What does f(3)=7 tell you?
Graphs, tables and rules describe the same input-output relationship.
Compare reasoning and find a refresher
Input 3 has output 7, so (3,7) is a point on the graph; it does not mean f(7)=3.
Functions in graphs, measurements and growth decisions
A school-entry practice unit connecting quantities, measurement accuracy, unfamiliar graphs and tables, transformations, inverses, sequences and growth decisions. Graph descriptions and sketches require rubric review; this closes specific representation gaps rather than claiming complete AP preparation.
Open any lesson. Completion records participation; automatic practice checks and human judgment of written arguments are separate. This unit retains its own versioned requirements.
Readiness and preparation
Check your starting point
Try these short questions on paper, then compare your reasoning. They are optional self-checks, not a placement score or a navigation requirement. Use a targeted review when useful and return with your browser’s Back control.
How does grouping change an expression?
Graph transformations act in a particular order.
Compare reasoning and find a refresher
In 2(x−3)+1, subtract 3 from the input first, multiply by 2, then add 1.
Can every real number be an input?
Context and formulas can both restrict a domain.
Compare reasoning and find a refresher
No. A count may require nonnegative integers; a denominator cannot be zero. State the applicable restriction rather than filling every graph position.
Loading saved mathematics work…
Representations and decisions: mixed evidence
The mixed assessment has separate response history and review criteria; it does not change lesson participation records.
Lesson sequence
Each linked lesson opens on its own page. Expand its details here when you want to check the objectives before choosing.
Report what measurements support
Lesson details and objectives
Convert instrument resolution into a defensible measurement interval.
- Convert instrument resolution into a defensible measurement interval.
- Propagate positive interval bounds through an area calculation.
- Choose report precision without claiming unsupported accuracy.
6 practice tasks.
Choose quantities, units and honest axes
Lesson details and objectives
Define a useful comparison quantity and its denominator.
- Define a useful comparison quantity and its denominator.
- Carry units through a multistep calculation.
- Defend axis origin, scale and labels for a supplied context.
6 practice tasks.
Read rates from uneven tables and graph scales
Lesson details and objectives
Calculate change per input unit on unequal intervals.
- Calculate change per input unit on unequal intervals.
- Estimate a secant rate from a supplied nonunit-scale graph.
- Distinguish average rate from output differences and local behavior.
6 practice tasks.
Compare functions across representations
Lesson details and objectives
Read contextual signs, extrema and domain from supplied graphs.
- Read contextual signs, extrema and domain from supplied graphs.
- Compare a graphed model with a formula or sampled table.
- Construct a contextual graph with justified landmarks and domain.
6 practice tasks.
Construct cube-root and absolute-value graphs
Lesson details and objectives
Build cube-root graphs from exact landmark coordinates.
- Build cube-root graphs from exact landmark coordinates.
- Construct absolute-value graphs with vertex, slopes and range.
- Distinguish smooth-root shape from a corner and justify domains.
7 practice tasks.
Graph ceiling steps and discrete decisions
Lesson details and objectives
Translate whole-unit charging rules into ceiling functions.
- Translate whole-unit charging rules into ceiling functions.
- Mark open and closed endpoints on step graphs.
- Distinguish discrete count domains from continuous measured inputs.
6 practice tasks.
Graph transformed exponentials and logarithms
Lesson details and objectives
Construct shifted exponential and logarithmic graphs with asymptotes.
- Construct shifted exponential and logarithmic graphs with asymptotes.
- Read domains, ranges, intercepts and end behavior.
- Explain the effect of a base between zero and one.
6 practice tasks.
Recover transformations from supplied graphs
Lesson details and objectives
Infer translation, reflection and scale from plotted landmarks.
- Infer translation, reflection and scale from plotted landmarks.
- Check an inferred transformation with a third point.
- Recognize when limited graph data do not determine a unique model.
6 practice tasks.
Read inverses without a formula
Lesson details and objectives
Read inverse values from unseen tables and supplied graphs.
- Read inverse values from unseen tables and supplied graphs.
- Reflect coordinates and interchange domain and range.
- Restrict a non-one-to-one graph before defining its inverse.
6 practice tasks.
Test even and odd symmetry with domains
Lesson details and objectives
Check domain symmetry before applying a parity identity.
- Check domain symmetry before applying a parity identity.
- Combine graph evidence with algebraic even/odd tests.
- Distinguish neither parity from missing or insufficient data.
6 practice tasks.
Translate recursive and explicit sequences
Lesson details and objectives
Convert arithmetic and geometric rules in both directions.
- Convert arithmetic and geometric rules in both directions.
- State index starts and integer domains.
- Explain why joining sequence points adds a continuous extension.
6 practice tasks.
Fit growth models and change time units
Lesson details and objectives
Recover exponential parameters from supplied graph landmarks.
- Recover exponential parameters from supplied graph landmarks.
- Convert equivalent time-unit expressions without changing predictions.
- Prove equal-interval difference and ratio rules for arbitrary intervals.
6 practice tasks.
Find delayed exponential crossovers
Lesson details and objectives
Compare exponential and polynomial values beyond a misleading initial window.
- Compare exponential and polynomial values beyond a misleading initial window.
- Locate and verify a discrete crossover.
- Distinguish finite graph evidence from a general eventual-growth argument.
6 practice tasks.
Solve target times and justify rounding
Lesson details and objectives
Solve contextual exponential target equations with logarithms.
- Solve contextual exponential target equations with logarithms.
- Evaluate noninteger times with units and domain checks.
- Choose and verify a ceiling or floor for discrete reporting decisions.
6 practice tasks.
Construct polynomial and rational graphs
Lesson details and objectives
Use factors, signs and domain exclusions to build branches rather than isolated plotted points.
- Construct polynomial graphs using multiplicity, sign intervals and end behavior.
- Distinguish a rational hole from a vertical asymptote and retain original exclusions.
- Read and justify intercepts and branches in supplied graphs.
6 practice tasks.
Combine meaningful functions into a quadratic decision
Lesson details and objectives
Build products and sums from units, then compare factored and vertex forms on a contextual domain.
- Interpret quadratic zeros and extrema through factored and completed-square forms.
- Construct revenue, cost and profit from meaningful component functions with compatible units.
- Filter continuous quadratic decisions through contextual and integer constraints.
6 practice tasks.