Chapter 1 Notes: Functions, Graphs, and Linear Functions

1.1 Functions and Models

• A function is a rule or correspondence that assigns to each element of the domain exactly one element of the range.
  • Domain: the set of inputs.
  • Range: the set of outputs.
  • A function can be defined by: a set of ordered pairs, a table, a graph, an equation, or a verbal description.
• Examples:
  • Body temperature example: Normal temperature ~ 98.6°F corresponds to 37°C on a Celsius thermometer. This illustrates a functional relationship between temperature readings in two units.
• Key definitions and ideas:
  • A graph represents a relation. It is a function if each x-value (input) corresponds to exactly one y-value (output).
  • Examples shown in the transcript include a domain and range determination from a graph.
• Domain and Range (from a graph example):
  • Given points: {(-5, -7), (-3, -2), (-1, 6), (0, -2), (3, 5), (5, -7)}
  • Domain: { -5, -3, -1, 0, 3, 5 }
  • Range: { -7, -2, 6, 5 }
  • Note: Repeated y-values are listed once in the range.
• Recognizing functions from data/tables/graphs:
  • Example: PC industry sales by year (domain: year x; range: sales S). For the table given, each year has a single sales value, so the relationship is a function; domain and range are defined accordingly.
• Function notation and evaluation:
  • If a function is defined by a graph, table, or equation, you can evaluate outputs using f(x).
  • Example-oriented tasks include evaluating f(-2) and f(4) from a graph.
• Vertical Line Test:
  • A set of points in the plane is the graph of a function if and only if no vertical line intersects the graph in more than one point.
  • Visual cue: if a vertical line crosses the graph at more than one point, the relation is not a function.
• Interpreting function values from a graph:
  • Example: Elderly men in the workforce: g(t) is a function giving millions of elderly men in the workforce for year t.
  • Given point (2020, 21) means g(2020) = 21 (million).
• Real-world modeling with functions:
  • Public health expenditures modeled as E(t) = 738.1(1.065)^t, where E(t) is in billions and t is years after 1990.
  • Application questions include estimating expenditures for 2010 (t = 20) and 2015 (t = 25), with a note on the uncertainty of predictions and model limitations.
• Section links:
  • The material links to online domain/range activities and to broader model-building ideas in applications.

1.2 Graphs of Functions

• Objectives:
  • Graph equations using point-plotting, graphing calculators, and Excel.
  • Align inputs and scales to model data; graph data points.
• Graphing by plotting points (example):
  • Graph y = 2x^2 by plotting integer x-values.
  • Points include: (0,0), (1,2), (2,8), (3,18) and symmetry with negative x: (-3,18), (-2,8), (-1,2).
  • Construct a smooth curve through these points.
• Complete graph: definition and example
  • A complete graph shows the basic shape, key turning points, axis intercepts, and suggests unseen portions.
  • Example: Graph x^3 - 5x. Turning points occur at two places, indicating a complete graph.
  • Points given: (-2, 2), (-1, 4), (0, 0), (1, -4), (2, -2).
• Graphing with a calculator:
  1) Write the function with x as the independent variable and y as the dependent variable; solve for y if necessary.
  2) Enter the function in the calculator’s editor (use parentheses as needed).
  3) Use GRAPH/ZOOM to view; standard window typically x ∈ [-10, 10], y ∈ [-10, 10].
  4) Use WINDOW to adjust viewing window to see parts not in the standard view.
• Example: Graphing a complete graph with a calculator window change (Window xmin = -10, xmax = 10, ymin = -25, ymax = 10) improves visibility.
• Example: Cost-Benefit model for pollution removal (p%)
  • p is a percentage: 0 ≤ p < 100 because p = 100 makes C undefined; C ≥ 0.
  • The cost curve C(p) is graphed over [0,100].
  • Point of interest: p = 90 gives (90, 48150) using TRACE, meaning a cost of $48,150.
• Example: Spreadsheet solution for C =
  • C =
    60000
    5350p
    100-p
  • This section shows a table of p versus C and how to use TABLE/TRACE to generate more values for a complete graph.
• Plotting real data and interpretation:
  • Example: Voting between 1950 and 2008 uses a model f(x) with x as years after 1950.
  • f(10) corresponds to 1960; f(58) corresponds to 2008; values approximate 65% and 57%, respectively.
• Aging workers model (graphing guidance):
  • Model: y = −0.000362x^3 + 0.0401x^2 − 1.39x + 21.7, with x in [20, 65].
  • Use y-values (0 to 10) for reasonable viewing windows; TABLE and TRACE can display outputs for specific ages (e.g., age 55, 64).
• U.S. diabetes data plotting:
  • Data points are years and thousands of adults with diabetes, projected to 2050.
  • Alignment steps: set x to years after 1980 (L1) and y to millions (L2) after converting from thousands by dividing by 1000.
  • Scatter plot illustrates non-perfect linear fit; used to illustrate arranging data in Excel/graphing utilities.

