All practice questions

Let the function $f$ be given by $f(x) = 3x - 5$. If a second function $g$ is defined in terms of $f$ as $g(x) = 2f(x + 1) + 4$, what is the value of $x$ for which $g(x) = 12$?

The preservation of wooden artifacts in waterlogged environments, such as ancient shipwrecks or peat bogs, presents a unique challenge to conservationists. The lack of oxygen in these settings prevents the growth of decay-causing bacteria, which would otherwise decompose organic matter. Consequently, the structural integrity of these fragile objects ______ largely intact over centuries, allowing researchers to study details that would typically be lost to time.

Which choice completes the text so that it conforms to the conventions of Standard English?

If the quadratic equation $x^2 - bx + 16 = 0$, where $b$ is a positive constant, has two real solutions such that one solution is $4$ times the other, what is the value of $b$?

While many cephalopods change their skin color and texture to blend into their surroundings, the mimic octopus (*Thaumoctopus mimicus*) employs a far more active defense mechanism by dynamically altering its physical shape to impersonate venomous marine animals like lionfish and sea snakes. Researchers initially hypothesized that this behavior was a generalized, hardwired reflex triggered by any nearby predator. ______ observations have revealed that the octopus selectively tailors its mimicry to the specific threat it encounters—for instance, adopting the shape of a predatory damselfish's natural enemy, the banded sea snake, only when attacked by damselfish.

Which choice completes the text with the most logical transition?

Text 1
Some biologists argue that the primary evolutionary benefit of a zebra’s stripes is thermoregulation. According to this model, the alternate absorption of heat by dark stripes and reflection by light stripes creates micro-convective air currents across the zebra's body. These currents supposedly cool the animal, allowing it to forage in intense equatorial heat where non-striped equids might suffer heat stress. Proponents of this view point to variations in stripe definition across different geographical regions, noting that stripe patterns are often most distinct in the hottest areas of a zebra's range.

Text 2
Experimental studies by Tim Caro and colleagues suggest that zebra stripes serve primarily to deter biting insects. By placing striped blankets on horses, researchers observed that tabanid flies approached striped surfaces at normal speeds but failed to decelerate properly, often bumping into the animals or veering away without landing. Caro argues that because biting flies transmit fatal diseases, the selective pressure to avoid them is far more critical to survival than thermoregulatory efficiency. Furthermore, he notes that several non-striped equid species thrive in equally hot African environments without experiencing significant heat stress.

Based on the passages, how would the author of Text 2 most likely respond to the geographic evidence cited by the proponents of the thermoregulation hypothesis in Text 1?

While preparing a presentation on historical writing materials, a student took the following notes:
* Papyrus, developed in ancient Egypt, is made from the fibrous pith of a river plant.
* Parchment, developed in the Mediterranean, is made from processed animal skins.
* Papyrus is highly sensitive to moisture and degrades quickly in damp environments.
* Parchment is much more durable and resistant to humidity than papyrus.
* Sheets of papyrus are usually written on only one side, while sheets of parchment can be written on both sides.

The student wants to contrast the durability and structural utility of papyrus and parchment. Which choice most effectively uses information from the given notes to accomplish this goal?

While preparing a presentation about dwarf planets, a student took the following notes:

* Eris is a dwarf planet located in the outer Solar System.
* It was discovered in January 2005 using images taken at the Palomar Observatory in California.
* The discovery was made by an astronomy research team led by Mike Brown.
* Eris is roughly the same size as Pluto but has $27\%$ more mass.

The student wants to emphasize the name of the scientist who led the discovery team. Which choice most effectively uses information from the given notes to accomplish this goal?

An agricultural drone sprays a liquid fertilizer at a constant rate of $120$ milliliters per second. The drone flies at a constant speed, covering $1.5$ hectares per hour. If the drone operates continuously at this rate, how many liters of fertilizer are sprayed per hectare?

(Note: $1$ liter = $1,000$ milliliters)

In medieval Europe, scribes favored iron-gall ink for its durability and deep black color, which was achieved through a chemical reaction between iron sulfate and tannic acid extracted from oak galls. Although the ink was highly resistant to fading, its acidic nature posed a significant threat over time. As the ink aged, it catalyzed the degradation of the parchment or paper underneath, a process known as ink-gall decay that often resulted in physical perforations where the writing once was. Consequently, modern conservators must employ specialized non-aqueous treatments to neutralize the acidity without dissolving the remaining ink. This allows libraries to preserve these historical documents for future scholarly analysis.

