Math II
Connect real and complex numbers, polynomials, quadratics, proof, circles, trigonometry, probability, and modeling.
- Problem types
- 786
- Practice variants
- 3,144
Page 8 of 22
Each problem type has four distinct practice variants. Open a preview to move among all four.
Write an absolute-value equation from a vertex, opening direction, and arm slope
Graph square-root, cube-root, absolute-value, step, and piecewise-defined functions, showing key features by hand in simple cases and with technology when the functions are more complicated.
Absolute-value vertex form assigns a separate job to each parameter. We’ll use the vertex for the horizontal and vertical shifts, use arm-slope magnitude and opening direction for the signed scale, …
Preview problemWrite a square-root equation from an endpoint, direction, and scale
Graph square-root, cube-root, absolute-value, step, and piecewise-defined functions, showing key features by hand in simple cases and with technology when the functions are more complicated.
A square-root equation can be reconstructed from its endpoint, direction, and scale. We’ll place the endpoint into transformed-root form, use the branch direction to fix the sign of the scale, …
Preview problemEvaluate a piecewise-defined function at a given x-value
Graph square-root, cube-root, absolute-value, step, and piecewise-defined functions, showing key features by hand in simple cases and with technology when the functions are more complicated.
Evaluating a piecewise function begins with interval membership, not substitution. We’ll test the target input against the conditions, use the inclusive or strict boundary symbol to identify exactly one rule, …
Preview problemClassify step versus piecewise-linear behavior from evidence
Graph square-root, cube-root, absolute-value, step, and piecewise-defined functions, showing key features by hand in simple cases and with technology when the functions are more complicated.
A function family is determined by how the output behaves within each interval, not merely by the number of pieces. We’ll check whether each band is constant or changing, look …
Preview problemTranslate a context into an exact piecewise graph record
Graph square-root, cube-root, absolute-value, step, and piecewise-defined functions, showing key features by hand in simple cases and with technology when the functions are more complicated.
A contextual piecewise graph starts by naming the input and output with units. We’ll translate each fee band into a rule and interval, turn threshold wording into open or closed …
Preview problemIdentify a parent family and domain from invariant shape
Graph square-root, cube-root, absolute-value, step, and piecewise-defined functions, showing key features by hand in simple cases and with technology when the functions are more complicated.
Parent families have invariant shape clues that survive changes of scale. We’ll look first for one-sided versus two-sided extent, then distinguish a curved endpoint branch from a V, an S-curve, …
Preview problemIdentify the zeros of a quadratic from factored form
Use factoring and completing the square to reveal zeros, extrema, and symmetry of quadratics.
Factored form exposes zeros through the zero-product property. We’ll set the product equal to zero, solve each factor equation separately with careful sign handling, and substitute both inputs to confirm …
Preview problemIdentify the axis of symmetry of a quadratic from factored form
Use factoring and completing the square to reveal zeros, extrema, and symmetry of quadratics.
A quadratic’s factored form locates two symmetric x-intercepts, and their midpoint gives the vertical symmetry axis. We’ll solve the factors for the roots, average those inputs, and use equal horizontal …
Preview problemFind the vertex of a quadratic from factored form
Use factoring and completing the square to reveal zeros, extrema, and symmetry of quadratics.
Factored form locates a quadratic’s vertex through symmetry without requiring expansion. We’ll read the roots, average them for the axis input, evaluate the original product there, and use the leading …
Preview problemIdentify the vertex from completed-square form of a quadratic
Use factoring and completing the square to reveal zeros, extrema, and symmetry of quadratics.
Completed-square form makes the vertex a parameter-reading problem. We’ll match the rule to a times the square of x minus h, plus k, reverse the sign inside the binomial, and …
Preview problemRead a quadratic extremum into separate fields
Use factoring and completing the square to reveal zeros, extrema, and symmetry of quadratics.
An extremum record separates three related facts: its type, the input where it occurs, and the output attained there. We’ll read the vertex from completed-square form, use the scale factor’s …
Preview problemFind the y-intercept of a quadratic from its equation
Use factoring and completing the square to reveal zeros, extrema, and symmetry of quadratics.
An intercept is found by fixing the coordinate that belongs to its axis. For the vertical-axis crossing, we’ll set the input to zero, simplify every term carefully, write the result …
Preview problemChoose the equivalent quadratic form that best reveals zeros
Use factoring and completing the square to reveal zeros, extrema, and symmetry of quadratics.
Equivalent quadratic forms are useful because each exposes a different feature. We’ll match the requested zeros to a product of linear factors, build that product from a signed pair with …
Preview problemFactor a quadratic and solve its zero factors
Use factoring and completing the square to reveal zeros, extrema, and symmetry of quadratics.
Factoring and solving are two distinct stages joined by the zero-product property. We’ll first match a signed integer pair to the constant product and linear sum, verify the factorization by …
Preview problemComplete the square and identify the extremum
Use factoring and completing the square to reveal zeros, extrema, and symmetry of quadratics.
Completing the square trades standard form for a form that displays symmetry and the extremum. We’ll square half the linear coefficient, add and subtract that same amount to preserve equivalence, …
Preview problemMatch quadratic forms to directly visible features
Use factoring and completing the square to reveal zeros, extrema, and symmetry of quadratics.
One parabola can carry several algebraic descriptions, each acting like a different lens. We’ll verify the forms are equivalent, then read the vertical intercept and opening from standard form, the …
Preview problemDerive an exact quadratic sketch fingerprint
Use factoring and completing the square to reveal zeros, extrema, and symmetry of quadratics.
