Math II
Connect real and complex numbers, polynomials, quadratics, proof, circles, trigonometry, probability, and modeling.
- Problem types
- 786
- Practice variants
- 3,144
Page 5 of 22
Each problem type has four distinct practice variants. Open a preview to move among all four.
Rewrite an exponential decay expression in negative-exponent form using \((b^k)^t=b^{kt}\)
Use exponent properties to transform exponential expressions and interpret growth/decay rates.
A reciprocal decay base can be expressed as a positive base with a negative exponent. We’ll rewrite the reciprocal using a negative-first power, apply the power-of-a-power rule without losing the …
Preview problemRewrite an exponential expression to reveal the initial value at input 0
Use exponent properties to transform exponential expressions and interpret growth/decay rates.
A constant shift in the exponent contributes a constant power factor that can be absorbed into the coefficient. We’ll split the exponent sum, evaluate that fixed power, combine only the …
Preview problemCompare exponential models across different time units using \((b^k)^t = b^{kt}\)
Use exponent properties to transform exponential expressions and interpret growth/decay rates.
Models using different time units must be compared at the same elapsed time, not at equal numerical inputs. We’ll relate the month and year counts, substitute the matched count into …
Preview problemEvaluate using exponent properties
Use exponent properties to transform exponential expressions and interpret growth/decay rates.
The outermost operation is multiplication of powers with the same base, so the product rule adds their exponents. We’ll identify the common base, combine the exponent counts before evaluating, compute …
Preview problemDetermine whether two exponential expressions are equivalent by rewriting with exponent rules
Use exponent properties to transform exponential expressions and interpret growth/decay rates.
Universal equivalence is established by rewriting both expressions into the same algebraic form, not by checking one input alone. We’ll split the exponent sum into a product of powers, evaluate …
Preview problemFind the multiplicative factor over a stated time interval from an exponential model
Use exponent properties to transform exponential expressions and interpret growth/decay rates.
The one-year multiplier must include every base interval contained in a year. We’ll interpret the exponent’s interval count, compare the model at the start and after one year, and divide …
Preview problemWrite a projectile height function from initial height and initial velocity
Build quadratic and exponential functions from context using explicit formulas, recursive processes, or calculation steps.
The projectile template assigns a distinct physical role to each term. We’ll keep gravity’s quadratic coefficient negative, use the upward initial velocity as the signed linear coefficient, place the launch …
Preview problemWrite a quadratic area function from variable side lengths
Build quadratic and exponential functions from context using explicit formulas, recursive processes, or calculation steps.
Rectangle area comes from multiplying the two side expressions, which naturally creates a quadratic and square units. We’ll write the factored model from width and length, expand it for an …
Preview problemBuild a quadratic revenue or profit function from a price-and-demand context
Build quadratic and exponential functions from context using explicit formulas, recursive processes, or calculation steps.
Revenue multiplies price per item by the number of items sold, so the demand expression must become a factor rather than an addend. We’ll form and expand that product, track …
Preview problemBuild a quadratic function in vertex form from a vertex and one point
Build quadratic and exponential functions from context using explicit formulas, recursive processes, or calculation steps.
A known vertex fixes h and k in vertex form, leaving only the vertical scale unknown. We’ll substitute the second graph point, square its full horizontal displacement before solving for …
Preview problemBuild a quadratic function from two zeros and one point
Build quadratic and exponential functions from context using explicit formulas, recursive processes, or calculation steps.
Two zeros determine the linear factors of the quadratic, while one more point determines their shared scale. We’ll build the factored template, substitute the extra point with both factor signs …
Preview problemBuild an exponential function from an initial value and a growth factor
Build quadratic and exponential functions from context using explicit formulas, recursive processes, or calculation steps.
In an exponential model, the initial value is the coefficient and the per-interval multiplier is the base. We’ll assign those parameters directly, verify the start with the zero-exponent rule, and …
Preview problemBuild an exponential function from a starting amount and percent change
Build quadratic and exponential functions from context using explicit formulas, recursive processes, or calculation steps.
A percent increase must be converted into the full multiplier that retains the original amount and adds the growth portion. We’ll change the percent to a decimal, add it to …
Preview problemBuild an exponential function from two data points
Build quadratic and exponential functions from context using explicit formulas, recursive processes, or calculation steps.
An input-zero point isolates the exponential coefficient because the base contributes a factor of one. We’ll use that point first, substitute the second point to solve the base as an …
Preview problemBuild a recursive exponential model from a context
Build quadratic and exponential functions from context using explicit formulas, recursive processes, or calculation steps.
A recursive exponential model needs both an indexed starting value and a rule that multiplies the preceding term. We’ll place the initial population at generation zero, translate doubling into a …
Preview problemWrite a recursive rule for a quadratic-looking sequence from its terms
Build quadratic and exponential functions from context using explicit formulas, recursive processes, or calculation steps.
The recurrence is driven by the changing first differences, so their pattern must be expressed using the destination index. We’ll compute the consecutive increases, match them to an odd-number formula …
Preview problemChoose whether a context is modeled by a quadratic or an exponential function
Build quadratic and exponential functions from context using explicit formulas, recursive processes, or calculation steps.
