half the linear coefficient: $4 \div 2 = 2$             Square the result: $2^2 = 4$             Multiply by $a$: $4 \times 1 = 4$
    • Adjust expression inside and outside parentheses:             $y = -(x^2 - 4x + 4) + 5 - (-4)$$y = -(x - 2)^2 + 9$
    • Vertex: $(h, k) = (2, 9)$
    • Since $a = -1 < 0$, the parabola opens downward, making $(2, 9)$ a maximum point.
  • Step 2 — Determine the Axis of Symmetry:
    • The vertical line through the vertex gives the axis of symmetry:             $x = 2$
  • Step 3 — Calculate the x-Intercepts:
    • Set $y = 0$ and solve:             $-x^2 + 4x + 5 = 0$
    • Multiply both sides by $-1$ to simplify factoring:             $x^2 - 4x - 5 = 0$
    • Factor by identifying two numbers that multiply to $-5$ and add to $-4$ ($-5$ and $1$):             $(x - 5)(x + 1) = 0$
    • Solve each factor:             $x - 5 = 0 \Rightarrow x = 5$$x + 1 = 0 \Rightarrow x = -1$
    • x-intercepts: $(5, 0)$ and $(-1, 0)$
  • Step 4 — Calculate the y-Intercept:
    • Set $x = 0$:             $y = -(0)^2 + 4(0) + 5 = 5$
    • y-intercept: $(0, 5)$ (which equals constant $c = 5$).
  • Step 5 — Graphing the Function:
    • Plot vertex $(2, 9)$, x-intercepts $(-1, 0)$ and $(5, 0)$, and y-intercept $(0, 5)$.
    • Utilize symmetry around $x = 2$: notice that $(0, 5)$ reflects across the axis of symmetry to point $(4, 5)$.
    • Sketch a smooth, symmetric, downward-opening parabola passing through all identified key points.

Absolute Value Functions

• Definition: An absolute value function produces a V-shaped graph represented in vertex form as $f(x) = a|x - h| + k$

• Core Concepts & Formulas:

  • Vertex Form: $f(x) = a|x - h| + k$, where $(h, k)$ is the vertex (the sharp corner tip of the V).
  • Axis of Symmetry: The vertical line passing through the vertex, defined as $x = h$
  • Direction of Opening & Width:
    • If $a > 0$, the graph opens upward (vertex is a minimum point).
    • If $a < 0$, the graph opens downward (vertex is a maximum point).
    • If $|a| > 1$, the V-shape is narrower than the parent function $f(x) = |x|$.
    • If $|a| < 1$, the V-shape is wider than the parent function $f(x) = |x|$.
  • Domain and Range:
    • Domain: Always all real numbers, $(-\infty, \infty)$.
    • Range: If $a > 0$, range is $y \ge k$. If $a < 0$, range is $y \le k$.

• Comprehensive Problem Walkthrough:

  • Problem Statement: Given $f(x) = -2|x + 2| + 8$, determine its vertex, axis of symmetry, x-intercepts, y-intercept, and sketch its graph.
  • Step 1 — Identify Parameters & Vertex:
    • Rewrite function to match form $a|x - h| + k$:             $f(x) = -2|x - (-2)| + 8$
    • Identify coefficients: $a = -2$, $h = -2$, $k = 8$
    • Vertex: $(-2, 8)$
    • Since $a = -2 < 0$, the graph opens downward (vertex is a maximum point).
    • Since $|a| = 2 > 1$, the V-shape is narrower than the parent function $f(x) = |x|$.
  • Step 2 — Determine Axis of Symmetry:
    • The vertical line through the vertex gives:             $x = -2$
  • Step 3 — Calculate the x-Intercepts:
    • Set $y = 0$:             $0 = -2|x + 2| + 8$
    • Isolate absolute value expression:             $2|x + 2| = 8$$|x + 2| = 4$
    • Split equation into two linear cases:             $x + 2 = 4 \quad \text{or} \quad x + 2 = -4$
    • Solve for $x$ in each case:             $x = 2 \quad \text{or} \quad x = -6$
    • x-intercepts: $(2, 0)$ and $(-6, 0)$
  • Step 4 — Calculate the y-Intercept:
    • Set $x = 0$:             $f(0) = -2|0 + 2| + 8 = -2(2) + 8 = -4 + 8 = 4$
    • y-intercept: $(0, 4)$
  • Step 5 — Graphing the Function:
    • Plot vertex $(-2, 8)$, x-intercepts $(2, 0)$ and $(-6, 0)$, and y-intercept $(0, 4)$.
    • From vertex $(-2, 8)$, draw two straight arms extending down-left and down-right.
    • Slope of arms: since $a = -2$, each arm drops $2$ units vertically for every $1$ unit moved horizontally away from the vertex.

