MAT-144 · Mathematical Reasoning
Topic 01 · Linear Functions
Vocabulary & key terms
Every term defined across this topic, grouped by lesson. Tap a lesson title to jump back to the page where the term was introduced.
39
terms in this topic. Skim before the review.
- Function
- The rule itself. We name it with a letter, usually f, but g and h show up too.
- Input
- The value you feed in. Almost always called x.
- Output
- What comes out. Often called y, or written as f(x).
- f(x)
- Read out loud as "f of x." It means: "the output of the function f when the input is x." It does not mean f times x.
- Evaluate
- To compute a function's output for a specific input. "Evaluate f at x = 7" means: substitute 7 wherever you see x in the rule, then simplify. Same idea as plugging a value in.
- Domain
- All inputs that are allowed. (Working 0 hours makes sense. Working −5 hours doesn't.)
- Range
- All possible outputs you can actually get out.
- Deductive
- Top-down. Apply a general rule to a specific case. "All large drinks are $5.50. This is large. So it costs $5.50."
- Inductive
- Bottom-up. Generalize a pattern from specific examples. "My large drink has been $5.50 every time. Probably it'll be $5.50 today."
- Conjecture
- A pattern-based guess from inductive reasoning. Not yet proven, but reasonable.
- Counterexample
- A single case that breaks a conjecture. One counterexample is enough, it only takes one black swan to disprove "all swans are white."
- Premise
- A starting statement assumed true. Deductive reasoning chains premises to a conclusion.
- Conclusion
- What you end up with. In deduction, it's certain. In induction, it's probable.
- Place value
- A digit's position. In 4,587: 4 is thousands, 5 is hundreds, 8 is tens, 7 is ones. To the right of the decimal point, the places keep going: in 7.4839, the 4 is tenths, 8 is hundredths, 3 is thousandths, 9 is ten-thousandths.
- Round up / down
- Replace a number with the nearest higher or lower "nice" value. Look at the digit just to the right of your target place, 5 or more, round up; less than 5, round down. The rule is the same whether you're rounding 4,587 to the nearest hundred or 7.4839 to the nearest hundredth.
- Estimate
- An approximate answer. Faster than the exact calculation, and good enough most of the time.
- Compatible numbers
- Numbers chosen because they're easy to combine in your head. 27 + 38 becomes 30 + 40 = 70.
- Order of magnitude
- The rough size of a number. Is the answer in the tens? Hundreds? Thousands? This is what your gut catches when something feels "way off."
- Sanity check
- A fast estimate you do after a calculation to confirm the answer isn't absurd. If your calculator says $4,892 for groceries, your sanity check should scream.
- Slope (m)
- A number that describes how steep a line is and which direction it tilts. The letter m is the standard symbol, there's no deep reason for it, it's just tradition.
- Rise
- The vertical change between two points. Up is positive, down is negative.
- Run
- The horizontal change between two points. Right is positive, left is negative.
- Rate of change
- The plain-English version of slope. "60 mph," "$15 per hour," "3 inches per year", all rates of change.
- Δ (delta)
- A Greek letter meaning "change in." So Δy / Δx reads as "change in y over change in x", slope, in two characters.
- Ordered pair
- A point on a graph, written (x, y). Slope needs two of them, that's why the formula has subscripts 1 and 2.
- Linear model
- An equation that describes how one quantity changes at a constant rate as another quantity changes. Always takes the form y = mx + b.
- Slope-intercept form
- The format y = mx + b. It's called this because m is the slope and b is the y-intercept, both ingredients are right there in the equation, no rearranging needed.
- y-intercept (b)
- Where the line crosses the y-axis, the value of y when x = 0. In word problems, it's the starting amount, base fee, or fixed cost.
- Independent variable
- The input. The thing you control or that "drives" the change. Goes on the x-axis.
- Dependent variable
- The output. The thing that depends on the input. Goes on the y-axis.
- Extrapolation
- Using your model to predict beyond your data. Powerful, but also where models go wrong if the underlying pattern changes.
- Cell
- A single box in the grid. Can hold text, a number, or a formula.
- Cell reference
- The address of a cell, like B2. Column letter first, row number second.
- Formula
- Anything that starts with =. Excel calculates the result. Examples: =A1+5, =B2*3, =A2*0.10+25.
- Function (Excel)
- A built-in shortcut, like =SUM(A1:A10), =AVERAGE(B2:B20), =SLOPE(y, x), or =INTERCEPT(y, x). Same framing as the math f(x) from Lesson 1 (give it inputs, get one output), but more specific: each Excel function is a named tool with a fixed behavior. Use them when you'd rather not write the math by hand.
- Range
- A group of cells, written with a colon: A1:A10 means "cells A1 through A10." Use ranges inside functions.
- Fill / drag-down
- Click a cell with a formula, grab the small square at its corner, drag it down. Excel copies the formula to every cell you drag over. adjusting cell references automatically.
- Trendline
- A best-fit line drawn through scattered data points on a chart. Right-click a data series → Add Trendline → choose Linear. Check Display Equation on chart to see y = mx + b printed alongside it. The fastest way to read slope and intercept off real data.
- Chart
- A graph generated from your cells. Select the data, click Insert Chart, choose Scatter or Line. Excel draws it for you.