How to Write Quadratic Function in Standard Form

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fra angelico, annunciation The Annunciation (Fig. 38.12) by Dominican friar and painter Fra Angelico (c. 1400– 1455) reveals a much more modest image than Gentile’s painting. In 1436, Fra Angelico moved to the monas- tery of San Marco. Soon thereafter Cosimo de’ Medici commissioned him to paint a series of frescoes there. The Annunciation fresco, located in the upstairs corridor of the monks’ dormitory, presents a simplified version of the story, in which the Archangel Gabriel informs the Virgin Mary that she is to be the mother of Jesus. The spare setting is rendered in perspective and with minimal, vernacular (everyday) details. The angel Gabriel kneels before a young Virgin Mary, who is seated on a simple wooden stool in the loggia of an enclosed garden. The thin, elegantly posed figures have calm facial expressions. They humble themselves with arms crossed and a slight bow toward each other. Their drapery is modeled in a limited color palette illumi- nated by a light source coming from behind Gabriel. There is a modest use of gold in Gabriel’s pink robe, while the Virgin’s cloak is an azurite blue over an off- white dress that blends into her skin tone. Gabriel’s beautiful wings in stripes of rich color lead the eye to the dark-green meadow behind him. The architecture frames the scene with whitewashed walls and slim Corinthian columns braced together with iron bars to reinforce the linear perspective of the cubed space, cre- ating a sense of harmony and order. A door opens into a dormitory cell beyond the portico; the cell is empty except for one window set high on the wall. The monastery had formerly belonged to Benedictines, followers of the early medieval St. Benedict who estab- lished the rules of monastic life, and was transferred to the Dominicans and rebuilt by Michelozzo di Bartolommeo under the sponsorship of Cosimo de’ Medici as part of the expansion of the area that also included Michelozzo’s Medici Palace (see Fig. 38.1). Michelozzo’s renovated mon- astery shows a spare Classicism similar to Fra Angelico’s frescoes, appropriate for the monks, who had taken a vow of poverty, as well as for Cosimo, who maintained a private cell to escape his worldly affairs. Because the friars—not the general public—were the primary intended audi- ence for the frescoes, no didactic details were needed to tell the story they already knew. Instead, the images were meant to inspire piety within the silence of a clois- tered monastery. The Language of Classicism The revival of Classicism not only played an impor- tant role in art patronage and the greater emphasis on naturalism, but it also expanded the artistic language to include a fuller symbolic and aesthetic repertoire. The humanist scholar Giovanni Pico della Mirandola (1463–1494) wrote Oration on the Dignity of Man in 1486 to explain humankind’s capacity for greatness through a quest for knowledge. Pico sought to bring together diverse branches of thought including Platonism, tied to the ancient philosopher Plato, and Christianity, which he argued held similar truths, and these ideas promoted an intellectual approach to art in the Renaissance. brunelleschi, the old sacristy During the period in which Brunelleschi was working on the Florence Cathedral dome, he also constructed several churches, chapels, and a hospital in Florence. Citizens marveled at the new Classical style he introduced. The Old Sacristy (Fig. 38.13), commissioned in 1421 by Cosimo de’ Medici, is an excellent example of Brunelleschi’s interest in Classicism adapted for a Christian building. This small, square room where priests prepared for services, accessed from inside the church of San Lorenzo, was designed with a dome set on top of a cube, in which the altar wall is subdivided into three rectangular sections of equal length by Corinthian fluted pilasters and molding made from pietra serena, a gray-green stone. Stone gridlines divide the floor into smaller squares. An elevated, square altar is located on the wall opposite the entrance, while an umbrella, or ribbed, dome with twelve ribs rises above the room. Triangular pendentives spring from the entablature to create arches that mark the transition from the square plan to the circular dome. In Classical antiquity, domes symbolized the heavenly realm, and they were used for mausolea, imperial homes, and civic buildings such as bath houses. The same sym- bolism appears in the Old Sacristy, which was used for Medici family tombs. Donatello completed the interior painted stucco reliefs, including corner medallions with scenes of the Four Evangelists (Matthew, Mark, Luke, and John, authors of the New Testament Gospels) above the Medici family coat-of-arms. The numbers three, four, and twelve are embedded in the chapel’s design principles in the form of the triangle, square, and ribs of the dome circle. These numbers also symbolize the Trinity (the Father, the Son, and the Holy Ghost), the Four Evangelists, and the Twelve Apostles (the disciples of Jesus). According to the ancient Greek philosopher Pythagoras (c. 570–c. 495 BCE), who was a major influence on Plato, numbers have symbolic meaning that can generate harmony, as found in musical ratios and proportion studies. For Brunelleschi, Classical numerology, together with medieval number symbolism, could be used to create a rich symbolic architectural language where harmony is achieved through numbers. donatello, david Art had long been commissioned by religious institutions, wealthy families, and civic groups for the public promotion of their ideology, but the rise of domestic art came about with the development of larger private family homes in the Renaissance. Historians studying household inventories have noted an increase in material possessions for the home during this time. One such possession is Donatello’s bronze David (Fig. 38.14), probably made in the 1440s for the courtyard of the Medici