Math I
Build connected understanding across quantities, equations, functions, coordinate geometry, congruence, and data.
- Problem types
- 659
- Practice variants
- 2,636
Page 6 of 19
Each problem type has four distinct practice variants. Open a preview to move among all four.
Choose the function operation that matches a context
Combine standard functions using arithmetic operations to build models.
The context determines how function outputs should be combined. We’ll identify whether the requested quantity is a total, difference, product, or per-unit comparison, require both fee outputs to refer to …
Preview problemWrite a recursive arithmetic sequence rule from terms
Write arithmetic and geometric sequences explicitly and recursively; translate between forms and model situations.
A recursive arithmetic rule needs both a starting term and a constant additive update. We’ll record the first listed value, compute every consecutive difference to verify that one change repeats, …
Preview problemWrite an explicit arithmetic sequence rule from terms
Write arithmetic and geometric sequences explicitly and recursively; translate between forms and model situations.
An explicit arithmetic rule reaches any term from its index without generating the earlier list. We’ll identify the first term and verify the common difference, then count how many equal …
Preview problemWrite a recursive rule for a geometric sequence from its terms
Write arithmetic and geometric sequences explicitly and recursively; translate between forms and model situations.
A recursive geometric rule scales the previous term by one constant factor. We’ll preserve the first listed value as the initial condition, divide consecutive terms to verify a common ratio, …
Preview problemWrite an explicit geometric sequence rule from terms
Write arithmetic and geometric sequences explicitly and recursively; translate between forms and model situations.
An explicit geometric rule compresses repeated multiplication into a power. We’ll identify the first term, verify the common ratio from consecutive quotients, and count how many ratio applications separate term …
Preview problemTranslate a recursive arithmetic rule into an explicit rule
Write arithmetic and geometric sequences explicitly and recursively; translate between forms and model situations.
Translating an arithmetic recurrence means replacing repeated updates with a direct count of those updates. We’ll read the initial term and additive change from the recursive rule, note that term …
Preview problemTranslate an explicit arithmetic rule into a recursive rule
Write arithmetic and geometric sequences explicitly and recursively; translate between forms and model situations.
To turn an explicit arithmetic rule into a recurrence, recover the information needed for one-step generation. We’ll evaluate the formula at the first index to obtain the initial term, read …
Preview problemTranslate a recursive geometric rule into an explicit rule
Write arithmetic and geometric sequences explicitly and recursively; translate between forms and model situations.
A geometric recurrence becomes explicit by counting repeated applications of its multiplier. We’ll extract the starting term and common ratio, observe that the first term uses zero extra factors, and …
Preview problemTranslate an explicit geometric rule into a recursive rule
Write arithmetic and geometric sequences explicitly and recursively; translate between forms and model situations.
An explicit geometric formula contains both ingredients of its recursive form. We’ll evaluate at the first index to recover the initial term, identify the exponential base as the common ratio, …
Preview problemModel an arithmetic situation with explicit and recursive rules
Write arithmetic and geometric sequences explicitly and recursively; translate between forms and model situations.
The context describes a discrete sequence with a starting row and a constant additive change. We’ll define the indexed seat count, translate the first row and per-row increase, and express …
Preview problemModel a geometric situation with explicit and recursive rules
Write arithmetic and geometric sequences explicitly and recursively; translate between forms and model situations.
The context describes a geometric sequence when each period scales the entire previous population by the same factor. We’ll define the indexed population, identify its starting value and common ratio, …
Preview problemClassify a sequence as arithmetic, geometric, both, or neither
Write arithmetic and geometric sequences explicitly and recursively; translate between forms and model situations.
Sequence classification requires testing definitions rather than judging whether the list merely looks regular. We’ll compute every consecutive difference for the arithmetic test and every consecutive ratio for the geometric …
Preview problemFind a specific term from a given arithmetic or geometric sequence rule
Write arithmetic and geometric sequences explicitly and recursively; translate between forms and model situations.
An explicit sequence rule gives direct access to a requested term through its index. We’ll replace the general index everywhere with the requested term number, preserve the grouped transition count, …
Preview problemDetermine which term of an explicit sequence first meets a given condition
Write arithmetic and geometric sequences explicitly and recursively; translate between forms and model situations.
To locate the term meeting a condition, turn the condition into an equation involving the explicit rule. We’ll set the term expression equal to the target, isolate the grouped transition …
Preview problemCompare two sequence representations at a specific term number
Write arithmetic and geometric sequences explicitly and recursively; translate between forms and model situations.
Two sequence representations can be compared only after both are evaluated at the same index. We’ll substitute the index directly into the explicit rule, count the one-fewer transitions needed by …
Preview problemDiagnose an indexing error in a sequence model
Write arithmetic and geometric sequences explicitly and recursively; translate between forms and model situations.
The fastest way to diagnose a sequence-indexing error is to test the stated first index. We’ll compare the proposed first output with the required starting value, verify the common difference …
Preview problemDescribe a vertical shift of a function
Analyze graph transformations caused by f(x)+k, kf(x), f(kx), and f(x+k); identify even/odd functions.
The location of a constant tells which coordinates a transformation changes. We’ll notice that this value is added after the function produces its output, map a general parent point to …
Preview problemDescribe a horizontal shift of a function
Analyze graph transformations caused by f(x)+k, kf(x), f(kx), and f(x+k); identify even/odd functions.
