College Algebra Study Notes: Quadratic Functions, Calculator Mechanics, Absolute Value, and Rational Functions

Graphing Quadratic Functions and Vertex Calculations

  • Standard Quadratic Form:

    • Quadratic equations follow the general form y=ax2+bx+cy = ax^2 + bx + c.

    • The values of aa, bb, and cc determine the shape, direction, and position of the parabola.

  • Determining Parabola Direction:

    • If a>0a > 0 (positive coefficient), the parabola opens upward.

    • If a<0a < 0 (negative coefficient), the parabola opens downward.

  • Finding the Vertex Algebraically:

    • The x-coordinate of the vertex is given by the formula x=−b2ax = \frac{-b}{2a}.

    • The y-coordinate of the vertex is found by evaluating the function at that x-value, represented as f(−b2a)f\left(\frac{-b}{2a}\right).

    • Full vertex coordinate formula: (−b2a,f(−b2a))\left(\frac{-b}{2a}, f\left(\frac{-b}{2a}\right)\right).

  • Step-by-Step Example: y=x2−4x−3y = x^2 - 4x - 3

    • Identify coefficients: a=1a = 1 (understood coefficient in front of x2x^2), b=−4b = -4, c=−3c = -3

    • Calculate x-coordinate of vertex:         x=−(−4)2(1)=42=2x = \frac{-(-4)}{2(1)} = \frac{4}{2} = 2

    • Calculate y-coordinate of vertex by substituting x=2x = 2:         y=(2)2−4(2)−3=4−8−3=−7y = (2)^2 - 4(2) - 3 = 4 - 8 - 3 = -7

    • Vertex location: (2,−7)(2, -7).

  • Graphing and Identifying Points on Calculator:

    • Enter equation under Y= menu: Y1 = x^2 - 4x - 3.

    • Press TRACE key, type 22, and press ENTER to verify the vertex at (2,−7)(2, -7).

    • Do not use the arrow keys to locate the vertex, as manual scrolling will not land exactly on integer coordinates.

    • To view the exact x−yx-y coordinate table, press SECOND + GRAPH (TABLE).

  • Symmetry and Table Values for y=x2−4x−3y = x^2 - 4x - 3:

    • Parabolas are symmetric across an axis of symmetry passing through the vertex.

    • Coordinate points plotted for complete accuracy (vertex plus two points on each side):

      • (−1,2)(-1, 2)

      • (0,−3)(0, -3) (Y-intercept)

      • (1,−6)(1, -6)

      • (2,−7)(2, -7) (Vertex)

      • (3,−6)(3, -6)

      • (4,−3)(4, -3)

  • Finding the Y-Intercept Algebraically:

    • The y-intercept occurs where x=0x = 0.

    • Evaluating y=x2−4x−3y = x^2 - 4x - 3 at x=0x = 0 cancels out all terms containing xx, leaving the constant term c=−3c = -3.

    • Y-intercept coordinate: (0,−3)(0, -3).

Calculator Mechanics and Handling Negative Coefficients

  • Distinction Between Negative Key and Subtraction Key:

    • The negative key (-) located at the bottom of the calculator keypad must be used for negative leading terms or factors.

    • The subtraction/minus key [-] located on the right side key column must be used for subtraction operations.

    • Entering a subtraction key in place of a negative key produces a syntax error.

    • Entering a negative key in place of a subtraction key causes the calculator to perform implicit multiplication by −1-1 rather than subtraction.

  • Correcting Equations Using Insert Mode:

    • To insert a missing character without retyping an entire equation, place the cursor over the position where the character belongs.

    • Press SECOND + DEL (INS mode) to insert new keys into the string.

    • Use DEL to remove extra spaces or erroneous characters.

  • Step-by-Step Example: y=−2x2+6x−1y = -2x^2 + 6x - 1

    • Key entry sequence: (-) 2 x^2 + 6 x - 1.

