Math I
Build connected understanding across quantities, equations, functions, coordinate geometry, congruence, and data.
- Problem types
- 659
- Practice variants
- 2,636
Page 9 of 19
Each problem type has four distinct practice variants. Open a preview to move among all four.
Identify the y-intercept of a quadratic
Graph linear and quadratic functions and show key features such as intercepts, maxima, and minima.
A y-intercept is found by using the input shared by every point on the vertical axis. We’ll substitute that input into the quadratic, simplify the output, and then write the …
Preview problemSketch a quadratic from key features
Graph linear and quadratic functions and show key features such as intercepts, maxima, and minima.
Given features determine a quadratic only after the scale factor is fixed. We’ll begin with vertex form, use the stated opening for the coefficient’s sign, and substitute the supplied intercept …
Preview problemCompare linear and quadratic graphs by key features
Graph linear and quadratic functions and show key features such as intercepts, maxima, and minima.
Comparing function families means looking beyond a shared intercept or coefficient. We’ll classify each rule by degree, identify its key graph features, and compare how outputs change over equal input …
Preview problemInterpret a quadratic graph feature in context
Graph linear and quadratic functions and show key features such as intercepts, maxima, and minima.
A contextual vertex combines an input event, an output quantity, and an extremum claim. We’ll assign the correct meanings and units to the two coordinates, translate the point into function …
Preview problemGraph an exponential growth function
Graph exponential functions and identify intercepts and end behavior; preview logarithmic and trigonometric graph features.
An exponential graph is built from a starting output and a repeated multiplicative factor. We’ll evaluate the zero input for the initial value, convert the factor into a percent change, …
Preview problemGraph an exponential decay function
Graph exponential functions and identify intercepts and end behavior; preview logarithmic and trigonometric graph features.
Exponential decay multiplies by the same retained fraction rather than subtracting a fixed amount. We’ll identify the initial output, translate the retained factor into a percent decrease, and plot several …
Preview problemIdentify the y-intercept of an exponential function
Graph exponential functions and identify intercepts and end behavior; preview logarithmic and trigonometric graph features.
The y-intercept comes from the zero input, and exponentials make that evaluation especially revealing. We’ll use the zero-exponent rule to separate the initial coefficient from the growth factor, then write …
Preview problemDescribe end behavior of exponential growth
Graph exponential functions and identify intercepts and end behavior; preview logarithmic and trigonometric graph features.
End behavior follows the repeated factor in opposite directions. We’ll track several outputs as inputs move right, where a growth base repeatedly multiplies them, and rewrite negative exponents as reciprocals …
Preview problemDescribe end behavior of exponential decay
Graph exponential functions and identify intercepts and end behavior; preview logarithmic and trigonometric graph features.
A decay base shrinks outputs as inputs move right, but its left-end behavior reverses that process. We’ll follow repeated multiplication by the retained fraction to the right and use negative …
Preview problemIdentify the horizontal asymptote of an exponential function
Graph exponential functions and identify intercepts and end behavior; preview logarithmic and trigonometric graph features.
For a transformed exponential, the horizontal asymptote comes from the outside vertical shift, not the coefficient or base. We’ll rewrite the rule in coefficient-base-shift form, distinguish a vertical stretch from …
Preview problemMatch an exponential equation to graph features
Graph exponential functions and identify intercepts and end behavior; preview logarithmic and trigonometric graph features.
Matching an exponential rule to a graph requires every major feature to agree. We’ll evaluate the zero input for the vertical intercept, combine the base and coefficient signs to determine …
Preview problemSketch an exponential graph from a table
Graph exponential functions and identify intercepts and end behavior; preview logarithmic and trigonometric graph features.
A table with equally spaced inputs is exponential when consecutive outputs share a constant ratio. We’ll test those ratios, use the zero-input row as the initial coefficient, and reconstruct a …
Preview problemInterpret exponential graph features in context
Graph exponential functions and identify intercepts and end behavior; preview logarithmic and trigonometric graph features.
An exponential model compresses several contextual ideas into its coefficient, base, and exponent. We’ll first attach the input and output to their real-world units, then use a zero input to …
Preview problemPreview a logarithmic graph as the inverse of an exponential graph
Graph exponential functions and identify intercepts and end behavior; preview logarithmic and trigonometric graph features.
Think of an inverse as the same relationship read backward: every input-output pair trades places. We’ll use that swap both algebraically and graphically, reflecting points across the line y equals …
Preview problemPreview trigonometric graph features
Graph exponential functions and identify intercepts and end behavior; preview logarithmic and trigonometric graph features.
A repeating wave can be summarized by its vertical center, vertical reach, and horizontal repeat length. We’ll use the maximum and minimum to find the midline, amplitude, and range, then …
Preview problemClassify graph shape as exponential, logarithmic preview, or trigonometric preview
Graph exponential functions and identify intercepts and end behavior; preview logarithmic and trigonometric graph features.
Graph families are best identified by structural fingerprints, not simply by whether a curve rises. We’ll inspect the domain, the orientation of any asymptote, the presence of an intercept, and …
Preview problemCompare linear functions by slope and y-intercept
Compare two functions represented in different ways: equations, graphs, tables, or verbal descriptions.
