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 13 of 18
Each problem type has four distinct practice variants. Open a preview to move among all four.
Specify cotangent zeros, asymptotes, and decreasing branches
Graph all six basic trigonometric functions.
Use cotangent's quotient form to separate its two landmark types: denominator zeros produce excluded asymptotes, while numerator zeros produce graph zeros when the denominator remains nonzero. Adjacent asymptotes bound one …
Preview problemDetermine the period of a basic trigonometric function
Graph all six basic trigonometric functions.
A period is the smallest horizontal shift that returns the entire graph to the same phase. Compare matching landmarks, such as zero crossings traveled in the same direction, and measure …
Preview problemState trig domain and range as separate sets
Graph all six basic trigonometric functions.
Keep domain and range tied to different questions: domain asks which horizontal inputs are allowed, while range asks which vertical outputs occur. Look for an expression that can become undefined …
Preview problemIdentify a basic trigonometric function from its graph features
Graph all six basic trigonometric functions.
Identify a parent trig graph by combining features instead of relying on one familiar point. First use boundedness, smoothness, asymptotes, and repeat length to narrow the function family. Then inspect …
Preview problemFind the vertical asymptotes of tangent, secant, and cosecant graphs
Graph all six basic trigonometric functions.
Find vertical asymptotes from the function's denominator, not from its zeros. Solve for every input that makes that denominator zero, express the repeating family with an integer parameter, and then …
Preview problemMap source trig features to reciprocal-graph features
Graph all six basic trigonometric functions.
A reciprocal graph is best traced from a few decisive source values. Source zeros make the reciprocal undefined and become vertical asymptotes, while source values of positive or negative one …
Preview problemFind the value of sin, cos, tan, csc, sec, or cot at a key angle
Graph all six basic trigonometric functions.
Locate the angle's terminal point on the unit circle before evaluating the function. Sine reads the vertical coordinate, cosine reads the horizontal coordinate, and the other trig functions come from …
Preview problemDetermine whether a basic trigonometric function is even, odd, or neither
Graph all six basic trigonometric functions.
Classify parity by replacing the input with its negative and comparing the resulting output with the original. An unchanged output corresponds to reflection across the vertical axis, while a sign-reversed …
Preview problemFind the amplitude of a sinusoid from its maximum and minimum values
Choose trig functions to model periodic phenomena using amplitude, frequency, and midline.
Amplitude measures the distance from a sinusoid's centerline to either extreme, not the entire top-to-bottom span. Subtract the minimum from the maximum and halve that vertical distance. As a check, …
Preview problemFind the midline of a sinusoidal function from its maximum and minimum values
Choose trig functions to model periodic phenomena using amplitude, frequency, and midline.
The midline is the horizontal level exactly halfway between a sinusoid's maximum and minimum. Average those two output values, then verify that the result is equally far from both extremes. …
Preview problemFind the period of a sinusoid from two matching graph features
Choose trig functions to model periodic phenomena using amplitude, frequency, and midline.
One period is the horizontal distance between consecutive points in the same phase of a sinusoid. Two neighboring maxima are especially useful because each marks the same position in successive …
Preview problemFind the frequency from the period of a periodic model
Choose trig functions to model periodic phenomena using amplitude, frequency, and midline.
Period and frequency describe reciprocal rates. Period tells how much input time one cycle takes, while frequency tells how many cycles occur per unit time. Invert the period value and …
Preview problemWrite a sine model from maximum and minimum values, period, midline, and the direction of a midline crossing
Choose trig functions to model periodic phenomena using amplitude, frequency, and midline.
A rising midline crossing is a natural phase anchor for a sine model. Use the amplitude for the vertical scale, the midline for the outside shift, and convert the desired …
Preview problemWrite a cosine model from amplitude or extreme values, period, midline, and the x-value of a maximum or minimum
Choose trig functions to model periodic phenomena using amplitude, frequency, and midline.
An extreme is a natural phase anchor for a cosine model. The sign of the cosine coefficient determines whether that anchor is a maximum or minimum, while its magnitude gives …
Preview problemChoose the basic trigonometric model type from a periodic situation's starting position
Choose trig functions to model periodic phenomena using amplitude, frequency, and midline.
Translate the physical starting position into displacement from the situation's midline, then record whether that displacement initially rises or falls. Parent sine models begin on the midline, whereas parent cosine …
Preview problemInterpret every sinusoidal parameter with units
Choose trig functions to model periodic phenomena using amplitude, frequency, and midline.
Match the equation to standard sinusoidal form before interpreting any parameter. The outside coefficient and constant control amplitude, midline, and vertical range; the inside coefficient and shift control cycle length, …
Preview problemFind and verify sinusoidal maximum and minimum values
Choose trig functions to model periodic phenomena using amplitude, frequency, and midline.
A sinusoid's vertical envelope is centered at its midline and extends one amplitude in each direction. Read amplitude as the absolute value of the trig coefficient, then add and subtract …
Preview problemFind when a sinusoidal model reaches its first maximum or minimum
Choose trig functions to model periodic phenomena using amplitude, frequency, and midline.
First determine which trig-factor value produces the requested extreme, taking the outside coefficient's sign into account. Set the model's entire phase equal to every angle where that factor value occurs …
Preview problemCompare sinusoidal models with parallel feature fields
Choose trig functions to model periodic phenomena using amplitude, frequency, and midline.