    • Because a=−2<0a = -2 < 0, the parabola opens downward.

    • Calculate x-coordinate of vertex:         x=−b2a=−62(−2)=−6−4=32=1.5x = \frac{-b}{2a} = \frac{-6}{2(-2)} = \frac{-6}{-4} = \frac{3}{2} = 1.5

    • Calculate y-coordinate of vertex:         y=−2(1.5)2+6(1.5)−1=−2(2.25)+9−1=−4.5+9−1=3.5y = -2(1.5)^2 + 6(1.5) - 1 = -2(2.25) + 9 - 1 = -4.5 + 9 - 1 = 3.5

    • Vertex location: (1.5,3.5)(1.5, 3.5).

  • Tracing Fractional Inputs:

    • While on the graph screen, press TRACE, type 3 / 2 (or 1.5), and press ENTER to move the cursor directly to the vertex coordinates (1.5,3.5)(1.5, 3.5).

  • Table Values for y=−2x2+6x−1y = -2x^2 + 6x - 1:

    • (−1,−9)(-1, -9)

    • (0,−1)(0, -1) (Y-intercept)

    • (1,3)(1, 3)

    • (1.5,3.5)(1.5, 3.5) (Vertex)

    • (2,3)(2, 3)

    • (3,−1)(3, -1)

Evaluating Functions and Function Notation

  • Function Notation Equivalent Terms:

    • f(x)f(x), g(x)g(x), and c(p)c(p) all represent dependent outputs equivalent to yy.

  • Evaluating Non-Integer Inputs on Calculator:

    • To evaluate f(3)f(\sqrt{3}) for the function y=−2x2+6x−1y = -2x^2 + 6x - 1:

      1. Ensure the equation is active in Y1.

      2. Press GRAPH to view the plot.

      3. Press TRACE key.

      4. Press SECOND + x^2 (to invoke \sqrt{}).

      5. Type 3 and press ENTER.

    • Output value: y≈3.39y \approx 3.39 (rounded to 3.43.4).

  • Screen Navigation Shortcut:

    • Press SECOND + MODE (QUIT) to exit any graph, table, or calculation menu and return to the main home screen.

Absolute Value Functions and Transformations

  • Absolute Value Graph Geometry:

    • Absolute value functions of the form y=a∣x−h∣+ky = a|x - h| + k yield V-shaped graphs.

  • Calculator Entry Sequence for Absolute Value:

    1. Press Y= and clear existing equations.

    2. Press MATH key.

    3. Right-arrow over to the NUM column header.

    4. Select Option 1: 1: abs(.

    5. Type the inner expression (e.g., x + 3).

    6. Right-arrow out of the vertical absolute value bars (or close parenthesis )).

    7. Type the remaining constant term outside the absolute value.

  • Analyzing y=∣x+3∣+2y = |x + 3| + 2

    • Set inner expression to zero to find vertex x-coordinate: x+3=0  ⟹  x=−3x + 3 = 0 \implies x = -3

    • The remaining constant provides vertex y-coordinate: y=2y = 2

    • Vertex coordinates: (−3,2)(-3, 2).

    • Slope of right side: m=1m = 1 (rises 11, runs 11).

    • Slope of left side: m=−1m = -1 (rises 11, runs −1-1).

  • Absolute Value Transformations:

    • Inserting a negative sign outside absolute value (y=−∣x+3∣+2y = -|x + 3| + 2) flips the V-shape downward.

    • Adding a coefficient inside or outside changes the slope/steepness (e.g., y=∣2x+3∣+2y = |2x + 3| + 2 changes the x-coordinate of the vertex to x=−1.5x = -1.5 and increases the slope to 22).

Rational Functions, Vertical Asymptotes, and Y-Intercepts

  • Parentheses Rules for Rational Expressions:

    • When entering rational functions into a calculator, the entire numerator and entire denominator must individually be enclosed in parentheses.

    • Formula format for input: (Numerator) / (Denominator).

