Math I
Build connected understanding across quantities, equations, functions, coordinate geometry, congruence, and data.
- Problem types
- 659
- Practice variants
- 2,636
Page 11 of 19
Each problem type has four distinct practice variants. Open a preview to move among all four.
Construct a recursive geometric sequence rule
Construct linear and exponential functions, including arithmetic/geometric sequences, from graphs, descriptions, or input-output pairs.
A recursive geometric rule needs a starting term and a multiplicative update that acts on the preceding term. We’ll translate the given first value into the initial condition and the …
Preview problemDecide whether to construct a linear or exponential model from table values
Construct linear and exponential functions, including arithmetic/geometric sequences, from graphs, descriptions, or input-output pairs.
Increasing outputs are not automatically exponential; the kind of repeated change is what identifies the model family. With equally spaced inputs, we’ll calculate every consecutive difference and compare the corresponding …
Preview problemConstruct a function when x = 0 is not shown
Construct linear and exponential functions, including arithmetic/geometric sequences, from graphs, descriptions, or input-output pairs.
A missing zero-input row hides the intercept, but it does not prevent us from recovering it. We’ll use multiple intervals to confirm a constant output change per input unit, then …
Preview problemConstruct a model and use it to predict
Construct linear and exponential functions, including arithmetic/geometric sequences, from graphs, descriptions, or input-output pairs.
A prediction is only as sound as both its fitted model and the assumption used to extend that model. We’ll derive the line from the observed points, verify it at …
Preview problemFind the input value when a linear or exponential model reaches a target output
Construct linear and exponential functions, including arithmetic/geometric sequences, from graphs, descriptions, or input-output pairs.
The target is an output, so finding its input means reversing the function’s operations in the opposite order. We’ll set the model equal to the target, undo the fixed addition, …
Preview problemCompare linear and exponential values over the same inputs
Use graphs/tables to see that exponential growth eventually exceeds linear, quadratic, and polynomial growth.
A linear pattern can stay ahead for several inputs even while an exponential pattern is preparing to pass it. We’ll derive each rule from its starting value and repeated change, …
Preview problemCompare quadratic and exponential values over the same inputs
Use graphs/tables to see that exponential growth eventually exceeds linear, quadratic, and polynomial growth.
Quadratic and exponential values can trade places more than once over a short input range, so growth type alone does not settle every row. We’ll compare the two outputs at …
Preview problemUse graph evidence to compare linear and exponential models
Use graphs/tables to see that exponential growth eventually exceeds linear, quadratic, and polynomial growth.
A graph supports a visual estimate of where two models exchange order, not an exact algebraic crossing. We’ll compare which curve is higher on each side of the visible intersection …
Preview problemCompare quadratic and exponential graphs using long-term behavior
Use graphs/tables to see that exponential growth eventually exceeds linear, quadratic, and polynomial growth.
A quadratic curve may begin above an exponential curve, yet their relative heights can reverse as multiplicative growth steepens. We’ll estimate the visible crossing, compare the curves immediately before and …
Preview problemFind the first interval where exponential values overtake comparison values
Use graphs/tables to see that exponential growth eventually exceeds linear, quadratic, and polynomial growth.
The first overtake is located by an order reversal across two consecutive rows, not by naming only the first row where one value is larger. We’ll compare matching outputs in …
Preview problemChoose a long-term model from accelerating changes
Use graphs/tables to see that exponential growth eventually exceeds linear, quadratic, and polynomial growth.
Accelerating differences can signal an exponential structure even when the raw outputs do not have a constant ratio. We’ll calculate the first differences, notice how those changes scale, and test …
Preview problemInterpret an overtaking point in context
Use graphs/tables to see that exponential growth eventually exceeds linear, quadratic, and polynomial growth.
An overtaking point has meaning in both coordinates: a time input and two approximately equal contextual outputs. We’ll read the axes and crossing, compare which model is higher on either …
Preview problemCompare exponential growth with fixed yearly increase
Use graphs/tables to see that exponential growth eventually exceeds linear, quadratic, and polynomial growth.
Repeated doubling and a fixed yearly addition create different model structures, but that does not automatically make their output magnitudes comparable. We’ll write each rule with an unspecified starting value, …
Preview problemExtend exponential table values
Use graphs/tables to see that exponential growth eventually exceeds linear, quadratic, and polynomial growth.
Doubling preserves a constant multiplier, not a constant amount added. We’ll apply the factor to each newly produced value for the required number of steps, then verify the continuation with …
Preview problemInterpret slope as rate of change with units
Interpret parameters in linear and exponential functions in context.
A slope interpretation must name its sign, its rate, and its output-per-input units. We’ll connect the variable coefficient to the cost change for one additional hour, then scale that rate …
Preview problemInterpret y-intercept as initial value
Interpret parameters in linear and exponential functions in context.
The vertical intercept is the model’s output at an input of zero, so its meaning comes from the zero-input context. We’ll substitute zero, express the result as a point with …
Preview problemInterpret the initial value in an exponential model
Interpret parameters in linear and exponential functions in context.
In an exponential model, the coefficient becomes the output at time zero because the powered factor becomes one. We’ll evaluate the zero-year input, state the resulting point with contextual units, …
Preview problemInterpret an exponential base in context
Interpret parameters in linear and exponential functions in context.
