Chapter 1.2
Function recap and basic definitions
A function is a rule that assigns to each input a unique output. In notation: y = f(x) where the input is the independent variable and the output is the dependent variable.
Key concepts to remember:
Domain and range of a function (inputs and possible outputs).
Function notation and interpretation of outputs for given inputs.
Practical goal from the lecture: develop a concise, flexible summary of definitions and concepts, using schematics to show relationships and structure.
Schematics and knowledge frameworks
Schematics are a framework for organizing ideas and showing interconnections.
Example: Kelowna / Okanagan region schematic shows interconnected components (e.g., Canada → BC → British Columbia → Canada). This represents a relationship structure, not just a list of items.
In broad terms, schematics help visualize relationships among elements in a system.
Indigenous cultural frameworks (e.g., medicine wheel) are cited as examples of schematics used to illustrate holistic relationships.
Purpose of schematics: illustrate relationships, hierarchies, and connections between elements in a system.
Recognizing graphs from verbal descriptions
You should be able to:
Draw or identify the graph of a function from a verbal description.
Determine whether a function is increasing or decreasing from verbal, tabular, or graphical information.
This involves translating verbal statements into mathematical form and then into a graph or algebraic representation.
Linear functions: forms and representations
A linear function can be represented in several equivalent forms:
Slope-intercept form: y = mx + b where m is the slope and b is the y-intercept.
Point-slope form: m = y - yo/x - xo where (xo, yo) is a point on the line.
Equation of line: y - yo = m(x - xo)
General insight: for a fixed slope, varying the intercept yields parallel lines.
Note on the role of parameters:
For the family of lines with same slope, varying the intercept b produces parallel lines.
For a fixed intercept, varying the slope m changes the steepness and orientation.
Example interpretation: the line represents a relationship where a change in x produces a proportional change in y, scaled by the slope m.
Intercepts and sketching lines
Intercepts of a line:
Y-intercept: when x = 0, the line crosses the y-axis at y = b (the intercept is the point $(0, b)$).
X-intercept: when y = 0, solve 0 = mx + b to get x = -\frac{b}{m} (the point (-\frac{b}{m}, 0)).
To sketch a line, you can:
Find the y-intercept and another point (e.g., using a chosen x-value) and plot them.
Alternatively, use the point-slope form to plot through a known point and then draw the line with slope m.
Across the lecture, emphasis on understanding intercepts as coordinates where the line crosses the axes, and how they relate to the equation parameters.
Families of parallel lines and slope interpretation
When treating lines of the form y = mx + b with fixed m but varying b, you obtain a family of parallel lines (same slope, different vertical shifts).
If you vary the slope m while keeping a common point or another constraint, you obtain different lines with different orientations.
Perpendicular lines:
If two lines have slopes m1 and m2, they are perpendicular when m1 = -1/m2 or m1m2 = -1 (i.e., slopes are negative reciprocals).
These concepts help in understanding how linear models behave under parameter changes and how to interpret graphs without computing every coordinate.
Example 1: Functional interpretation from a real-world setup (linear relationship intuition)
A classic setup is a linear relationship in a real scenario, where equal changes in input produce constant changes in output, i.e., constant slope.
In the lecture, an analogy is drawn with a scenario where you can line up team members parallel at equal distance; the key takeaway is that equal spacing yields a consistent, linear pattern of some measured quantity (e.g., coverage, detection probability) as a function of spacing.
The essential idea: from a verbal description, identify that the quantity changes at a constant rate with respect to the input, implying a linear model with a constant slope.
Example 2: Vehicle depreciation and repair costs (linear modeling)
Given: initial cost V_0 = 25{,}000 dollars.
Depreciation: value decreases by 2{,}000 dollars per year. Thus the value as a function of age in years t is
V(t) = 25{,}000 - 2000 t.
Repair costs accumulate at 1{,}500 dollars per year. Thus the repair cost as a function of age is
C(t) = 1500 t.