Square Root Functions

• Definition: A square root function represents a single-directional curved ray defined in standard form as $f(x) = a\sqrt{x - h} + k$
• Core Concepts & Formulas:
  • Starting Point Form: $f(x) = a\sqrt{x - h} + k$, where $(h, k)$ is the starting point (endpoint) of the curve.
  • Domain Restriction: The radicand must be non-negative ($x - h \ge 0$), which restricts domain to x \ge h$.\n * **Range**:\n * If a > 0$, the graph curves upward:$y \ge k.\n * If a < 0$, the graph curves downward:$y \le k.\n * **y-Intercept Domain Check**: Because the domain is restricted, always check whether x = 0 is in the domain before asserting that a y-intercept exists.\n\n* **Comprehensive Problem Walkthrough**:\n * **Problem Statement**: Given f(x) = -\sqrt{x + 1} + 2, find its domain, range, x-intercept, y-intercept, and sketch its graph.\n * **Step 1 — Identify Parameters, Starting Point, and Domain**:\n * Rewrite in standard form: f(x) = -\sqrt{x - (-1)} + 2\n * Identify parameters: a = -1$,$h = -1$,$k = 2\n * Starting point: (-1, 2)\n * Set radicand non-negative to solve for domain:\n            x + 1 \ge 0 \Rightarrow x \ge -1\n * Domain: x \ge -1$or$[-1, \infty)\n * **Step 2 — Determine Range**:\n * Since a = -1 < 0$, the curve falls downward from the starting point$k = 2.\n * Range: y \le 2$or$(-\infty, 2]\n * **Step 3 — Calculate the x-Intercept**:\n * Set y = 0:\n            0 = -\sqrt{x + 1} + 2\n * Isolate radical expression:\n            \sqrt{x + 1} = 2\n * Square both sides:\n            x + 1 = 4 \Rightarrow x = 3\n * x-intercept: (3, 0)\n * **Step 4 — Calculate the y-Intercept**:\n * Verify x = 0$in domain:$0 \ge -1 is valid, so a y-intercept exists.\n * Set x = 0:\n            f(0) = -\sqrt{0 + 1} + 2 = -1 + 2 = 1\n * y-intercept: (0, 1)\n * **Step 5 — Table of Values & Graphing**:\n * Select x$-values starting from$h = -1 that produce perfect square radicands:\n * For x = -1$:$f(-1) = -\sqrt{0} + 2 = 2 \rightarrow (-1, 2)\n * For x = 0$:$f(0) = -\sqrt{1} + 2 = 1 \rightarrow (0, 1)\n * For x = 3$:$f(3) = -\sqrt{4} + 2 = 0 \rightarrow (3, 0)\n * For x = 8$:$f(8) = -\sqrt{9} + 2 = -1 \rightarrow (8, -1)\n * Plot points (-1, 2)$,$(0, 1)$,$(3, 0)$, and$(8, -1).\n * Connect points with a smooth curve starting exactly at (-1, 2) and curving downward toward the right.\n\n# Cube Root Functions\n\n* **Definition**: A cube root function produces an elongated S-shaped curve defined as f(x) = a\sqrt[3]{x - h} + k\n* **Core Concepts & Formulas**:\n * **Inflection Point Form**: f(x) = a\sqrt[3]{x - h} + k$, where$(h, k) is the inflection point (the center of the S-curve).\n * **Domain and Range**: Both domain and range are unrestricted and equal to all real numbers, (-\infty, \infty)$, because odd roots of negative numbers are mathematically defined (e.g.,$\sqrt[3]{-8} = -2).\n * **Curve Behavior**:\n * If a > 0, the curve is strictly increasing (rises left to right).\n * If a < 0, the curve is strictly decreasing (falls left to right).\n\n* **Comprehensive Problem Walkthrough**:\n * **Problem Statement**: Given f(x) = -\sqrt[3]{x - 8} - 1$, identify$a, h, k, find its inflection point, x-intercept, y-intercept, table of values, and sketch its graph.\n * **Step 1 — Identify Parameters & Inflection Point**:\n * Compare equation to standard form f(x) = a\sqrt[3]{x - h} + k:\n            a = -1$,$h = 8$,$k = -1\n * Inflection point: (8, -1)\n * Since a = -1 < 0, the S-curve is strictly decreasing.