Which choice best describes the function of the underlined sentence in the text as a whole?

In the high-altitude alpine ecosystems of the Pamir Mountains, lichenometry serves as a critical tool for dating glacial retreats and environmental shifts. Specifically, the slow growth rate of crustose lichens, along with their high sensitivity to minute fluctuations in atmospheric chemistry, ______ researchers to reconstruct historical pollution patterns with remarkable chronological precision.

Which choice completes the text so that it conforms to the conventions of Standard English?

The polynomial function $p$ is defined by $p(x) = x^3 - 4x^2 - kx + 36$, where $k$ is a constant. In the $xy$-plane, the graph of $y = p(x)$ has an $x$-intercept at $(3, 0)$. What is the value of $k$?

For decades, linguistic anthropologists assumed that the structural features of a language had no influence on the spatial reasoning of its speakers. However, recent studies of languages that use absolute cardinal directions (like north and south) instead of relative terms (like left and right) have demonstrated that language can indeed shape cognitive mapping. By showing that speakers of these absolute languages maintain an orientation that is remarkably stable across unfamiliar environments, researchers have ______ the long-held belief that language structure and spatial cognition are completely separate.

Which choice completes the text with the most logical and precise word or phrase?

One of the solutions to the quadratic equation $x^2 + 8x + c = 0$, where $c$ is a constant, is $x = -4 + \sqrt{5}$. What is the value of $c$?

Which transition word or phrase, when inserted into the blank, logically establishes a relationship of similarity between the two research findings?

Roman dodecahedrons—small, hollow bronze objects with twelve pentagonal faces and circular holes of varying sizes—have puzzled archaeologists for centuries. Various hypotheses suggest they functioned as candle holders, surveying instruments, or even knitting tools. However, the discovery of a dodecahedron in a buried hoard alongside valuable coins and jewelry suggests that these objects held significant monetary or status value. This finding has led some researchers to propose that, regardless of their practical utility, dodecahedrons were prized possessions, possibly serving as symbols of wealth or prestige. Consequently, studies focusing solely on their mechanical utility may overlook the socio-economic context of their preservation.

Which choice best describes the function of the underlined sentence in the text as a whole?

The graph of the quadratic function $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, is a parabola in the $xy$-plane that passes through the points $(-3, 22)$ and $(9, 22)$. If the minimum value of $f(x)$ is $4$, what is the value of $f(1)$?

In the field of cognitive linguistics, the acquisition of sign languages by young children _______ a fascinating look into the human brain's natural capacity for grammar. Researchers have observed that deaf children exposed to sign language from birth hit linguistic milestones at the same rate as hearing children learning spoken languages. This suggests that the biological foundations for language development are independent of sensory modality.

Which choice completes the text so that it conforms to the conventions of Standard English?

A transformation is applied to the exponential function $f(x) = 2^x - 3$ to obtain a new function $g$, where $g(x) = -f(x - 1) + 2$. What is the $y$-intercept of the graph of $y = g(x)$ in the $xy$-plane?

Urban heat islands, characterized by elevated temperatures in city centers compared to rural surroundings, have been shown to accelerate the phenology of many plant species, causing them to leaf out earlier in the spring. However, researcher Elena Vance and colleagues observed that urban populations of the cherry plum (*Prunus cerasifera*) did not leaf out earlier than rural populations, despite experiencing warmer spring temperatures. Vance hypothesized that this anomaly occurs because *P. cerasifera* requires a minimum threshold of winter chilling to break bud dormancy. Because milder urban winters delay the completion of this chilling requirement, which choice most logically completes the text?

The height of a diver, in meters, above the pool surface $t$ seconds after leaving the diving board is modeled by the function $f(t) = -4.9(t - 1)^2 + 10$. If the diving board is moved so that the diver's trajectory is shifted $0.5$ seconds later in time and the maximum height is increased by $2$ meters, which of the following functions $g$ models the diver's new trajectory?