An exponential base is the factor multiplying the current amount during each input interval. We’ll express that action as a recurrence, convert the factor to a retained percentage, and separate …
Preview problemConvert an exponential base to percent change
Interpret parameters in linear and exponential functions in context.
A base above one represents the entire retained amount plus a growth portion, so the base itself is not the percent increase. We’ll compare the factor with one, subtract the …
Preview problemConvert percent change to an exponential base
Interpret parameters in linear and exponential functions in context.
A percent increase contributes only the added portion, while the exponential base must also retain the original whole. We’ll convert the percent to a decimal and add it to one …
Preview problemInterpret a shifted linear model in context
Interpret parameters in linear and exponential functions in context.
A shifted linear form is anchored at the input that makes its parenthesized expression zero. We’ll interpret that anchor, the number of units beyond it, and the rate applied to …
Preview problemInterpret a shifted exponential cooling model
Interpret parameters in linear and exponential functions in context.
A shifted cooling model separates the surrounding baseline from the temperature excess above it. We’ll evaluate the initial excess and total, interpret the decay factor as acting only on that …
Preview problemCompare parameter meanings in two linear models
Interpret parameters in linear and exponential functions in context.
Comparing two linear costs requires treating starting fees and hourly rates as separate advantages. We’ll compare both parameter pairs, set the models equal to locate their crossover, and verify the …
Preview problemCompare parameter meanings in two exponential models
Interpret parameters in linear and exponential functions in context.
A larger exponential starting value and a larger growth factor are different advantages, so neither parameter alone determines the long-term comparison. We’ll convert both bases to percent rates, equate the …
Preview problemIdentify units of linear model parameters
Interpret parameters in linear and exponential functions in context.
Parameter units follow their roles in the model, not merely the numbers attached to them. We’ll identify the input and output units, treat the intercept as an output at zero …
Preview problemCorrect a parameter interpretation error
Interpret parameters in linear and exponential functions in context.
A parameter’s algebraic position determines its meaning: the variable coefficient is a rate, while the constant is the zero-input amount. We’ll diagnose the swapped roles, restore the correct contextual units, …
Preview problemIdentify a plane from a geometric description
Use precise definitions of angle, circle, perpendicular line, parallel line, and line segment from point/line/distance ideas.
The object’s dimension and extent matter more than the finite boundary used to draw it. We’ll match a flat two-dimensional surface extending without end to its precise geometric definition, then …
Preview problemDefine an angle from two rays
Use precise definitions of angle, circle, perpendicular line, parallel line, and line segment from point/line/distance ideas.
An angle is defined by two rays sharing one endpoint, so the ray notation reveals the vertex before the angle is named. We’ll trace each ray from its first letter, …
Preview problemName an angle using three-point notation
Use precise definitions of angle, circle, perpendicular line, parallel line, and line segment from point/line/distance ideas.
Three-point angle notation encodes the vertex by putting it in the middle, while the outer letters identify points on the two sides. We’ll locate the common endpoint in the diagram, …
Preview problemDefine a circle using center and radius
Use precise definitions of angle, circle, perpendicular line, parallel line, and line segment from point/line/distance ideas.
A circle is a set of coplanar points satisfying an exact distance condition from one center. We’ll read the center and radius from the diagram, then express membership using an …
Preview problemClassify a circle-related segment
Use precise definitions of angle, circle, perpendicular line, parallel line, and line segment from point/line/distance ideas.
Circle-segment names come from where the endpoints sit, not from how the picture happens to look. We’ll track whether the segment starts at the center and whether its other endpoint …
Preview problemDetermine whether lines are perpendicular
Use precise definitions of angle, circle, perpendicular line, parallel line, and line segment from point/line/distance ideas.
A line relationship must be justified by marked or measured geometry, not by orientation on the page. We’ll translate the angle marking at the intersection into its exact measure, then …
Preview problemIdentify parallel lines from a definition
Use precise definitions of angle, circle, perpendicular line, parallel line, and line segment from point/line/distance ideas.
In three-dimensional geometry, never meeting is not quite enough to classify two lines. We’ll check distinctness, coplanarity, and nonintersection as a package; the plane condition is what separates the intended …
Preview problemDefine a line segment using endpoints
Use precise definitions of angle, circle, perpendicular line, parallel line, and line segment from point/line/distance ideas.
When geometry asks for a precise definition, every phrase has a job: name the ambient line, the boundary points, and the membership condition. We’ll build the statement from A and …
Preview problemUse definitions instead of visual assumptions
Use precise definitions of angle, circle, perpendicular line, parallel line, and line segment from point/line/distance ideas.
A diagram can suggest a conjecture, but definitions decide whether that conjecture is proved. We’ll separate visual appearance from certified evidence and look for a right-angle mark, an angle measure, …
Preview problemWrite a precise definition for perpendicular lines
Use precise definitions of angle, circle, perpendicular line, parallel line, and line segment from point/line/distance ideas.
An example–nonexample pair helps isolate the condition that actually changes the classification. We’ll compare the marked angle with the measured acute angle, express the decisive condition as an if-and-only-if statement, …
Preview problem