Math III
Go further with polynomial and rational expressions, advanced functions, trigonometry, geometric modeling, and statistical inference.
- Problem types
- 641
- Practice variants
- 2,564
Page 10 of 18
Each problem type has four distinct practice variants. Open a preview to move among all four.
Graph one cosine cycle from quarter-period anchors
Graph exponential, logarithmic, and trigonometric functions with key features.
Compute the cosine cycle’s vertical geometry from the midline and amplitude, and its horizontal geometry from the period and quarter-period. The starting pattern matters: with a positive coefficient, cosine begins …
Preview problemGraph tangent from center, zeros, and repeated asymptotes
Graph exponential, logarithmic, and trigonometric functions with key features.
For tangent, anchor the analysis at the center and compute its period before locating anything else. The nearest vertical asymptotes lie half a period on either side, and both the …
Preview problemExtract only identifiable graph-family features
Graph exponential, logarithmic, and trigonometric functions with key features.
Treat this as an evidence audit rather than a request to guess a unique curve. The horizontal asymptote fixes the outside shift, and the point at input zero fixes the …
Preview problemIdentify key graph features from an exponential, logarithmic, or sinusoidal equation
Graph exponential, logarithmic, and trigonometric functions with key features.
Identify the family from where the variable appears: a variable in the exponent signals an exponential graph. Next inspect the base to distinguish growth from decay. Finally read the additive …
Preview problemWrite an exponential equation from a horizontal asymptote, an initial value, and a growth or decay factor
Graph exponential, logarithmic, and trigonometric functions with key features.
Use the form with an initial scale, a repeated factor, and a vertical shift. The horizontal asymptote determines the shift, the stated growth factor becomes the base, and substituting input …
Preview problemConstruct a unique unit-scale logarithmic equation
Graph exponential, logarithmic, and trigonometric functions with key features.
Write the natural-log model with unit vertical scale, a horizontal shift inside the argument, and a possible vertical shift outside. The vertical asymptote fixes the horizontal-shift parameter; substituting the given …
Preview problemWrite a sinusoidal equation from maximum, minimum, period, and starting position
Graph exponential, logarithmic, and trigonometric functions with key features.
Recover the sinusoid’s vertical parameters from the extrema: their average is the midline, and half their difference is the amplitude. Convert the stated period into the inside coefficient with the …
Preview problemInterpret exponential, logarithmic, and sinusoidal features in context
Graph exponential, logarithmic, and trigonometric functions with key features.
Interpret the exponential asymptote as a limiting boundary, not automatically as a value the model reaches. The decay factor explains the long-term approach, while positivity of the coefficient and exponential …
Preview problemRewrite a polynomial in factored form to reveal its zeros
Rewrite functions in equivalent forms to reveal and explain useful properties.
Factored form is useful here because each linear factor exposes one zero directly. For a monic quadratic, find two constants whose product matches the constant term and whose sum matches …
Preview problemDetermine polynomial end behavior directly from the leading term
Rewrite functions in equivalent forms to reveal and explain useful properties.
End behavior does not require expanding the entire product. Take only the leading part of each factor and multiply those pieces to obtain the polynomial’s leading term. Its degree parity …
Preview problemRewrite a rational function and classify all graph features
Rewrite functions in equivalent forms to reveal and explain useful properties.
Record every original denominator restriction before simplifying the rational expression. Factoring reveals whether a denominator zero cancels or remains: a canceled factor creates a removable hole, while an uncanceled one …
Preview problemRewrite an exponential function to reveal the growth or decay factor for a different time interval
Rewrite functions in equivalent forms to reveal and explain useful properties.
Match the exponential factor to the time unit requested. Regroup the product in the exponent with the power rule so one new base represents all of the smaller compounding intervals …
Preview problemConvert exponential and logarithmic forms and solve
Rewrite functions in equivalent forms to reveal and explain useful properties.
Label the three roles in the exponential equation before converting: the base stays the logarithmic base, the exponential result becomes the logarithm’s argument, and the exponent becomes the logarithm’s output. …
Preview problemIdentify root graph features directly
Rewrite functions in equivalent forms to reveal and explain useful properties.
First identify the root family, since a square root has a one-sided endpoint and requires a nonnegative radicand. Setting the radicand to zero locates that endpoint, while the inside and …
Preview problemNormalize a sine or cosine equation and read features
Rewrite functions in equivalent forms to reveal and explain useful properties.
Normalize the entire angle into the form with an inside coefficient multiplying a shifted input before reading any horizontal feature. Then assign the parameters by role: the outside coefficient gives …
Preview problemChoose the equivalent form that best reveals a requested feature of a function
Rewrite functions in equivalent forms to reveal and explain useful properties.
Equivalent forms are useful for different features, so start with the feature the prompt asks to expose. Zeros are most visible in factored form because each factor can be set …
Preview problemRead the feature exposed by an equivalent form
Rewrite functions in equivalent forms to reveal and explain useful properties.
Factored form packages several polynomial features into the factors themselves. Set each factor equal to zero to locate the zeros, then read the overall degree and leading coefficient from the …
Preview problemVerify whether two function forms are equivalent on the same domain
Rewrite functions in equivalent forms to reveal and explain useful properties.