Comparing linear functions means keeping two features in separate lanes: slope measures change, while the y-intercept measures the starting height. Because both rules are already in slope-intercept form, we can …
Preview problemCompare a linear equation to a graph description
Compare two functions represented in different ways: equations, graphs, tables, or verbal descriptions.
An equation and a graph can be compared once they speak the same language: slope and y-intercept. We’ll read those features from the formula, then recover them from the graph …
Preview problemCompare a linear equation to a table
Compare two functions represented in different ways: equations, graphs, tables, or verbal descriptions.
To compare a linear equation with a table, extract the same two defining features from each representation. The equation reveals its slope and intercept by form; the table reveals its …
Preview problemCompare a graph description to a verbal function description
Compare two functions represented in different ways: equations, graphs, tables, or verbal descriptions.
A graph and a verbal model may look different while encoding the same starting value and rate. We’ll read the graph’s vertical intercept and rise per input unit, then translate …
Preview problemCompare two function values at the same input
Compare two functions represented in different ways: equations, graphs, tables, or verbal descriptions.
A fair function-value comparison holds the input fixed and lets only the outputs differ. We’ll substitute the requested input into each rule, respecting multiplication before addition and recognizing when a …
Preview problemCompare average rates of change over the same interval
Compare two functions represented in different ways: equations, graphs, tables, or verbal descriptions.
Average rates are comparable only when both functions use the same interval endpoints. We’ll evaluate each function at those endpoints, find its output change, and divide by the shared input …
Preview problemCompare intercepts of two functions
Compare two functions represented in different ways: equations, graphs, tables, or verbal descriptions.
The two kinds of intercept answer opposite questions: a y-intercept fixes the input at zero, while an x-intercept fixes the output at zero. We’ll apply those conditions to both functions …
Preview problemDistinguish linear and exponential tables
Compare two functions represented in different ways: equations, graphs, tables, or verbal descriptions.
Increasing outputs alone do not identify a function family; the kind of repeated change does. Since the inputs advance by equal steps, we’ll test consecutive outputs in two ways: subtraction …
Preview problemCompare linear and quadratic graph features
Compare two functions represented in different ways: equations, graphs, tables, or verbal descriptions.
A line and a parabola differ most clearly in how their rates behave across the entire domain. We’ll connect the equations to the graphs, checking for constant versus changing rate, …
Preview problemCompare which function is greater over an interval
Compare two functions represented in different ways: equations, graphs, tables, or verbal descriptions.
A larger slope tells which line is catching up, but not which one is already higher. We’ll subtract the functions so their comparison becomes a sign question, then use the …
Preview problemMatch equivalent functions across representations
Compare two functions represented in different ways: equations, graphs, tables, or verbal descriptions.
Different representations can describe one function when they generate the same input-output relationship. We’ll use the equation as a common language, test every displayed table row against it, and read …
Preview problemChoose a model type from real-world behavior
Compare two functions represented in different ways: equations, graphs, tables, or verbal descriptions.
The key modeling question is whether equal time steps produce repeated addition or repeated multiplication. We’ll translate the monthly description into a recurrence, extend it to a rule after any …
Preview problemCompare domains and ranges across representations
Compare two functions represented in different ways: equations, graphs, tables, or verbal descriptions.
Domain and range come from projecting a relation horizontally and vertically, but the graph’s inclusion type matters just as much as its endpoints. We’ll read x-values for domain and y-values …
Preview problemCompare long-term behavior of two functions
Compare two functions represented in different ways: equations, graphs, tables, or verbal descriptions.
Long-term behavior cannot be decided by comparing one early pair of outputs or the visible constants alone. We’ll contrast a fixed amount added per step with a fixed factor applied …
Preview problemWrite an evidence-based comparison of two functions
Compare two functions represented in different ways: equations, graphs, tables, or verbal descriptions.
A complete comparison of two lines should connect their starting gap, their relative rate, and their crossover. We’ll read the intercepts and slopes, determine how quickly the initial gap closes, …
Preview problemUse first differences to identify a linear pattern
Distinguish linear from exponential situations by equal differences versus equal factors over equal intervals.
First differences measure how much the output changes from one row to the next. Because the input steps are equal, we’ll subtract each earlier output from the following output in …
Preview problemUse ratios to identify an exponential pattern
Distinguish linear from exponential situations by equal differences versus equal factors over equal intervals.
A multiplicative pattern is detected by ratios only after confirming that the input steps are equal. We’ll divide each later output by the one immediately before it, keeping the order …
Preview problemClassify a verbal pattern as linear or exponential
Distinguish linear from exponential situations by equal differences versus equal factors over equal intervals.
The units in a verbal pattern reveal whether the change is additive or multiplicative. We’ll distinguish a fixed number of units per week from a percent or unitless factor, then …
Preview problemClassify a graph description as linear or exponential
Distinguish linear from exponential situations by equal differences versus equal factors over equal intervals.
Graph shape is really a picture of how the rate behaves. We’ll check whether equal horizontal moves produce equal vertical changes, which appears as one straight path, or whether the …
Preview problemCheck input spacing before comparing changes
Distinguish linear from exponential situations by equal differences versus equal factors over equal intervals.
Before comparing output changes, we need to know whether those changes occurred over equal input distances. We’ll temporarily ignore the output column and subtract consecutive input values. Equal gaps allow …
Preview problem