The location of the variable determines the function family more reliably than the mere presence of multiplication. We’ll represent both changing sides as linear expressions, expand their one-time area product …
Preview problemWrite an explicit function model from a context and choose a reasonable domain
Build quadratic and exponential functions from context using explicit formulas, recursive processes, or calculation steps.
The algebraic roots of the height equation are candidate boundary times, but the motion context begins at launch. We’ll solve the ground-level equation exactly, classify both roots relative to time …
Preview problemWrite an explicit function model from a verbal situation and evaluate it
Build quadratic and exponential functions from context using explicit formulas, recursive processes, or calculation steps.
The verbal model supplies an initial amount, a repeated factor, and an interval count. We’ll place those pieces into exponential form, substitute the requested elapsed time, interpret the output with …
Preview problemBuild an explicit model from context and solve for the input that gives a target output
Build quadratic and exponential functions from context using explicit formulas, recursive processes, or calculation steps.
Ground events occur when the height output is zero, and the variable greatest common factor must be kept because it represents one of those times. We’ll factor the equation, solve …
Preview problemForm and interpret the sum of two functions, (f+g)(x)
Combine standard function types arithmetically to create models.
A function sum adds two outputs evaluated at the same input. We’ll substitute both complete rules with their signs grouped, combine only like terms, and test a simple input in …
Preview problemForm and interpret \((f-g)(x)\)
Combine standard function types arithmetically to create models.
A net-value difference preserves the stated order: revenue first, then the entire cost output. We’ll substitute both rules with full grouping, remove the constant grouping without moving it inside the …
Preview problemForm and interpret the product of two functions in context
Combine standard function types arithmetically to create models.
The product function multiplies both side-length outputs at the same input, matching the rectangle area relationship. We’ll substitute the dimension rules as factors, expand to check every cross term, and …
Preview problemForm and interpret a quotient function to model an average or unit rate
Combine standard function types arithmetically to create models.
Average cost divides the entire total-cost output by the number of orders. We’ll group the full numerator, simplify both terms for nonzero input, derive dollars-per-order units, distinguish the algebraic restriction …
Preview problemBuild a contextual sum or difference of functions
Combine standard function types arithmetically to create models.
Profit is a directed difference, so subtracting the cost function must reverse every term inside its grouping. We’ll form revenue minus the complete cost, distribute the subtraction, simplify the quadratic, …
Preview problemCombine two functions by addition or subtraction in context
Combine standard function types arithmetically to create models.
Function outputs may be added only when they use the same input and compatible units. We’ll confirm the shared time and dollar units, substitute both rules into the total, keep …
Preview problemEvaluate an arithmetic combination of functions
Combine standard function types arithmetically to create models.
Evaluating a function combination means using the same allowed input in each component before performing the outside operation. We’ll confirm the shared domain, calculate the two outputs separately, and add …
Preview problemEvaluate an arithmetic combination of functions from a table
Combine standard function types arithmetically to create models.
A table lookup for a combined function begins by fixing one input row. We’ll locate the requested row, read both operands without moving to another input, translate the notation into …
Preview problemDetermine the domain of a combined function such as (f+g)(x)
Combine standard function types arithmetically to create models.
A function sum needs both component outputs, so its domain is the intersection rather than the union of their domains. We’ll compare the two allowed-input sets, retain only their overlap, …
Preview problemChoose the arithmetic operation that combines two functions in context
Combine standard function types arithmetically to create models.
The requested total must include both attendance counts once. We’ll verify that both functions use the same event input and person units, identify the operation that combines compatible counts into …
Preview problemInterpret a named feature of a combined function model in context
Combine standard function types arithmetically to create models.
A zero belongs to the combined profit output, not to the revenue component by itself. We’ll set profit equal to zero, substitute the difference model, rearrange it to compare revenue …
Preview problemInterpret \(f(x)+k\) as a vertical shift of a quadratic or absolute-value function
Analyze transformations of quadratic and absolute-value functions and identify even/odd functions.
A constant added outside the function changes every output while leaving each input fixed. We’ll express the transformed rule as the parent output plus a constant, use its sign for …
Preview problemIdentify the horizontal shift of a quadratic or absolute-value function
Analyze transformations of quadratic and absolute-value functions and identify even/odd functions.
A change inside the input creates a horizontal shift whose direction is opposite the visible sign. We’ll match the rule with f of x minus h, solve the correspondence between …
Preview problemIdentify a vertical stretch or compression from the coefficient of a quadratic or absolute-value function
Analyze transformations of quadratic and absolute-value functions and identify even/odd functions.
An outside coefficient multiplies outputs rather than shifting inputs or outputs by a fixed amount. We’ll compare its magnitude with one, apply the point mapping from y to a times …
Preview problemIdentify a reflection across the x-axis from a transformed quadratic or absolute-value function
Analyze transformations of quadratic and absolute-value functions and identify even/odd functions.
A negative sign outside the function negates completed outputs while preserving inputs. We’ll translate that rule into the point mapping from y to negative y, identify the corresponding reflection and …
Preview problemDetermine the effect of replacing \(x\) with \(-x\) in a function rule
Analyze transformations of quadratic and absolute-value functions and identify even/odd functions.
Replacing x with negative x changes inputs before the parent rule is evaluated, but the visible result depends on the parent’s symmetry. We’ll simplify the transformed rule with the parentheses …
Preview problem