DSST Math for Liberal Arts: Algebra, Graphs, and Functions in Real-Life Applications

Expect to translate a word situation into an expression, equation, function, or graph, then interpret the slope, intercepts, and domain in context.

A variable names an unknown quantity. An expression has no equals sign; an equation states that two expressions are equal.
A function assigns exactly one output to each input; f(3) means substitute 3, while solving f(x) = 3 means find the input that produces output 3.
A linear function y = mx + b has slope m, the constant rate of change, and intercept b, the starting value. Slope from two points is (y₂ − y₁)/(x₂ − x₁), or rise over run.
Direct variation y = kx is linear with b = 0; k is the unit rate.
Domain is the set of allowed inputs. In applications it is restricted by the situation: time cannot be negative, quantities are often whole numbers, volume cannot exceed capacity.
A quadratic y = ax² + bx + c opens upward when a > 0; its vertex is the maximum or minimum, which is why quadratics model revenue, area, and projectile height.
Graph reading: identify increasing and decreasing intervals, intercepts, and the input where one model overtakes another.

A 500-liter tank drains at a steady 12 L per minute. Model the volume: V(t) = 500 − 12t.

Starting volume: V(0) = 500 L, the y-intercept.
After 20 minutes: V(20) = 500 − 240 = 260 L.
Empty when 500 − 12t = 0, so t = 500/12 ≈ 41.7 minutes.
The slope is −12: the volume drops 12 L each minute, and the negative sign carries that meaning.
The practical domain is 0 ≤ t ≤ 41.7, because negative time is meaningless and beyond 41.7 the model predicts a negative volume.

Confusing f(x) = 3 (an output) with x = 3 (an input).
Dropping the sign or units from a slope, then misreading the answer.
Swapping the intercepts: the y-intercept is usually the starting value, while the x-intercept is the zero or break-even point.
Extending a linear model past the point where the real situation changes.
Ignoring axis scales when reading a graph.