Compare sinusoidal models one feature field at a time. Outside coefficients control vertical scale, outside constants control the common centerline, and inside coefficients and shifts control cycle length and phase. …
Preview problemWrite a sinusoidal model from maximum, minimum, period, and starting-position clues
Choose trig functions to model periodic phenomena using amplitude, frequency, and midline.
Build the model from vertical and horizontal evidence separately. The extreme values determine amplitude and midline, while the time from a maximum to the next minimum is only half a …
Preview problemDecide whether a periodic model is appropriate
Choose trig functions to model periodic phenomena using amplitude, frequency, and midline.
A periodic model needs more than an up-and-down appearance: identify a physical mechanism that recurs after a roughly stable time interval. Separate systematic cycles at different time scales from irregular …
Preview problemPredict an output value from a sine or cosine model
Choose trig functions to model periodic phenomena using amplitude, frequency, and midline.
Evaluate a sinusoidal model from the inside out. Substitute the requested input, simplify the entire angle, find the exact trig value, and only then apply the outside scale and vertical …
Preview problemIdentify the cross-section formed when a prism is sliced parallel to its base
Identify cross-sections of 3D objects and solids generated by rotating 2D objects.
Treat a prism as a stack of congruent layers parallel to its base. A slicing plane with that same orientation meets corresponding lateral edges at the same relative height, so …
Preview problemIdentify the cross-section of a right prism cut by a plane perpendicular to its base
Identify cross-sections of 3D objects and solids generated by rotating 2D objects.
A plane perpendicular to a right prism's base sweeps vertically through the solid. One pair of section edges follows the prism's height, while the other pair comes from the segment …
Preview problemClassify a fully specified cylinder cross-section
Identify cross-sections of 3D objects and solids generated by rotating 2D objects.
Classify a cylinder section from both the plane's orientation and the surfaces it meets. An interior plane parallel to the bases lies at one height, and the cylinder wall stays …
Preview problemIdentify the cross-section formed by slicing a cone
Identify cross-sections of 3D objects and solids generated by rotating 2D objects.
The cutting plane's relation to a cone's base determines the section family. A parallel cut stays at one height, where every point on the cone's boundary is the same distance …
Preview problemIdentify the cross-section formed when a plane slices a sphere
Identify cross-sections of 3D objects and solids generated by rotating 2D objects.
Every boundary point of a sphere is the same distance from its center. When the slicing plane contains that center, the intersection keeps the sphere's full radius within the plane; …
Preview problemIdentify the solid formed by revolving a rectangle around a parallel line
Identify cross-sections of 3D objects and solids generated by rotating 2D objects.
Track the entire filled rectangle, not just its boundary, as one side remains fixed. At each position along that axis, the perpendicular segment sweeps a filled disk whose radius is …
Preview problemDetermine cone dimensions from a rotated right-triangle region
Identify cross-sections of 3D objects and solids generated by rotating 2D objects.
Map each side of the right triangle by its relation to the rotation axis. The leg on the axis becomes the solid's axial dimension, the perpendicular leg sweeps the base …
Preview problemIdentify the solid formed by revolving a 2D figure around a line in its plane
Identify cross-sections of 3D objects and solids generated by rotating 2D objects.
Track both the curved boundary and the filled interior as the semicircular region turns about its diameter. The arc sweeps a boundary at a fixed distance from the midpoint, while …
Preview problemMatch a rotated 2D figure to the 3D solid it forms
Identify cross-sections of 3D objects and solids generated by rotating 2D objects.
Fix the stated side as the axis and examine crosswise segments of the filled rectangle. Each has the same length, so every one sweeps a filled disk of the same …
Preview problemIdentify the cross-section shape from a described plane slice of a 3D solid
Identify cross-sections of 3D objects and solids generated by rotating 2D objects.
Use both orientation and location to describe a cone's cross-section. A plane parallel to the base meets the cone at one constant height and preserves the base's rotational symmetry, while …
Preview problemIdentify a solid from a fully specified generating region and axis
Identify cross-sections of 3D objects and solids generated by rotating 2D objects.
Read the coordinate bounds relative to the exact rotation axis. For each fixed vertical coordinate, the filled horizontal segment begins on the axis and sweeps a disk whose radius is …
Preview problemIdentify the type of conic by inspecting squared terms and signs in the equation
Complete the square for general quadratic conic equations; identify and graph circles, ellipses, parabolas, or hyperbolas.
Classify a conic by inventorying its squared-variable terms before doing any graphing. One squared variable signals one family, opposite signs on two squares signal another, and same-sign squares narrow the …
Preview problemComplete squares and report every circle parameter
Complete the square for general quadratic conic equations; identify and graph circles, ellipses, parabolas, or hyperbolas.
Group the terms by variable and complete each square using the square of half its linear coefficient. Every amount added to form a perfect square must also be added to …
Preview problemComplete squares and extract ellipse axes
Complete the square for general quadratic conic equations; identify and graph circles, ellipses, parabolas, or hyperbolas.
Factor each quadratic coefficient before completing the square inside its variable group, then move the compensating constants carefully. Normalize the completed equation so the right side is one. The binomial …
Preview problem