An exact sketch fingerprint should derive enough checked landmarks to determine one parabola unambiguously. We’ll combine roots and their midpoint with the vertex, opening, vertical intercept, and a reflected point …
Preview problemFind a missing parameter in a quadratic from its structured form and a given feature
Use factoring and completing the square to reveal zeros, extrema, and symmetry of quadratics.
A stated graph feature becomes an equation for the missing parameter. We’ll translate the named intercept into its fixed input-output condition, substitute that point into the structured quadratic, solve the …
Preview problemDecide quadratic equivalence by coefficient comparison
Use factoring and completing the square to reveal zeros, extrema, and symmetry of quadratics.
Polynomial equivalence is an all-input claim, so one matching test value is not enough. We’ll rewrite both quadratics in the same standard form, compare the coefficients of every power of …
Preview problemInterpret the base of an exponential growth model as a percent increase
Use exponent properties to interpret exponential expressions, including percent growth and decay.
An exponential growth base combines the retained whole with the added fraction. We’ll identify the repeated multiplier, subtract one to isolate only the gain, convert that decimal to percent, and …
Preview problemConvert an exponential decay base to a percent decrease
Use exponent properties to interpret exponential expressions, including percent growth and decay.
A decay base tells us the fraction retained, not the fraction lost. We’ll subtract the repeated multiplier from one to find the loss, convert that difference to percent, and keep …
Preview problemRewrite an exponential model so the input uses a different time unit
Use exponent properties to interpret exponential expressions, including percent growth and decay.
Changing an exponential model’s time unit changes how many old intervals the exponent counts, while preserving the same function. We’ll convert the new time variable into old intervals, substitute that …
Preview problemRewrite an exponential expression with a shifted exponent to show the initial value at t = 0
Use exponent properties to interpret exponential expressions, including percent growth and decay.
A shift in the exponent hides a constant power of the base inside the expression. We’ll split the exponent sum into variable and constant powers, combine the constant power with …
Preview problemCompare two exponential expressions for equivalence by rewriting one with exponent rules
Use exponent properties to interpret exponential expressions, including percent growth and decay.
Exponential equivalence becomes visible after both expressions share the same variable power. We’ll split the shifted exponent with the product rule, evaluate and combine the constant factors, and compare the …
Preview problemInterpret doubling time or half-life from an exponential function
Use exponent properties to interpret exponential expressions, including percent growth and decay.
Doubling time and half-life connect a special multiplier to the input interval that applies it. We’ll identify the factor produced by a one-unit increase in the exponent, interpret whether it …
Preview problemFind the cumulative percent change from an exponential multiplier over several periods
Use exponent properties to interpret exponential expressions, including percent growth and decay.
Repeated percent changes compound through multipliers rather than by adding the period rates. We’ll raise the one-period factor to the number of periods, subtract the original whole from the cumulative …
Preview problemFind the single-period percent change from a multi-period exponential factor
Use exponent properties to interpret exponential expressions, including percent growth and decay.
Recovering one period from a multi-period factor reverses compounding. We’ll model the total factor as a power of the unknown period multiplier, take the applicable positive root, convert the recovered …
Preview problemInterpret exponential-model parameters with units
Use exponent properties to interpret exponential expressions, including percent growth and decay.
Interpreting an exponential model means attaching quantities, units, and intervals to every parameter. We’ll evaluate at zero for the initial output, interpret the unitless base per one input interval, test …
Preview problemChoose the equivalent exponential form that makes the requested feature easiest to read
Use exponent properties to interpret exponential expressions, including percent growth and decay.
Equivalent exponential forms can reveal different features of the same function. We’ll rewrite the variable exponent to be exactly the input, absorb the fixed power into the coefficient, and verify …
Preview problemValidate an exponential rewrite with a rule and check
Use exponent properties to interpret exponential expressions, including percent growth and decay.
A valid exponential rewrite needs a rule that proves equality for every input, not just a matching example. We’ll identify the exponent operation, apply the corresponding identity symbolically, simplify the …
Preview problemCompare two quadratic vertex records
Compare function properties across representations, especially quadratic comparisons.
A vertex comparison has separate horizontal, vertical, and extremum-type dimensions. We’ll read each vertex from its form, use the squared-term sign to classify its extremum, compare both coordinates, and avoid …
Preview problemCompare two quadratic real-zero sets
Compare function properties across representations, especially quadratic comparisons.
Comparing zeros means comparing their values, multiplicities, and set relationship rather than only their counts. We’ll solve every factor equation, preserve repeats in multisets, form the distinct-zero sets, and then …
Preview problemCompare two y-intercepts at x=0
Compare function properties across representations, especially quadratic comparisons.
A y-intercept comparison holds the input fixed at zero for both functions. We’ll evaluate each complete rule at that shared input, especially any grouped expression, and compare the resulting outputs …
Preview problemCompare two quadratic axes of symmetry
Compare function properties across representations, especially quadratic comparisons.
Axes of symmetry can be extracted differently from different quadratic forms but compared as the same kind of vertical line. We’ll average factored-form roots, read the vertex-form center with the …
Preview problemCompare constant linear rate with changing quadratic rate
Compare function properties across representations, especially quadratic comparisons.
Linear and quadratic rates must be compared over equal input steps and at the same level of change. We’ll identify the line’s constant first difference, derive the quadratic’s input-dependent first …
Preview problemCompare two function values at the same input across representations
Compare function properties across representations, especially quadratic comparisons.
A pointwise comparison is fair only when both functions use the same allowable input. We’ll verify domain membership, evaluate each complete representation at that input, and compare the outputs and …
Preview problem