Math II
Connect real and complex numbers, polynomials, quadratics, proof, circles, trigonometry, probability, and modeling.
- Problem types
- 786
- Practice variants
- 3,144
Page 9 of 22
Each problem type has four distinct practice variants. Open a preview to move among all four.
Compare the average rate of change of two functions on the same interval
Compare function properties across representations, especially quadratic comparisons.
Average rates are secant slopes, so both functions must use identical interval endpoints. We’ll find each pair of endpoint outputs, divide the signed output change by the common input change, …
Preview problemCompare two domain and range set records
Compare function properties across representations, especially quadratic comparisons.
Domain and range comparisons require four independent set records before any relation is named. We’ll project each representation onto its input and output axes, preserve endpoint inclusion, then compare domains …
Preview problemDecide whether an equation, graph, and table represent the same quadratic by comparing extrema and other key features
Compare function properties across representations, especially quadratic comparisons.
Agreement across an equation, graph, and table requires matching structure as well as a few values. We’ll derive the zeros, symmetry input, and vertex height from the equation, then test …
Preview problemChoose which function better fits a maximize-or-minimize goal by comparing extrema
Compare function properties across representations, especially quadratic comparisons.
A goal-based comparison begins by identifying which coordinate measures the objective. We’ll confirm that each vertex represents a maximum, compare the vertex output values for revenue, and keep the input …
Preview problemEvaluate a function-comparison claim
Compare function properties across representations, especially quadratic comparisons.
A comparison claim is supported only by evidence about the exact feature it names. We’ll use opening direction to interpret each vertex output as a maximum, compare those heights on …
Preview problemCompare linear and exponential table growth precisely
Compare growth of exponential, linear, quadratic, and other polynomial models using graphs/tables.
Growth patterns reveal themselves through different invariants: constant differences for linear models and constant ratios for exponential models. We’ll identify both patterns, compare corresponding rows for leads or ties, and …
Preview problemCompare quadratic and exponential table growth
Compare growth of exponential, linear, quadratic, and other polynomial models using graphs/tables.
Quadratic and exponential tables can cross more than once, so early leadership does not settle long-run growth. We’ll use first and second differences to identify the quadratic, ratios to identify …
Preview problemCompare exponential and polynomial growth from a graph
Compare growth of exponential, linear, quadratic, and other polynomial models using graphs/tables.
Long-run growth should be judged from later common inputs and the mechanisms that generate each model, not one early lead. We’ll verify an ordering reversal with exact values, measure the …
Preview problemIdentify and verify the first overtaking row
Compare growth of exponential, linear, quadratic, and other polynomial models using graphs/tables.
The word first turns an overtake into an ordered-search problem. We’ll scan same-input comparisons from left to right, find the first strict reversal, and verify the immediately preceding row still …
Preview problemSelect a usable window around a computed crossover
Compare growth of exponential, linear, quadratic, and other polynomial models using graphs/tables.
A useful crossover window must include evidence on both sides of the lead change while keeping that evidence visually separated. We’ll compute the bracketing inputs and outputs, test each window …
Preview problemUse model comparison to answer a decision context
Compare growth of exponential, linear, quadratic, and other polynomial models using graphs/tables.
A decision context fixes the comparison horizon and units before model type matters. We’ll translate each plan into its correct linear or exponential expression, evaluate both at the same time, …
Preview problemCompare two exact average rates of change
Compare growth of exponential, linear, quadratic, and other polynomial models using graphs/tables.
Exact average-rate comparison uses the same ordered interval and divides each output change by its width. We’ll evaluate both endpoint pairs, preserve the fractional rate instead of rounding, and subtract …
Preview problemCompare repeated multiplication with power growth
Compare growth of exponential, linear, quadratic, and other polynomial models using graphs/tables.
Shared outputs do not make two growth patterns the same, because classification depends on how each row changes. We’ll compare consecutive ratios with first and second differences, then connect those …
Preview problemSeparate initial value from growth behavior
Compare growth of exponential, linear, quadratic, and other polynomial models using graphs/tables.
A larger output can come from a larger starting value without implying a faster growth rate. We’ll evaluate both models at zero, interpret their shared multiplier, and simplify a same-input …
Preview problemAudit a growth claim outside its graph window
Compare growth of exponential, linear, quadratic, and other polynomial models using graphs/tables.
A graph window supports claims only over the inputs it actually displays. We’ll verify the ordering at its boundary, extend both formulas to the same later input, and use any …
Preview problemIdentify the initial height from a projectile function
Apply quadratic functions to physical situations such as projectile motion under gravity.
Initial height means the model’s output at the launch instant, so the decisive input is zero. We’ll substitute that input, observe which terms vanish, and distinguish the remaining position value …
Preview problemIdentify the initial velocity in a projectile function
Apply quadratic functions to physical situations such as projectile motion under gravity.
In a standard projectile model, each coefficient has a physical role rather than merely being a number to compute with. We’ll match the linear time coefficient to initial vertical velocity, …
Preview problemFind the maximum height of a projectile from its quadratic model
Apply quadratic functions to physical situations such as projectile motion under gravity.
A negative leading coefficient makes the projectile’s vertex a peak, but the vertex has both a time coordinate and a height coordinate. We’ll locate the peak time, evaluate the model …
Preview problemFind when a projectile reaches its maximum height from a quadratic model
Apply quadratic functions to physical situations such as projectile motion under gravity.