To decide whether two formulas define the same function, compare both their algebra and their allowed inputs. Expanding the product checks that the output rules match, while recognizing that both …
Preview problemRewrite to a specified solvable form and finish the solution
Rewrite functions in equivalent forms to reveal and explain useful properties.
The repeated powers make this look like a quadratic in disguise. Substitute a single variable for the squared expression, factor the resulting quadratic, and then translate each result back to …
Preview problemRewrite to expose and interpret a specified parameter
Rewrite functions in equivalent forms to reveal and explain useful properties.
Separate the numerator term by term before interpreting the average. That rewrite distinguishes the fixed cost spread across all items from the per-item contribution, and it also makes the long-run …
Preview problemChoose the form that reveals one named graph feature
Rewrite functions in equivalent forms to reveal and explain useful properties.
Equivalent forms can emphasize different features of the same quadratic. When the goal is to read the zeros, factored form is the efficient representation because each zero comes directly from …
Preview problemDecide whether a proposed rewrite keeps the same function and the same restrictions
Rewrite functions in equivalent forms to reveal and explain useful properties.
Cancellation preserves the outputs only where the original rational expression is defined. Record the denominator restriction before simplifying, then compare that restricted line with the unrestricted linear function. The missing …
Preview problemCompare the value of two functions at a given input from different representations
Compare functions represented algebraically, graphically, numerically, or verbally.
A fair function comparison uses the same input in both representations. Evaluate the formula at the stated input and read the other output from the matching table row. Once those …
Preview problemCompare function domains and ranges as sets
Compare functions represented algebraically, graphically, numerically, or verbally.
Build each domain and range from its own defining restriction before comparing the sets. The square root permits its boundary input and produces only nonnegative outputs, while the logarithm requires …
Preview problemCompare zeros and intercepts in parallel rows
Compare functions represented algebraically, graphically, numerically, or verbally.
Shared zeros tell you that two graphs cross the x-axis at the same places, but they do not determine the rest of either function. Compare another easy feature, such as …
Preview problemCompare asymptotes and end behavior in parallel fields
Compare functions represented algebraically, graphically, numerically, or verbally.
Analyze the two rational models in layers. The reciprocal term controls the shared vertical asymptote and the one-sided behavior near it, while the constant term gives each horizontal asymptote as …
Preview problemCompare average rates across representations
Compare functions represented algebraically, graphically, numerically, or verbally.
Average rates are comparable only when they use the same interval. Find each function's output change between the shared endpoints, divide by the common input change, and keep the endpoint …
Preview problemCompare extrema and turning behavior in parallel
Compare functions represented algebraically, graphically, numerically, or verbally.
Compare extrema and turning behavior as two related but distinct features. For each graph, identify whether a highest or lowest point exists, then trace the graph from left to right …
Preview problemCompare periodic features in parallel parameter rows
Compare functions represented algebraically, graphically, numerically, or verbally.
Organize the comparison by parameter rather than describing one entire wave at a time. The outside coefficient controls amplitude and reflection, the vertical shift sets the midline and range, and …
Preview problemCompare the long-term behavior of polynomial and exponential functions across representations
Compare functions represented algebraically, graphically, numerically, or verbally.
Long-run growth is about what persists after any early crossings, not which graph happens to be higher at one input. Compare how each output changes when the input increases by …
Preview problemCompare polynomial and rational behavior in parallel
Compare functions represented algebraically, graphically, numerically, or verbally.
Use the function families to build parallel feature lists. A polynomial's domain, continuity, and end behavior come from its leading term, while a rational function also requires checking denominator restrictions, …
Preview problemCompare an exponential function and a logarithmic function to determine whether they are inverses
Compare functions represented algebraically, graphically, numerically, or verbally.
Test the inverse relationship by composing the functions in both orders on their proper domains. If each composition returns its input, the swapped domain and range, reflected points, and exchanged …
Preview problemDecide whether two different representations show the same function
Compare functions represented algebraically, graphically, numerically, or verbally.
Matching zeros identifies the factors of a quadratic, but it does not yet fix the vertical scale. Write the general factored form with an unknown leading multiplier, then use the …
Preview problemDetermine what information is missing to compare functions
Compare functions represented algebraically, graphically, numerically, or verbally.
Match the scope of the conclusion to the scope of the evidence. One table row supports a comparison at one input, while a statement about every input requires information covering …
Preview problemWrite an evidence-scoped comparison without overgeneralizing
Compare functions represented algebraically, graphically, numerically, or verbally.
Separate what the evidence proves from what it leaves unknown. Matching zeros support matching x-intercepts, while one differing output is enough to rule out identical functions. Without rules, domains, or …
Preview problemSolve an exponential equation of the form a(b)^(ct)=d by isolating the exponential factor and taking logarithms
Use logarithms to solve exponential equations of the form ab^(ct)=d and evaluate with technology.
Isolate the exponential expression before doing anything to the exponent. If the remaining constant can be rewritten with the same base, equate the exponents and solve the resulting linear equation; …
Preview problem