Math III
Go further with polynomial and rational expressions, advanced functions, trigonometry, geometric modeling, and statistical inference.
- Problem types
- 641
- Practice variants
- 2,564
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Each problem type has four distinct practice variants. Open a preview to move among all four.
Interpret a graph feature using input, output, units, and event meaning
Create equations in two or more variables, graph them, and interpret relationships with labels and scales.
Interpret a graph point in the order the axes define it: horizontal input first, vertical output second, with units attached to both. An x-intercept is not automatically a maximum or …
Preview problemSeparate algebraic and contextual domain and range restrictions
Create equations in two or more variables, graph them, and interpret relationships with labels and scales.
Keep algebraic restrictions separate from contextual ones. The reciprocal formula first excludes values that make division impossible and outputs the formula can never reach; only afterward does the travel setting …
Preview problemFind and interpret every intersection of competing model graphs
Create equations in two or more variables, graph them, and interpret relationships with labels and scales.
An intersection means both models produce the same cost at the same billing period, so set their outputs equal and search the entire nonnegative domain. Because the number of crossings …
Preview problemWrite a unique simplest equation from specified graph features and scale
Create equations in two or more variables, graph them, and interpret relationships with labels and scales.
Translate each zero into a factor using x minus the zero, paying close attention to the sign. Simple zeros need one copy of each factor, which also gives the least …
Preview problemWrite an equation that matches advanced table values
Create equations in two or more variables, graph them, and interpret relationships with labels and scales.
Test how the outputs change when the input increases by one. A stable ratio points to repeated multiplication, while the output at input zero supplies the front coefficient because any …
Preview problemInterpret a contextual asymptote limit or radical endpoint
Create equations in two or more variables, graph them, and interpret relationships with labels and scales.
A vertical asymptote is an excluded input boundary, not a point the model reaches. Because speed is positive, examine the reciprocal only as speed approaches the boundary from the right. …
Preview problemSolve a fixed-quantity formula and select its feasible graph features
Create equations in two or more variables, graph them, and interpret relationships with labels and scales.
Substitute the fixed volume and isolate the variable assigned to the vertical axis before thinking about the graph. Positive cylinder dimensions retain only the first-quadrant branch. Then use the squared …
Preview problemWrite a polynomial constraint for a design or revenue context
Represent constraints and systems, then interpret viable and non-viable solutions in modeling contexts.
Keep the two design dimensions visible as factors, since their product is the modeled area. Translate “at least” with an inclusive lower bound, then separately require each dimension to stay …
Preview problemWrite a rational constraint for a rate or resource context
Represent constraints and systems, then interpret viable and non-viable solutions in modeling contexts.
Preserve the average-cost expression exactly, including which term is divided by the resource quantity. The phrase “at most” creates an inclusive upper bound. State the positive resource domain before manipulating …
Preview problemWrite a radical constraint for a physical or geometry context
Represent constraints and systems, then interpret viable and non-viable solutions in modeling contexts.
Keep the modeled distance in radical form and translate “at most” as an inclusive upper bound. Before adding a domain condition, inspect the radicand itself: a square is never negative, …
Preview problemWrite an exponential or logarithmic constraint from a growth context
Represent constraints and systems, then interpret viable and non-viable solutions in modeling contexts.
Use the supplied growth model as the population expression, then translate the target wording separately. “Must exceed” is strict, so the boundary value itself does not satisfy the requirement. Pair …
Preview problemJudge whether a candidate solution is viable under a model and its domain
Represent constraints and systems, then interpret viable and non-viable solutions in modeling contexts.
A candidate must belong to the original domain before it can be tested as a solution. Inspect the denominator first and substitute the candidate there; if the expression becomes undefined, …
Preview problemInterpret a modeling boundary with equality, inclusion, and feasible side
Represent constraints and systems, then interpret viable and non-viable solutions in modeling contexts.
A boundary pairs the input where the cost reaches the budget with that exact output and its units. The less-than-or-equal condition includes equality, while the increasing model determines that smaller …
Preview problemWrite a system for a model and a target output condition
Represent constraints and systems, then interpret viable and non-viable solutions in modeling contexts.
Keep a system as two separate relationships: one equation describes the changing cost, and the other describes the fixed output threshold. Their common point would satisfy both at once, but …
Preview problemFind the feasible solution set of a one-variable constraint inequality
Represent constraints and systems, then interpret viable and non-viable solutions in modeling contexts.
The zeros of the factors are sign-change boundaries, so use them to split the number line into intervals. Test the factor signs on each interval rather than assuming the region …
Preview problemAudit an algebraic candidate for equation and context viability
Represent constraints and systems, then interpret viable and non-viable solutions in modeling contexts.
A solution of a transformed equation is only a candidate for the original one. Recover the original denominator restriction and test the candidate there before doing anything else. If denominator …
Preview problemChoose the optimal feasible candidate after checking all constraints
Represent constraints and systems, then interpret viable and non-viable solutions in modeling contexts.
Feasibility and optimization are separate filters. Once every listed design is confirmed viable, compare them only on the stated objective—in this case, minimizing a common dollar cost. Order the costs …
Preview problemIdentify missing constraints that make a model feasible
Represent constraints and systems, then interpret viable and non-viable solutions in modeling contexts.
