Comprehensive Study Guide: Functions, Tables, Graphs, and Algebraic Foundations

Course Logistics and Announcements

• Class Schedule Adjustments:

  • There is no class on Monday of next week due to the Labor Day holiday, providing a three-day weekend.
  • Office hours schedule:
    • Tomorrow: $1:30\text{ PM}$ to $3:00\text{ PM}$.
    • Tuesday: $1:30\text{ PM}$ to $3:00\text{ PM}$.
    • No office hours will be held after Tuesday until Wednesday of next week.

• Class Materials and Printing Recommendations:

  • Always review the course schedule in advance to identify which lesson notes to print off.
  • Current coverage: Completing Lesson 1 and Lesson 2 today. Lesson 3 will begin on Wednesday of next week.
  • It is recommended to print off all lesson objectives and notes ahead of time (all A notes, B notes, and C notes) to ensure all materials are readily available.

• Quiz 1 Details:

  • Date & Location: Administered next week Thursday during recitation.
  • Duration: Last $20\text{ minutes}$ of class.
  • Coverage: Objectives $A1$ through $A3$ (Lessons 1 and 2).
    • Lesson 1 notes cover objectives $A1$ and $A2$.
    • Lesson 2 notes cover objective $A3$.

Objective A1: Identifying Functions

Tables, Graphs, and Sets of Ordered Pairs

• Tables:

  • First Table: Represents a valid function.
  • Second Table: Represents a valid function.

• Graphs & Vertical Line Test:

  • Graph 1: Represents a function because it passes the vertical line test (any vertical line intersects the graph at most once).
  • Graph 2 (Deceptive Graph with Overlapping Endpoints): Represents a function.
    • The graph features a vertical alignment with an upper closed circle and a lower open circle.
    • An open circle indicates that the specific location is not included/touched. Thus, a vertical line passing through that section technically only intersects the graph once at the closed circle.
    • Special Cases / Exceptions:
      • If both points were open circles, it would still be a function because a vertical line would intersect $0$ points.
      • If both points were closed circles, it would not be a function because a vertical line would intersect $2$ points.
  • Graph 3 (Bottom Left): Not a function due to vertical line segments causing multiple intersections.
  • Graph 4: Represents a function.

• Ordered Pair Sets:

  • Set 1 ($Q_1, 70$, $Q_2, 73$, etc.): Represents a function.
  • Set 2: Does not represent a function.

Equations

• $y^2 = 9 - x^2$ (Not a function):

  • The output variable $y$ has an even exponent ($y^2$).
  • Isolating $y$ requires taking the square root: y = \text{\textpm}\text{\textsqrt{}}9 - x^2.
  • The \text{\textpm} symbol indicates two outputs for a single input. For example, setting $x = 0$ yields y = \text{\textsqrt{}}9 = \text{\textpm}3, mapping input $0$ to outputs $3$ and $-3$

• $x + y^2 = 9$ (Not a function):

  • The output $y$ has an even exponent ($y^2$), creating a \text{\textpm} upon isolation.

• $x^2 + y = 9$ (Function):

  • Isolating $y$ yields $y = 9 - x^2$. No \text{\textpm} is introduced, ensuring a single output for every input.

• $2x = 4$ (Not a function):

  • Isolating the single variable gives $x = 2$.
  • This restricts the input strictly to $2$ while the output can be any value (e.g., $(2, 1)$, $(2, 2)$, $(2, 3)$). A single input mapping to multiple outputs fails the definition of a function.

• $5y = 15$ (Function):

  • Isolating $y$ gives $y = 3$.
  • This specifies that the output must always be $3$, while the input can be any real number (e.g., $(1, 3)$, $(-2, 3)$, $(12, 3)$). Repeating outputs for different inputs is completely valid for a function.

• $x^3 = 8$ (Not a function):

  • Simplifies to $x = 2$ (a vertical line), which fails the function test.

• $y^3 = 8$ (Function):

  • Taking the cube root yields $y = 2$ (a horizontal line), which passes the function test.

• $x^2 + 3y + 7 = 0$ (Function):

  • Isolating $y$ yields y = \text{\textfrac{}}-x^2 - 7{3}. No \text{\textpm} is introduced.

Verbal and Contextual Representations

• Scenario 1: "At the beginning of each month $x$ in the past year, Sam checks the balance $y$ in his savings account."