  • Step-by-Step Example: f(x)=x2+6x−4x2−3x−18f(x) = \frac{x^2 + 6x - 4}{x^2 - 3x - 18}

    • Correct calculator key entry sequence:         ( x^2 + 6 x - 4 ) / ( x^2 - 3 x - 18 )

    • Failure to parenthesize will cause the calculator to divide only the last term of the numerator by the first term of the denominator due to standard order of operations.

  • Window Settings Setup:

    • Press WINDOW to set visual bounds:

      • Xmin = −10-10

      • Xmax = 1010

      • Xscl = 11

      • Ymin = −10-10

      • Ymax = 1010

      • Yscl = 11

  • Determining Vertical Asymptotes Algebraically:

    • Vertical asymptotes occur at x-values that force the denominator to equal zero (denominator=0\text{denominator} = 0).

    • Set denominator to zero: x2−3x−18=0x^2 - 3x - 18 = 0

    • Sign rules for quadratic factoring ax2+bx+cax^2 + bx + c:

      • + and + yields (x+term)(x+term)(x + \text{term})(x + \text{term})

      • + and - yields (x+term)(x−term)(x + \text{term})(x - \text{term})

      • - and - yields (x+term)(x−term)(x + \text{term})(x - \text{term})

      • - and + yields (x−term)(x−term)(x - \text{term})(x - \text{term})

    • Factor pairs of 1818:

      • 1×181 \times 18

      • 2×92 \times 9

      • 3×63 \times 6

    • Select factor pair that subtracts to yield middle coefficient −3-3:

      • Selected numbers: 33 and 66

      • Assign signs: (x+3)(x−6)=0(x + 3)(x - 6) = 0

    • Solve each factor:

      • x+3=0  ⟹  x=−3x + 3 = 0 \implies x = -3

      • x−6=0  ⟹  x=6x - 6 = 0 \implies x = 6

    • Vertical Asymptotes are defined by vertical lines at x=−3x = -3 and x=6x = 6.

    • Graphs cannot cross vertical asymptotes.

    • Older graphing calculator models (such as TI-83) display vertical asymptotes as connected pixel vertical lines resembling EKG graphs.

  • Finding Rational Y-Intercepts:

    • Substitute x=0x = 0 into the complete function:         f(0)=(0)2+6(0)−4(0)2−3(0)−18=−4−18=29f(0) = \frac{(0)^2 + 6(0) - 4}{(0)^2 - 3(0) - 18} = \frac{-4}{-18} = \frac{2}{9}

    • Decimal equivalent: 0.2‾0.\overline{2}.

    • Y-intercept coordinate: (0,29)\left(0, \frac{2}{9}\right).

Fraction Simplification and Calculator Utility Functions

  • Converting Decimals to Simplified Fractions:

    • Step 1: Perform numerical division on main screen (e.g., enter 2128 / 4816).

    • Decimal output: 0.4418588...0.4418588...

    • Step 2: Press MATH key.

    • Step 3: Select Option 1 (1: >Frac).

    • Step 4: Press ENTER twice.

    • Resulting simplified fraction: 1543\frac{15}{43}.

Course Structure, Recitation, and Assessment Details

  • Recitation Location & Time:

    • Recitation Room: Room 307a (changed from Room 307c).

    • Recitation Hours: 2:00 PM to 3:00 PM.

  • Grading Structure:

    • Main College Algebra course consists of homework assignments, quizzes, and unit exams administered in the TLC (Testing Learning Center).

    • Recitation section grade is purely homework-based; there are no quizzes or tests in recitation.

  • Course Material & Homework Platform:

    • Online platform: Cengage accessed via Canvas modules.

    • First assignment topic: "Graphs and Graphing Utilities" (due August 30).

    • Calculator requirement for testing: Permitted to use personal TI-84 Plus, TI-83, or TI-30 graphing/scientific calculators on all course exams.