Math I
Build connected understanding across quantities, equations, functions, coordinate geometry, congruence, and data.
- Problem types
- 659
- Practice variants
- 2,636
Page 8 of 19
Each problem type has four distinct practice variants. Open a preview to move among all four.
Recognize periodic behavior from repeated values
Interpret and sketch key graph/table features in context: intercepts, intervals, maxima/minima, symmetry, end behavior, and periodicity.
Periodicity means an entire output pattern returns after a fixed input shift. We’ll read the values in input order, compare matching blocks, and test whether the same shift preserves every …
Preview problemSketch a graph from verbal features
Interpret and sketch key graph/table features in context: intercepts, intervals, maxima/minima, symmetry, end behavior, and periodicity.
A reliable sketch starts with features that the rule fixes exactly, not with a guessed curve. We’ll use useful forms of the quadratic to locate its zeros, symmetry line, and …
Preview problemMatch graph features to a contextual story
Interpret and sketch key graph/table features in context: intercepts, intervals, maxima/minima, symmetry, end behavior, and periodicity.
Treat the graph as a short story told from left to right. We’ll translate its starting level, steady direction of change, and ending level into what a real quantity would …
Preview problemRead features from a piecewise graph description
Interpret and sketch key graph/table features in context: intercepts, intervals, maxima/minima, symmetry, end behavior, and periodicity.
For a piecewise graph, read each segment on its own before combining the results. We’ll track the horizontal and vertical values contributed by each piece, paying special attention to open …
Preview problemJudge whether a graph feature is meaningful in context
Interpret and sketch key graph/table features in context: intercepts, intervals, maxima/minima, symmetry, end behavior, and periodicity.
A point can belong to an unrestricted mathematical graph without making sense in the situation being modeled. We’ll interpret the input and output coordinates separately, attach their contextual meanings, and …
Preview problemCompare graph features with table evidence
Interpret and sketch key graph/table features in context: intercepts, intervals, maxima/minima, symmetry, end behavior, and periodicity.
Table evidence has to be matched to the scope of the graph claim. We’ll compare the claimed feature’s sampled output with the nearby sampled outputs, which can support a local …
Preview problemFind the domain of a continuous graph
Relate a function's domain to its graph and to the real-world quantities it describes.
Domain is the graph’s horizontal footprint—the input values reached as you scan from left to right. We’ll identify the two horizontal endpoints, check whether the graph fills every input between …
Preview problemIdentify the domain of a discrete graph by listing the x-values
Relate a function's domain to its graph and to the real-world quantities it describes.
For a discrete graph, the domain comes from the horizontal coordinate of each plotted point. We’ll project every point down to the input axis, record each distinct landing value, and …
Preview problemDetermine a contextual domain for counted inputs
Relate a function's domain to its graph and to the real-world quantities it describes.
The formula tells us how an input produces a cost, but the situation tells us which inputs are possible. We’ll interpret the input as a ticket count, apply the capacity …
Preview problemFind the mathematical domain of a rational function
Relate a function's domain to its graph and to the real-world quantities it describes.
A rational expression accepts every real input except those that make a denominator zero. We’ll isolate the denominator, solve the equation that makes it vanish, and verify the restriction by …
Preview problemCompare mathematical domain and contextual domain
Relate a function's domain to its graph and to the real-world quantities it describes.
Mathematical and contextual domains answer different questions. We’ll first inspect the formula alone for algebraic restrictions, then restore the meaning of the input and apply the real-world rules for sign, …
Preview problemInterpret domain endpoints in context
Relate a function's domain to its graph and to the real-world quantities it describes.
A contextual interval describes both a span of inputs and what its boundaries mean. We’ll read the highlighted time segment continuously, inspect each endpoint marker for inclusion, and translate the …
Preview problemMatch interval notation to a graph description
Relate a function's domain to its graph and to the real-world quantities it describes.
Interval notation packages three graph decisions into two delimiters. We’ll decode the left and right symbols separately to determine whether each boundary is included, convert those decisions into strict or …
Preview problemChoose valid input constraints for a situation
Relate a function's domain to its graph and to the real-world quantities it describes.
A valid contextual domain must capture the input type as well as its numerical bounds. We’ll use the fact that tickets are counted to require whole numbers, then interpret the …
Preview problemFind the range over a restricted domain
Relate a function's domain to its graph and to the real-world quantities it describes.
The restricted domain supplies inputs, while the range records the outputs those inputs produce. We’ll use the sign of the linear coefficient to determine which domain endpoint gives the minimum …
Preview problemIdentify an input outside the domain
Relate a function's domain to its graph and to the real-world quantities it describes.
To diagnose a proposed input, test it against the operation that can make the rule undefined. We’ll write the nonzero-denominator condition, substitute the proposed value into that denominator, and interpret …
Preview problemCorrect a domain error in a model statement
Relate a function's domain to its graph and to the real-world quantities it describes.
Correcting a model’s domain starts with what the input measures, not just what the formula can calculate. We’ll test the original statement with fractional and negative ticket counts, then decide …
Preview problemCalculate average rate of change from two points
Calculate, estimate, and interpret average rate of change over an interval.