The two lines intersect when the current value equals cumulative repair costs: solve
V(t) = C(t) \ 25000 - 2000t = 1500t
Solve for t: t = \frac{25000}{3500} = \frac{50}{7} \approx 7.143\text{ years}.
At the intersection time, the common value is
V\left(\frac{50}{7}\right) = 25000 - 2000\left(\frac{50}{7}\right) = \frac{75000}{7} \approx 10714.29\$.
This equals the repair cost at that time: C\left(\frac{50}{7}\right) = 1500\left(\frac{50}{7}\right) = \frac{75000}{7}\$.
Interpretation: The moment when the accumulated repair costs equal the resale value of the vehicle under this simple model; if you sell at that time, the net outcome aligns with the model’s intersection point.
Additionally, consider a practical threshold: when the resale value is a certain percentage of the original value, the model can guide replacement timing.
Example threshold: when the value is 60% of original: V(t) = 0.60 \cdot 25000 = 15000
Solve: 25000 - 2000t = 15000 \Rightarrow 2000t = 10000 \Rightarrow t = 5\text{ years}.
Another simple sketch result: time when value hits zero (the x-intercept of the value line):
Solve for 0 = 25000 - 2000t to get
t = \frac{25000}{2000} = 12.5\text{ years}.
Graph interpretation points:
The x-intercept of the value line (t-axis) is at t = 12.5 years.
The intersection of the two lines (value and repair cost) is at t = \frac{50}{7} \approx 7.143\text{ years} with common value \approx 10714.29\$.
The resale/adjusted value at that time is the same as the cumulative repair cost under this linear model.
Sketching and interpreting linear graphs (practical tips)
To sketch a line, pick two points and plot them; connect with a straight line.
For a line in slope-intercept form, you can directly read the intercepts:
Y-intercept: b (point $(0, b)$).
X-intercept: x = -\frac{b}{m} (point $(-\frac{b}{m}, 0)$).
The slope determines steepness and direction:
If m > 0, the line is increasing; if m < 0, it is decreasing.
The concept of a family of lines with the same slope but different intercepts emphasizes that changing the intercept translates the line vertically without rotating it (parallel lines).
Practice problems and problem-solving strategies (referenced in the lecture)
Suggested practice problems (section 51.2, problems 3, 8, 9, 14, 16, 19, 29, 36, 37, 41, 42, 50) focus on:
Identifying slope and intercept from data or descriptions.
Writing equations in slope-intercept and point-slope forms.
Analyzing linear relationships from tables, verbal descriptions, and graphs.
Key steps when solving linear-model problems:
1) Identify the dependent and independent variables and what the slope represents (rate of change).
2) Choose a convenient form (slope-intercept or point-slope) for clarity.
3) Use a given point or intercept to determine constants.
4) Solve for any required quantities (e.g., intercepts, time to reach a threshold, intersection points).
5) Interpret the result in the real-world context and consider model limitations.
Connections to prior material, principles, and real-world relevance
Link to foundational algebra: understanding linear relationships is a cornerstone for more advanced topics (exponential/logarithmic models, systems of equations).
Real-world relevance: depreciation, maintenance costs, and decision-making about replacement are common applications of linear models in economics and engineering.
Ethical and practical implications: the model assumes constant rates (e.g., depreciation, repair costs); in reality, rates may change due to depreciation schedules, maintenance, or market conditions, so you should consider model validity and the potential impact of using a simplified linear model for long-term forecasting.
Quick recap and takeaways
A linear function has a constant rate of change; represented as y = m x + b or equivalently in point-slope form y - y0 = m(x - x0).
Intercepts provide quick graphing anchors: y$-intercept is b; x$-intercept is -\frac{b}{m} (assuming m \neq 0).
A family of lines with the same slope is a family of parallel lines; perpendicular lines satisfy m1 m2 = -1.$$
Real-world examples (e.g., vehicle value and repair costs) illustrate how to translate a practical situation into a linear model, solve for key quantities, and interpret intersections as critical decision points.