\n * **Step 2 — Calculate the x-Intercept**:\n * Set y = 0:\n            0 = -\sqrt[3]{x - 8} - 1\n * Isolate cube root expression:\n            \sqrt[3]{x - 8} = -1\n * Cube both sides:\n            x - 8 = (-1)^3 = -1\n            x = 7\n * x-intercept: (7, 0)\n * **Step 3 — Calculate the y-Intercept**:\n * Set x = 0:\n            f(0) = -\sqrt[3]{0 - 8} - 1 = -\sqrt[3]{-8} - 1\n * Evaluate \sqrt[3]{-8} = -2:\n            f(0) = -(-2) - 1 = 2 - 1 = 1\n * y-intercept: (0, 1)\n * **Step 4 — Generate Table of Values**:\n * Select x$-values around$h = 8$that make$(x - 8)$small perfect cubes ($-8, -1, 0, 1):\n * For x = 0$:$x - 8 = -8 \Rightarrow f(0) = -(-2) - 1 = 1 \rightarrow (0, 1)\n * For x = 7$:$x - 8 = -1 \Rightarrow f(7) = -(-1) - 1 = 0 \rightarrow (7, 0)\n * For x = 8$:$x - 8 = 0 \Rightarrow f(8) = -(0) - 1 = -1 \rightarrow (8, -1)\n * For x = 9$:$x - 8 = 1 \Rightarrow f(9) = -(1) - 1 = -2 \rightarrow (9, -2)\n * **Step 5 — Graphing the Function**:\n * Plot points (0, 1)$,$(7, 0)$,$(8, -1)$, and$(9, -2).\n * Connect points with a smooth S-shaped curve passing through inflection point (8, -1), decreasing smoothly from upper-left to lower-right.\n\n# Function Family Quick Reference & Graphing Methodology\n\n* **Quick Reference Summary Table**:\n * **Linear Function**:\n * Key Form: y = mx + b\n * Key Point: y-intercept (0, b)$, Slope$m\n * Domain: (-\infty, \infty)\n * Range: (-\infty, \infty)\n * **Quadratic Function**:\n * Key Form: y = a(x - h)^2 + k\n * Key Point: Vertex (h, k)\n * Domain: (-\infty, \infty)\n * Range: y \ge k$(if$a > 0$) or$y \le k$(if$a < 0)\n * **Absolute Value Function**:\n * Key Form: y = a|x - h| + k\n * Key Point: Vertex (h, k)\n * Domain: (-\infty, \infty)\n * Range: y \ge k$(if$a > 0$) or$y \le k$(if$a < 0)\n * **Square Root Function**:\n * Key Form: y = a\sqrt{x - h} + k\n * Key Point: Starting Point (h, k)\n * Domain: x \ge h$(or$x \le h depending on orientation)\n * Range: y \ge k$(if$a > 0$) or$y \le k$(if$a < 0)\n * **Cube Root Function**:\n * Key Form: y = a\sqrt[3]{x - h} + k\n * Key Point: Inflection Point (h, k)\n * Domain: (-\infty, \infty)\n * Range: (-\infty, \infty)\n\n* **Universal 5-Step Graphing Checklist**:\n 1. **Identify Parameters**: Extract values for a$,$h$, and$k from the given equation (algebraically convert to vertex or starting-point form if presented in standard form).\n 2. **Determine Key Points & Symmetry**: Locate the primary key point (Vertex, Starting Point, or Inflection Point). For quadratic and absolute value functions, identify the axis of symmetry equation (x = h).\n 3. **Calculate Intercepts**:\n * Find x-intercept(s) by setting y = 0$and solving for$x\n * Find y-intercept by setting x = 0$and solving for$y$(verify first that$x = 0 is in the domain for square root functions).\n 4. **Establish Domain and Range**: State domain and range based on function type and directional opening/extension (a > 0$vs.$a < 0$$).
  1. Plot and Draw Graph: Plot all calculated key points and intercepts, then draw the distinct structural curve:
     • Linear: Straight line with arrows on both ends.
     • Quadratic: Symmetric U-shaped parabola or dome.
     • Absolute Value: Symmetric V-shape with two straight linear arms.
     • Square Root: Single-directional smooth curve radiating out from the starting point.
     • Cube Root: Continuous S-shaped curve extending infinitely in both directions through the central inflection point.