  • Classification: Function.
  • Explanation: Inputs ($x$) are the specific months (January through December). Outputs ($y$) are the account balances on the 1st day of each month at a specific time (e.g., $7:00\text{ AM}$).
  • Sample Data Table:
    • $x = \text{January} \rightarrow y = 2\text{ dollars}$
    • $x = \text{February} \rightarrow y = 5\text{ dollars}$
    • $x = \text{March} \rightarrow y = 5\text{ dollars}$
  • On the first of any month, an account has exactly one balance. Even if balances repeat across months, each input month maps to exactly one output value.

• Scenario 2: "At the beginning of each month $y$ in the past year, Sam checks his savings account balance $x$."

  • Classification: Not a Function.
  • Explanation: Inputs ($x$) are dollar balances, and outputs ($y$) are months. If the account balance was $5\text{ dollars}$ in both February and March, the single input $5\text{ dollars}$ maps to two outputs (February and March).
  • Universal Rule for Verbal Problems: A contextual relationship is only a function if it satisfies the requirement for every single possibility, not just a specially constructed setup.

• Terminology Convention ("blank as a function of blank"):

  • In the phrase "$V$ as a function of $t$", $V$ represents the output and $t$ represents the input.

Objective A2: Tables and Graphs from Symbolic and Verbal Models

Core Definitions: Domain and Range

• Domain: The set of all possible input values ($x$-values). Graphically, it describes the horizontal extent of the graph (how far left to how far right it extends).
• Range: The set of all possible output values ($y$-values). Graphically, it describes the vertical extent of the graph (how far down to how far up it extends).

Symbolic Example: $y = x^2 - 6$

• Given Domain Constraint: Evaluated for $x$ from $-4$ to $4$.

• Table Calculations:

  • $x = -4 \rightarrow y = (-4)^2 - 6 = 16 - 6 = 10$
  • $x = -3 \rightarrow y = (-3)^2 - 6 = 9 - 6 = 3$
  • $x = -2 \rightarrow y = (-2)^2 - 6 = 4 - 6 = -2$
  • $x = -1 \rightarrow y = (-1)^2 - 6 = 1 - 6 = -5$
  • $x = 0 \rightarrow y = (0)^2 - 6 = 0 - 6 = -6$
  • $x = 1 \rightarrow y = (1)^2 - 6 = 1 - 6 = -5$
  • $x = 2 \rightarrow y = (2)^2 - 6 = 4 - 6 = -2$
  • $x = 3 \rightarrow y = (3)^2 - 6 = 9 - 6 = 3$
  • $x = 4 \rightarrow y = (4)^2 - 6 = 16 - 6 = 10$

• Calculator Syntax Warning:

  • Entering $-4^2$ without parentheses evaluates as $-(4^2) = -16$.
  • Always use parentheses around negative inputs: $(-4)^2 = 16$.

• Graphing Guidelines:

  • Axis Placement: The $x$-values range symmetrically from $-4$ to $4$, so center the vertical $y$-axis. The $y$-values range from $-6$ to $10$; lowering the horizontal $x$-axis slightly provides more visual room for positive $y$-values.
  • Scale Selection: Use an $x$-scale of $1$ unit per grid line. Use a $y$-scale of $2$ units per grid line to comfortably fit the maximum ($10$) and minimum ($-6$) values.
  • Shape & Extension: Plot points and connect them in a smooth, U-shaped parabola. Include arrows at the ends to signify that the equation continues infinitely beyond the plotted domain interval.

Verbal Example: Maple Tree Growth

• Problem Statement: Amy planted a sugar maple tree $6\text{ feet}$ tall. The average growth is $1.5\text{ feet}$ per year. Create a table and graph representing tree height ($y$ in feet) $x$ years after planting.
• Determining Starting Time:
  • Initial time is $x = 0$ (the moment of planting), not $x = 1$.
  • Negative time values are omitted as physical context does not extend backwards.
• Table Setup (Eliminating Decimals):
  • By evaluating every $2\text{ years}$, growth increases by $2 \times 1.5\text{ feet} = 3\text{ feet}$, avoiding fractional values:
    • $x = 0\text{ years} \rightarrow y = 6\text{ feet}$
    • $x = 2\text{ years} \rightarrow y = 9\text{ feet}$
    • $x = 4\text{ years} \rightarrow y = 12\text{ feet}$
    • $x = 6\text{ years} \rightarrow y = 15\text{ feet}$
• Graphing Details:
  • Use Quadrant I only (x \text{\textge} 0, y \text{\textge} 0).
  • Scale the $x$-axis by $2$s and the $y$-axis by $3$s.
  • Arrows: Draw a single ray starting strictly at $(0, 6)$ and extending upward to the right with an arrow. Do not place an arrow pointing left past $x = 0$.