1.3 Linear Functions

• What is a linear function?
  • A linear function can be written in the form $f(x) = ax + b$ where a and b are constants.
  • Domain and range: all real numbers.
• Identifying linear vs non-linear relations:
  • Example: $xy = 5$ is not a linear function because x and y are not in the form y = mx + b.
  • Example: $0 = 3t - s + 5$ can be solved for s: $s = 3t + 5$, which is linear; domain and range are all real numbers.
  • The equation $y = 9$ is linear with slope $a = 0$ and intercept $b = 9$; domain is all real numbers; range is {9}.
• Intercepts (how to find them algebraically):
  • y-intercept: set x = 0, solve for y; point is (0, b).
  • x-intercept(s): set y = 0, solve for x; point is (a, 0).
• Example: Intercepts for $4x - 8y = 16$
  • x-intercept: set y = 0 ⇒ 4x = 16 ⇒ x = 4.
  • y-intercept: set x = 0 ⇒ -8y = 16 ⇒ y = -2.
  • Intercepts: (4, 0) and (0, -2).
• Loan balance example (linear function):
  • The balance after x payments: $y = 76500 - 1275x$ (note: the transcript says a loan plus interest of 76,500; interpret as starting amount 76,500 with payments reducing it by 1,275 each month).
  • Slope: $m = -1275$; y-intercept: $b = 76500$.
  • Interpretation: each month, the amount owed decreases by $1,275.
• Slope concepts:
  • Slope formula: $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$
  • Vertical lines have undefined slope (division by zero).
  • Horizontal lines have slope 0.
  • Positive slope means the line rises to the right; negative slope means it falls.
• Orientation and constant rate of change:
  • For a linear function $f(x) = mx + b$, the slope m is the constant rate of change.
• Slope and intercept details:
  • Example: y = 7x − 12 → slope m = 7; y-intercept b = −12.
  • Example: 2x − 3y = 12 → y = (2/3)x − 4 → slope m = 2/3; y-intercept = −4.
• Hispanics in the United States (linear model):
  • Model: $H(x) = 0.224x + 9.01$ where x is years after 1990.
  • Slope 0.224 indicates the percentage of Hispanics increases by 0.224 percentage points per year.
• Marginal revenue and marginal profit (linear functions):
  • Revenue: $R(x) = 89.50x$
  • Cost: $C(x) = 54.36x + 6790$
  • Profit: $P(x) = R(x) - C(x) = (89.50 - 54.36)x - 6790 = 35.14x - 6790$
  • Marginal revenue: the slope of the revenue function, $MR = 89.50$ per unit.
  • Marginal profit: the slope of the profit function, $MP = 35.14$ per unit.