Because the question asks when the maximum occurs, the needed quantity is the vertex input rather than its output. We’ll identify the quadratic coefficients, apply the axis formula with careful …
Preview problemFind when a projectile reaches the ground
Apply quadratic functions to physical situations such as projectile motion under gravity.
Reaching the ground translates to an output condition, so we begin by setting the height equal to zero. We’ll solve for every intercept time, then use the flight context to …
Preview problemFind when a projectile reaches a target height
Apply quadratic functions to physical situations such as projectile motion under gravity.
A target height becomes an equation between the projectile model and that output level. We’ll solve the resulting quadratic, verify every root in the physical flight interval, and interpret the …
Preview problemInterpret equal-height projectile times by symmetry
Apply quadratic functions to physical situations such as projectile motion under gravity.
Equal outputs on a projectile parabola come from inputs equally spaced around its symmetry axis. We’ll verify the shared height, average the two times to recover the peak time, and …
Preview problemDetermine the meaningful domain for a projectile model
Apply quadratic functions to physical situations such as projectile motion under gravity.
A quadratic accepts every real input algebraically, but a projectile’s meaningful time interval is bounded by physical events. We’ll solve for the ground contacts, identify which one is launch and …
Preview problemDetermine the meaningful range for a projectile model over a given domain
Apply quadratic functions to physical situations such as projectile motion under gravity.
The meaningful range is built from height outputs over the stated flight-time domain, not from the time endpoints themselves. We’ll evaluate both endpoints and the interior vertex, identify the attained …
Preview problemBuild a complete projectile sketch record
Apply quadratic functions to physical situations such as projectile motion under gravity.
A reliable projectile sketch begins with a complete feature record rather than a guessed arc. We’ll find the ground contacts, use their midpoint and a function value for the vertex, …
Preview problemEvaluate a projectile design requirement
Apply quadratic functions to physical situations such as projectile motion under gravity.
A design requirement becomes a precise comparison at a specified physical time. We’ll confirm that time lies within the flight, evaluate the height there, test the required strict inequality, and …
Preview problemRead named projectile features from a graph or table
Apply quadratic functions to physical situations such as projectile motion under gravity.
A key-time table encodes the entire physical trajectory when each column is read as a time-height pair. We’ll identify the endpoint and peak rows, use equal-height symmetry to locate the …
Preview problemUse the Pythagorean identity to find sine when cosine and the quadrant are known
Prove and use sin^2(theta)+cos^2(theta)=1 to find trig values with quadrant information.
The Pythagorean identity determines the missing trig magnitude, while the quadrant determines its sign. We’ll substitute the known cosine, isolate sine squared, take both algebraic square roots, and then use …
Preview problemUse the Pythagorean identity to find cosine when sine and the quadrant are known
Prove and use sin^2(theta)+cos^2(theta)=1 to find trig values with quadrant information.
Finding a missing cosine has two stages: calculate its magnitude from the identity, then assign its sign from location. We’ll substitute the known sine, isolate cosine squared, take the square …
Preview problemDetermine the signs of sine and cosine when the quadrant is known
Prove and use sin^2(theta)+cos^2(theta)=1 to find trig values with quadrant information.
Trig signs come directly from coordinates on the unit circle: cosine follows the horizontal coordinate and sine follows the vertical coordinate. We’ll locate the quadrant relative to both axes and …
Preview problemVerify whether a sine value and cosine value can belong to the same angle
Prove and use sin^2(theta)+cos^2(theta)=1 to find trig values with quadrant information.
A proposed sine-cosine pair belongs to one angle only if it lies on the unit circle. We’ll square both values exactly, add them under the Pythagorean identity, and use whether …
Preview problemFind an exact trigonometric value using the identity when one trig value and the quadrant are known
Prove and use sin^2(theta)+cos^2(theta)=1 to find trig values with quadrant information.
Exact trig work keeps the square-root magnitude separate from the quadrant sign. We’ll use the identity to find the missing square, simplify the radical without decimal approximation, and then use …
Preview problemFind a missing unit-circle coordinate using the Pythagorean identity and quadrant information
Prove and use sin^2(theta)+cos^2(theta)=1 to find trig values with quadrant information.
A point lies on the unit circle only when its squared coordinates add to one, which determines a missing coordinate’s magnitude. We’ll substitute the known coordinate, take both square roots, …
Preview problemSimplify a trigonometric expression using the Pythagorean identity
Prove and use sin^2(theta)+cos^2(theta)=1 to find trig values with quadrant information.
Identity simplification starts by matching the entire expression’s structure, not by manipulating its terms separately. We’ll recognize the complete sum of the two squared trig functions and replace that whole …
Preview problemDetermine whether given sine and cosine values are possible for the same angle
Prove and use sin^2(theta)+cos^2(theta)=1 to find trig values with quadrant information.
A proposed sine-cosine pair must pass both a magnitude test and any sign constraints. We’ll confirm each value is in range, add their squares rather than the values themselves, and …
Preview problemRead separate sine and cosine values from a unit-circle point
Prove and use sin^2(theta)+cos^2(theta)=1 to find trig values with quadrant information.
Unit-circle coordinates have a fixed trig order: the horizontal coordinate is cosine and the vertical coordinate is sine. We’ll read the two values without swapping them, check their signs against …
Preview problem