MAT-144 · Mathematical Reasoning
Topic 01 · Linear Functions
Study card
The formulas, the moves, and the traps for Topic 1, in one printable page.
Key formulas
The moves to memorize. Each one shows up multiple times across the topic.
Function notation
f(x) = the rule applied to input x
To evaluate
f(2), substitute 2 wherever you see x in the rule. f(x) = 3x + 5 means f(2) = 3(2) + 5 = 11. Same input, same output, every time — that's the definition of a function.Slope (rate of change)
m = y2 − y1x2 − x1
=
riserun
Pick any two points on the line. Slope is "how much y changes for every one-unit change in x." Positive m = rising left-to-right; negative m = falling.
Slope-intercept form
y = mx + b
m is the slope (multiplier of x). b is the y-intercept (where the line crosses the y-axis). Plug both in and you've described a line.
Three forms of a linear equation
Slope-intercept: y = mx + b
Point-slope: y − y1 = m(x − x1)
Standard: Ax + By = C
Point-slope: y − y1 = m(x − x1)
Standard: Ax + By = C
Slope-intercept is the easiest to graph. Point-slope is best when you have a slope and one point. Standard is what ALEKS sometimes asks you to convert TO or FROM.
Rounding
digit is 5 or more → round up
digit is 4 or less → round down
digit is 4 or less → round down
Find the rounding place first, then look ONE digit to its right. The digit to the right decides; the rest of the trailing digits don't matter.
Ratio
"A to B" → AB
then reduce by GCF
Order matters. "Green to red" puts green on top, not red. After writing the ratio, reduce by dividing top and bottom by their greatest common factor (6/9 → 2/3).
Common mistakes
The traps that cost real points. Memorize these as much as the formulas.
- Calling a non-function a function. If one input maps to two different outputs (like {(1, 2), (1, 5)}), it's NOT a function. Same x = same y, every time.
- Looking at the wrong digit when rounding. "Round to the tenth" means look at the hundredths digit (the one after tenths). Identify the rounding place first, then look one step right.
- Slope sign mistakes. A line that falls left-to-right has NEGATIVE slope. If your computed slope is positive but the line clearly falls, you swapped (y2−y1) order.
- Rise/run reversed. Rise is the vertical change (Δy). Run is the horizontal change (Δx). If your slope is "1/4" but should be "4," check which one you put on top.
- Confusing slope and y-intercept. In
y = 2x + 5, slope is 2 and y-intercept is 5. The slope is the multiplier of x; the intercept is the constant added on. - Plotting slope wrong from the intercept. Start at the y-intercept, then go UP THE RISE and OVER THE RUN to find a second point. Negative slopes go DOWN and over.
- Ratio in the wrong order. "Green to red" is green/red, not red/green. Read the prompt twice before writing.
- Forgetting to reduce a fraction. 6/9 is correct as a raw ratio but if the prompt says "lowest terms," divide top and bottom by their GCF: 2/3.
- Function evaluation parentheses.
f(2x)is NOT2·f(x). Substitute carefully — wherever you see x in the rule, replace it with the entire input. - Excel: forgot the = sign. Every formula in Excel starts with
=.A2+B2is just text;=A2+B2is the calculation.
Quick reference
The visual + Excel patterns worth keeping at hand.
Special slopes
m = 0 → horizontal line (no rise)
m = undefined → vertical line (no run, division by zero)
m > 0 → line rises left-to-right
m < 0 → line falls left-to-right
m = undefined → vertical line (no run, division by zero)
m > 0 → line rises left-to-right
m < 0 → line falls left-to-right
Reading slope from a graph or table
From a graph: pick two clear lattice points. Count rise (vertical) and run (horizontal). Slope = rise/run, with sign matching the direction.
From a table: for any two rows, slope = (Δy)/(Δx). If x-values are equally spaced, slope = the constant difference in y divided by that x-step.
From a table: for any two rows, slope = (Δy)/(Δx). If x-values are equally spaced, slope = the constant difference in y divided by that x-step.
Function vocabulary
Function — the rule, named with a letter (f, g, h)
Input — the value you feed in (typically x)
Output — what comes out (typically y or f(x))
Evaluate — plug a specific value in for x
Domain — the set of valid inputs
Range — the set of resulting outputs
Input — the value you feed in (typically x)
Output — what comes out (typically y or f(x))
Evaluate — plug a specific value in for x
Domain — the set of valid inputs
Range — the set of resulting outputs
Excel basics
=A2+B2 sum two cells
=SUM(A2:A10) sum a range
=AVERAGE(B2:B10) average a range
=SLOPE(y_range, x_range) best-fit slope
=INTERCEPT(y_range, x_range) best-fit y-intercept
$A$1 absolute reference (won't shift on copy)
=SUM(A2:A10) sum a range
=AVERAGE(B2:B10) average a range
=SLOPE(y_range, x_range) best-fit slope
=INTERCEPT(y_range, x_range) best-fit y-intercept
$A$1 absolute reference (won't shift on copy)
Two ways to reason
Deductive — apply a known rule to get a guaranteed answer (the slope formula, plugging into a function)
Inductive — spot a pattern from examples and predict (figuring out a function rule from a table of values)
Inductive — spot a pattern from examples and predict (figuring out a function rule from a table of values)