Objective A3: Fundamental Algebra Review

Equations vs. Expressions

• Equations: Contain an equality sign ($=$. You can perform inverse operations to isolate and solve for a specific variable value.
  • Example: $10p - 12 = 28 \rightarrow 10p = 40 \rightarrow p = 4$
  • Example: $10w - 3 = 7 + w \rightarrow 9w = 10 \rightarrow w = 1$
• Expressions: Do not contain an equality sign ($=$). They cannot be solved for a single numerical value; they can only be simplified by combining like terms.
  • Example: Simplify $5x - 8 - 3y + 12 + 2x - y$
    • Combine $x$-terms: $5x + 2x = 7x$
    • Combine $y$-terms: $-3y - y = -4y$
    • Combine constants: $-8 + 12 = 4$
    • Final Simplified Expression: $7x - 4y + 4$
  • Ordering Convention: Writing terms in alphabetical order ($7x - 4y + 4$) is standard, but term rearrangement does not alter mathematical correctness.

Distribution and Negative Signs

• Example: Simplify $4(x - 3) - 5(x + 2) - (x - 5)$
  • Distribute $4$: $4x - 12$
  • Distribute $-5$: $-5x - 10$
  • Distribute $-1$: $-x + 5$
  • Group terms: $(4x - 5x - x) + (-12 - 10 + 5) = -2x - 17$

Clearing Fractions in Linear Equations

• Example: Solve \text{\textfrac{}}3x{8} - 1 = \text{\textfrac{}}x - 7{6}
  • Method: Multiply every term on both sides by the Least Common Multiple (LCM) of the denominators $8$ and $6$, which is $24$.
  • Step 1 (Distribute LCM):         24 \times \text{\textfrac{}}3x{8} - 24 \times 1 = 24 \times \text{\textfrac{}}x - 7{6}
  • Step 2 (Simplify Coefficients):$3(3x) - 24 = 4(x - 7)$$9x - 24 = 4x - 28$
  • Step 3 (Isolate Variable):$9x - 4x = -28 + 24$$5x = -4$         x = -\text{\textfrac{}}4{5}
  • Verification: Always substitute the final solution back into the original equation to ensure both sides balance.

Solving Literal Equations (Multiple Variables)

• Example 1: Isolate $y$ in $2x + y = 10$
  • Subtract $2x$ from both sides: $y = 10 - 2x$
• Example 2: Isolate $r$ in $5rv + 3t - 10 = w$
  • Add $10$ and subtract $3t$: $5rv = w + 10 - 3t$
  • Divide by $5v$: r = \text{\textfrac{}}w + 10 - 3t{5v}

Interval Notation and Inequalities

Structural Rules

• Parentheses ():
  • Used for strict inequalities ($<\text{ or }>$).
  • Corresponds to open circles on a number line.
  • Indicates that the endpoint is excluded.
• Brackets []:
  • Used for inclusive inequalities ($ ext{ extle} ext{ or } ext{ extge}$).
  • Corresponds to closed circles on a number line.
  • Indicates that the endpoint is included.
• Infinities (-\text{\textinfty}\text{ or }\text{\textinfty}):
  • ALWAYS use parentheses ().
  • Brackets are never used with infinity because infinity is an unbounded concept, not a reachable number.

Examples and Transformations

• Example 1: $x < 8$

  • Number Line: Open circle at $8$, shaded infinitely to the left.
  • Interval Notation: (-\text{\textinfty}, 8)

• Example 2: -4 \text{\textle} x < 4

  • Number Line: Closed circle at $-4$, open circle at $4$, shaded in between.
  • Interval Notation: $[-4, 4)$

• Example 3: $0 < x\text{ or }x < 8$ (i.e., $x > 0\text{ or }x < 8$)

  • Analysis: The condition $x > 0$ covers all numbers from $0$ to \text{\textinfty}. The condition $x < 8$ covers all numbers from -\text{\textinfty} to $8$.
  • Since it is an "or" statement, any real number that satisfies at least one condition is included.
  • Combined Coverage: Covers the entire real number line.
  • Interval Notation: (-\text{\textinfty}, \text{\textinfty})