Read every physical dimension from the cut-and-fold diagram before forming the constraint. The cut size, the shortened width, and the shortened length must all be positive, with each corner cut …
Preview problemRepresent an entire contextual feasible set with correct domain type
Represent constraints and systems, then interpret viable and non-viable solutions in modeling contexts.
Represent the entire set, not just its two boundary values. Non-strict inequalities include both endpoints, and the word “continuous” means every real length between them is permitted—not merely whole-meter measurements. …
Preview problemClassify why a constraint-set intersection is empty
Represent constraints and systems, then interpret viable and non-viable solutions in modeling contexts.
Separate the radical's input domain from its output range. A real principal square root accepts only inputs with a nonnegative radicand and returns only nonnegative outputs. Compare that actual output …
Preview problemRearrange a formula to isolate a linear target variable in one term
Rearrange formulas to highlight a chosen quantity across the expression types studied.
Identify the operation directly attached to the target variable and undo it with the same operation on both sides. Here the target is one factor in a product, so divide …
Preview problemIsolate a powered variable with explicit real branches and conditions
Rearrange formulas to highlight a chosen quantity across the expression types studied.
Undo an even power with a square root only after recognizing its real-domain condition. Algebraically, a squared variable can produce two opposite branches, but the variable's meaning may narrow them. …
Preview problemRearrange a radical formula with branches and real conditions
Rearrange formulas to highlight a chosen quantity across the expression types studied.
Record the principal square root's nonnegative output condition before reversing the formula. Squaring then removes the radical, and a final subtraction isolates the original input. No plus-or-minus branch belongs on …
Preview problemRearrange a rational formula to isolate a variable in the denominator
Rearrange formulas to highlight a chosen quantity across the expression types studied.
Start by recording that the original denominator variable cannot be zero. Clear that denominator to expose a product, then divide by the other factor to isolate the target. The new …
Preview problemRearrange a formula when the target variable is mixed across terms
Rearrange formulas to highlight a chosen quantity across the expression types studied.
When the target appears in several terms, do not try to undo each term separately. Factor the target first so the entire remaining sum becomes one multiplier, then divide by …
Preview problemIsolate an exponential quantity under positive-base real conditions
Rearrange formulas to highlight a chosen quantity across the expression types studied.
Treat the exponential quantity as one multiplier attached to the target. A positive base raised to a real exponent stays positive, so that multiplier is nonzero and division by it …
Preview problemRearrange a logarithmic formula to isolate a chosen quantity
Rearrange formulas to highlight a chosen quantity across the expression types studied.
Isolate the logarithm first, keeping careful track of the negative sign outside it. Then use the inverse relationship between a common logarithm and a power of ten, so the isolated …
Preview problemIsolate a triangle side or principal angle with valid trig conditions
Rearrange formulas to highlight a chosen quantity across the expression types studied.
In the sine ratio, the requested side is already the numerator, so clear the hypotenuse denominator by multiplication. The triangle conditions matter after the rearrangement: an acute angle has positive …
Preview problemIdentify restrictions introduced or exposed by a rearranged formula
Rearrange formulas to highlight a chosen quantity across the expression types studied.
Look at the rearranged formula for any operation that can fail. Here division is by an entire sum, so that sum—not the numerator, output, or each addend separately—must be nonzero. …
Preview problemDecide whether two formula rearrangements are equivalent
Rearrange formulas to highlight a chosen quantity across the expression types studied.
Test equivalence by transforming one formula into the other with a reversible operation on both sides. The subtraction occurs outside the power, so adding the same constant to both sides …
Preview problemUse a rearranged formula to compute a requested value
Rearrange formulas to highlight a chosen quantity across the expression types studied.
Rearrange the area formula symbolically before substituting measurements. Isolating the missing side makes the required operation clear: divide the known area by the known nonzero width. After evaluating, multiply the …
Preview problemInterpret an isolated formula by output, known inputs, conditions, and units
Rearrange formulas to highlight a chosen quantity across the expression types studied.
Interpret an isolated formula by separating roles: the variable alone is the output, and the remaining quantities are known inputs. Carry their units through the operation—area units divided by width …
Preview problemDecide which variable to isolate from a formula based on the question
Rearrange formulas to highlight a chosen quantity across the expression types studied.
Let the question determine the target before doing any algebra. Match the quantity being requested to its symbol, then plan inverse operations that leave that symbol alone. This keeps a …
Preview problemDecide whether a target variable can be isolated by elementary algebra, and isolate it when possible
Rearrange formulas to highlight a chosen quantity across the expression types studied.
First inspect every place the unknown appears and classify how it is used. We’ll test what happens when terms are moved and when an inverse function is applied, checking after …
Preview problemEstimate solutions by reading intersections of a polynomial graph and a line
Solve f(x)=g(x) approximately using intersections of polynomial, rational, radical, absolute-value, exponential, and logarithmic graphs.
Treat the equation as a search for inputs where the two graphs share an output. Scan the entire displayed window for every crossing, then project each crossing vertically to the …
Preview problemRead approximate solutions to f(x)=g(x) from graph intersections
Solve f(x)=g(x) approximately using intersections of polynomial, rational, radical, absolute-value, exponential, and logarithmic graphs.
A rational graph may be split into separate branches by an excluded input, so search each branch independently for crossings with the other graph. Treat the break or asymptote as …
Preview problem