Average rate of change is the slope of the secant line through the two given points. We’ll compute the output change and input change using the same subtraction order, then …
Preview problemCalculate average rate of change from a table on a given interval
Calculate, estimate, and interpret average rate of change over an interval.
The stated interval tells us exactly which two table rows belong in the average-rate calculation. We’ll pair each endpoint input with its output, ignore rows outside the interval, and form …
Preview problemEstimate average rate of change from graph endpoints
Calculate, estimate, and interpret average rate of change over an interval.
Estimating a rate from a graph begins with reading both endpoint coordinates from the axis scales. We’ll treat those estimates as the endpoints of a secant, calculate approximate rise over …
Preview problemCalculate the average rate of change of a function on an interval
Calculate, estimate, and interpret average rate of change over an interval.
When a function rule is given, the interval endpoints provide the two inputs for a secant slope. We’ll evaluate the rule at each endpoint first, then divide the difference in …
Preview problemInterpret average rate of change with units
Calculate, estimate, and interpret average rate of change over an interval.
An average rate and a total change describe related but different quantities. We’ll find the interval’s elapsed time, read the rate’s sign and output-per-input units, and multiply rate by elapsed …
Preview problemCompare average rates of change over two intervals
Calculate, estimate, and interpret average rate of change over an interval.
The same nonlinear function can have different average rates on different intervals. We’ll evaluate every endpoint needed, build a separate output-change-over-input-change quotient for each interval, and compare the resulting rates …
Preview problemIdentify the interval with the greatest average rate of change from a table
Calculate, estimate, and interpret average rate of change over an interval.
The greatest ending output does not necessarily identify the greatest average rate. We’ll form a rate for each consecutive pair of table rows, using its own output change divided by …
Preview problemDetermine the sign of average rate of change
Calculate, estimate, and interpret average rate of change over an interval.
The sign of an average rate comes from the directed change between the interval’s endpoints. We’ll read the initial and final points from left to right, determine the signs of …
Preview problemUse average rate of change to find a missing endpoint value
Calculate, estimate, and interpret average rate of change over an interval.
A known average rate converts the interval’s input change into an output change. We’ll write the endpoint change quotient with the missing output, substitute the known endpoint value, and solve …
Preview problemMatch an average-rate statement to interval notation
Calculate, estimate, and interpret average rate of change over an interval.
Turning an average-rate statement into interval notation requires two decisions: endpoint order and endpoint inclusion. We’ll place the smaller input on the left, interpret the word “through” to determine whether …
Preview problemGraph a line from slope-intercept form
Graph linear and quadratic functions and show key features such as intercepts, maxima, and minima.
Slope-intercept form gives a graphing plan directly: the constant locates the vertical intercept, and the coefficient gives rise over run. We’ll plot the intercept, use the signed slope to generate …
Preview problemGraph a line using intercepts
Graph linear and quadratic functions and show key features such as intercepts, maxima, and minima.
Intercepts turn a two-variable equation into two easy one-variable checks. We’ll set the other coordinate to zero for each axis crossing, keep the resulting coordinates in the correct order, and …
Preview problemGraph a line through two points
Graph linear and quadratic functions and show key features such as intercepts, maxima, and minima.
Two distinct points determine exactly one line. We’ll plot both coordinates, compute vertical change over horizontal change in a consistent direction, and use one point with that slope to recover …
Preview problemIdentify key features of an unrestricted linear graph
Graph linear and quadratic functions and show key features such as intercepts, maxima, and minima.
A complete line description combines algebraic features with what the arrows mean. We’ll read slope and the vertical intercept from slope-intercept form, solve for the horizontal intercept, and use the …
Preview problemGraph a quadratic from vertex form
Graph linear and quadratic functions and show key features such as intercepts, maxima, and minima.
Vertex form reveals the parabola’s structural anchor before any plotting begins. We’ll decode the vertex and symmetry axis, use the leading coefficient for opening and width, and evaluate inputs equally …
Preview problemGraph a quadratic from intercept form
Graph linear and quadratic functions and show key features such as intercepts, maxima, and minima.
Intercept form gives the zeros immediately, including how the graph behaves at each one. We’ll solve each factor, use multiplicity to decide crossing or touching, and place the symmetry axis …
Preview problemGraph a quadratic from standard form using key features
Graph linear and quadratic functions and show key features such as intercepts, maxima, and minima.
Standard form spreads the key features across several calculations, so we’ll organize them around symmetry. We’ll read the coefficients, use them to find the axis and vertex, then use the …
Preview problemIdentify the maximum or minimum of a quadratic
Graph linear and quadratic functions and show key features such as intercepts, maxima, and minima.
The vertex tells both where an extremum occurs and what its extreme output is, but those are different coordinates. We’ll read the vertex from vertex form, use the leading coefficient …
Preview problemIdentify quadratic x-intercepts and zeros
Graph linear and quadratic functions and show key features such as intercepts, maxima, and minima.
Zeros are input values, while x-intercepts are the corresponding points on the axis. We’ll set the factored output equal to zero, solve each factor with careful attention to its sign, …
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