A constant added inside the function changes inputs, so the horizontal direction cannot be read from its visible sign alone. We’ll match a transformed input to an original input, solve …
Preview problemDescribe a vertical scale or reflection
Analyze graph transformations caused by f(x)+k, kf(x), f(kx), and f(x+k); identify even/odd functions.
A multiplier outside the function acts on every output while leaving each input fixed. We’ll write the coordinate mapping, compare corresponding points on the parent and transformed graphs, and use …
Preview problemDescribe a horizontal scale or reflection
Analyze graph transformations caused by f(x)+k, kf(x), f(kx), and f(x+k); identify even/odd functions.
A multiplier inside the function changes horizontal coordinates by its reciprocal. We’ll match a transformed input to the original feature input, solve for the new x-coordinate, and verify the result …
Preview problemDescribe reflection across the x-axis
Analyze graph transformations caused by f(x)+k, kf(x), f(kx), and f(x+k); identify even/odd functions.
First locate the negative sign relative to the function, because an outside negative acts on outputs rather than inputs. We’ll translate that effect into a point-mapping rule, then use the …
Preview problemDescribe reflection across the y-axis
Analyze graph transformations caused by f(x)+k, kf(x), f(kx), and f(x+k); identify even/odd functions.
A negative inside the function changes the input before the output is produced. We’ll express that as a coordinate mapping and compare the graph’s paired points, watching for the coordinate …
Preview problemWrite a transformed function from a verbal transformation description
Analyze graph transformations caused by f(x)+k, kf(x), f(kx), and f(x+k); identify even/odd functions.
Treat the two movements separately before combining them. Horizontal changes belong inside the function’s input and use the sign opposite the graph’s motion, while vertical changes are applied outside to …
Preview problemMatch a transformed equation to a graph description
Analyze graph transformations caused by f(x)+k, kf(x), f(kx), and f(x+k); identify even/odd functions.
Start from the parent parabola’s vertex and opening direction, then read the equation by where each change occurs. The expression inside the square controls horizontal placement, the term outside controls …
Preview problemTransform a table using a vertical function transformation
Analyze graph transformations caused by f(x)+k, kf(x), f(kx), and f(x+k); identify even/odd functions.
Because the operation is outside the function, it transforms outputs while leaving the inputs paired with them unchanged. We’ll read each original output from the table, apply the same vertical …
Preview problemDetermine whether a function is even
Analyze graph transformations caused by f(x)+k, kf(x), f(kx), and f(x+k); identify even/odd functions.
Evenness is a statement about every pair of opposite inputs, so a few numerical checks are not enough. We’ll substitute negative x into the rule, simplify carefully, and compare the …
Preview problemDetermine whether a function is odd
Analyze graph transformations caused by f(x)+k, kf(x), f(kx), and f(x+k); identify even/odd functions.
Oddness requires more than getting different outputs at opposite inputs: the outputs must be exact opposites for every x. We’ll compute the function at negative x, separately negate the original …
Preview problemClassify a function as even, odd, both, or neither
Analyze graph transformations caused by f(x)+k, kf(x), f(kx), and f(x+k); identify even/odd functions.
The four-way classification comes from testing two identities independently, not from stopping after one fails. We’ll calculate the function at negative x once, then compare that expression with both the …
Preview problemCompare two transformations of the same parent function
Analyze graph transformations caused by f(x)+k, kf(x), f(kx), and f(x+k); identify even/odd functions.
Decode each transformed rule on its own before comparing them. Changes outside the function move outputs vertically, while changes inside the input move graph features horizontally with the opposite-sign convention. …
Preview problemFind and correct a transformation notation error
Analyze graph transformations caused by f(x)+k, kf(x), f(kx), and f(x+k); identify even/odd functions.
Before correcting the proposal, determine what it actually does. An input change moves features horizontally, and solving when the altered input equals an original landmark exposes the true direction without …
Preview problemInterpret a function transformation in context
Analyze graph transformations caused by f(x)+k, kf(x), f(kx), and f(x+k); identify even/odd functions.
Compare the two pricing rules term by term while keeping the contextual units attached. The coefficient of the item count represents the per-item rate, and the constant represents the fixed …
Preview problemDecide whether an ordered-pair set represents a function
Understand functions as mappings from inputs to exactly one output; connect f(x) to graph y=f(x).
Function status is controlled by the first coordinates, since those are the inputs. We’ll scan the relation input by input and look for a single input paired with two different …
Preview problemDecide whether a mapping diagram represents a function
Understand functions as mappings from inputs to exactly one output; connect f(x) to graph y=f(x).
Read the mapping diagram from the input side and count arrows leaving each input. A function requires exactly one outgoing arrow from every input, while the number of arrows entering …
Preview problemUse the vertical line test to decide whether a graph is a function
Understand functions as mappings from inputs to exactly one output; connect f(x) to graph y=f(x).
The vertical-line test asks whether one fixed x-value can produce more than one y-value. We’ll imagine a general test line x equals c, use the graph’s equation to determine its …
Preview problemDetermine whether a table represents a function
Understand functions as mappings from inputs to exactly one output; connect f(x) to graph y=f(x).
Treat each table row as an ordered pair, with x as the input and y as its output. We’ll group rows by their x-values and check whether any one input …
Preview problemIdentify input and output quantities in a context
Understand functions as mappings from inputs to exactly one output; connect f(x) to graph y=f(x).
Turn the phrase “depends on” into a one-way arrow between the two quantities. The quantity supplied or controlled first is the input, and the quantity determined in response is the …
Preview problem