1.4 Equations of Lines

• Key forms:
  • Slope-intercept form: $y = mx + b$ where m is the slope and b is the y-intercept.
  • Point-slope form: $y - y<em>1 = m\bigl(x - x</em>1\bigr)$ where (x1, y1) is a point on the line.
  • General form: $ax + by = c$ where a, b, c are real numbers and not both a and b are zero.
  • Horizontal line: $y = b$ where b is the constant y-value; slope = 0.
  • Vertical line: $x = a$ where a is the constant x-value; slope is undefined.
• Parallel and perpendicular lines:
  • Parallel lines have the same slope.
  • Perpendicular lines have slopes that are negative reciprocals (if one slope is m, the perpendicular slope is ��-1/m).
• Average rate of change and secants:
  • Average rate of change of f between x = a and x = b (a < b):
    $ext{ARC} = \frac{f(b) - f(a)}{b - a}.$
  • Slope of the secant line between two points on a graph corresponds to this average rate of change.
• Difference quotient (instantaneous rate approximator):
  • For f, the difference quotient is
    $\frac{f(x+h) - f(x)}{h} ext{ for } h <br />\neq 0.$
• Examples:
  • Appliance repair (linear model): A plumber charges a service call of $65 plus $30 per hour. If x is hours, y = 30x + 65.
  • Point-slope to write equation: Through (−4, 2) with slope 1/3, using y − y1 = m(x − x1) gives the equation of the line through that point with slope 1/3.
  • Blood alcohol percent (linear model): Table shows increasing BA% with drinks; rate of change is 0.02 percentage points per drink.
  • Slope = 0.02; using point (5, 0.11): y − 0.11 = 0.02(x − 5) ⇒ y = 0.02x + 0.01.
• Inmate population modeling (example):
  • Given data: 2001 → 1.345 million; 2013 → 1.570 million.
  • Slope: m = (1.570 − 1.345) / (2013 − 2001) = 0.225 / 12 = 0.01875 million per year.
  • Linear model: using point (2001, 1.345): N(x) = 1.345 + 0.01875(x − 2001).
  • Prediction: N(2017) = 1.645 million (matches the given 1.645).
• Special cases: using lines for horizontal and vertical through a given point and slope.
• Public School Enrollment (linear approximation):
  • Data show enrollment in thousands for select years. A linear model can approximate the growth.
  • Average rate of change computed as 237.8 thousand per year.
  • Equation using point (1980, 41,651) and slope m = 237.8:
    $y - 41{,}651 = 237.8(x - 1980)$
    $y = 237.8x - 429{,}193.$

Section 2: Additional Notes and Formulas

• Important generic forms and interpretations:
  • Slope-intercept form: $y = mx + b$; slope m is the rate of change; intercept b is the value when x = 0.
  • Point-slope form: $y - y<em>1 = m(x - x</em>1).$
  • General form: $ax + by = c.$
  • Horizontal line: $y = b;$ slope is 0.
  • Vertical line: $x = a;$ slope is undefined.
• Key concepts to master for exams:
  • Distinguishing functions from non-functions using the vertical line test.
  • Computing and interpreting domain and range from graphs and data.
  • Converting between equation forms (slope-intercept, point-slope, and general form).
  • Finding intercepts algebraically and interpreting them in real-world contexts.
  • Understanding slope as a constant rate of change and relating it to real-world quantities (revenue, costs, population, etc.).
  • Using average rate of change and the difference quotient to analyze changes over intervals.
  • Interpreting and using linear models to approximate real-world data, including potential limitations.

Quick reference formulas (LaTeX)

• Linear function form: $f(x) = ax + b$
• Slope-intercept: $y = mx + b$
• Point-slope: $y - y<em>1 = m(x - x</em>1)$
• General form: $ax + by = c$
• Slope (two points): $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$
• Intercepts:
  • Y-intercept: $b = f(0)$
  • X-intercept: solve $f(x) = 0$ for x.
• Difference quotient: $\frac{f(x+h) - f(x)}{h}$
• Average rate of change over [a, b]: $ext{ARC} = \frac{f(b) - f(a)}{b - a}$
• Exponential model example: $E(t) = 738.1(1.065)^t.$
• Example revenue model: $R(x) = 89.50x,\, C(x) = 54.36x + 6790, \, P(x) = R(x) - C(x) = 35.14x - 6790.$