Palace. It was the first freestanding nude sculpture since antiquity, and it reflects Cosimo de’ Medici’s Classical interests. In the Old Testament, the young shepherd David, who eventually became King of the Israelites, was challenged to fight the undefeated Philistine giant warrior Goliath. David was victorious, and this story became a popular subject in Florence, a city that championed its morals and intellect over brute strength and its cultural superiority over the lingering feudal-era military rule epitomized by rivals such as Milan. Vasari noted an inscription beneath the figure, now gone, that stated: “Behold, a boy overcame the great tyrant. Conquer, o citizens!” David stands in a Classical curvilinear pose, looking down upon the severed head of Goliath lying sideways on the ground in front of his sandaled feet. The scene is not violent. The nude David wears a cap decorated with laurel (denoting both victory and perseverance) and holds a heavy sword in one hand while anchoring his other hand against his swayed hips. A feather from Goliath’s helmet rests sensually against the inside of David’s thigh. Rather than a heroic, victorious nude, here David is young, with soft flesh, and many scholars have speculated on the rationale for such a seemingly erotic version of this biblical hero. It is possible that Donatello was inspired by Plato, who argued in his Symposium that a person’s physical beauty can inspire the viewer to contemplate divine love, which is represented in the supreme, ideal beauty of the youthful male figure. 38.14 Donatello, David, c. 1440. Bronze, height 5 ft. 2⅛ in. (1.58 m). Museo Nazionale del Bargello, Florence, Italy. zanobi di domenico, nerli cassone In addition to sculptures, artists created and decorated a wide range of objects for the home, including furniture, tapestries, and ceramics. Leon Battista Alberti’s text On the Family (Libri della famiglia) from 1440 provides a description of the life of a wealthy homeowner and, more importantly, a glimpse into the elusive private lives of women at this time when men dominated public spaces and shaped cultural discourse. A young woman brought a dowry into her arranged marriage and was responsible for the upkeep of the home and the early moral education of the children, while her husband oversaw the family business. A wooden marriage chest for the personal goods of a bride is called a cassone. In September 1472, the Florentine merchant Lorenzo Morelli paid Zanobi di Domenico (active c. 1464–74) 21 florins to create a pair of cassoni in anticipation of his arranged marriage to Vaggia Nerli, who brought a large dowry from her wealthy family. This cassone (Fig. 38.15, p. 639) was painted by Zanobi and two other artists. It is a rare surviving example, with its spal- liera (backboard) intact, as well as documentation of its original ownership. Such rich furnishings were often decorated with hunting scenes, jousting tournaments, and stories from Classical antiquity, and this cassone illustrates a scene from a history book of ancient Rome written by Titus Livius, known as Livy (c. 64 BCE–c. 17 CE), which highlights an example of brave patriotism. Such scenes provided a new repertoire of moralizing stories to remind the bride and groom of the importance of family and societal loyalty. antonio del pollaiuolo, hercules and antaeus Another domestic work, this one found in the Palazzo Medici, is Antonio del Pollaiuolo’s (c. 1431–1498) Hercules and Antaeus (Fig. 38.16), from around 1475. Standing about 18 inches tall, this small bronze statuette is a versatile, portable object that can be turned and viewed from all sides. Pollaiuolo was interested in movement, specifically of the human body, and here he shows the Greco-Roman mythological wrestling match between Antaeus, the son of an earth goddess, and Hercules, the son of Zeus. Both were known for their physical strength, but Antaeus derived his strength from the earth, so Hercules had to hoist him off the ground to gain the upper hand. Hercules’ spine bends dramatically as he adjusts his weight backward and wraps his arms around Antaeus, while Antaeus cries in agony and tries to push Hercules away. Their bodies are tense and angular, with bulging calf muscles and exaggerated poses. Antaeus’s legs flail in the air, encouraging viewers to look at the sculpture from multiple vantage points, thereby solving the problem of the one-point view. Such statuettes reflect a growing interest in small domestic art, featuring Classical sub- jects, that could be collected, studied, and displayed in various locations in the home. sandro botticelli, birth of venus Classical sub- jects also appear in Sandro Botticelli’s (1444–1510) Birth of Venus (Fig. 38.17), dated around 1482. This large painting of Venus, the goddess of love and beauty, was created in tempera on canvas, and may have been commissioned by the Medici. Canvas paintings were rare at this time, and generally they were found not in urban palaces but in villas (country houses), where this painting could have functioned as a less expensive alternative to a tapestry. Although its original location is not known, Vasari saw this painting at a Medici villa in the 1530s. Large mythological paintings and nude female images were unprecedented in Italy, yet they provide a subject suited to wealthy families such as the Medici, who sought to promote their humanist aspirations through such art- works: In Birth of Venus, the display of this nude, Classical figure expresses Platonic notions about the elevation of physical love into a higher spiritual or divine love, through an artistic idealization of the female form. Thus, while a naked female figure would have been unacceptable by Renaissance standards, a nude goddess of love, with its detachment from earthly concerns, was deemed intel- lectually uplifting. The scene in Birth of Venus, perhaps taken from a poem written by humanist scholar Angelo Poliziano (1454–1494) and described in an ancient Greek Homeric hymn, shows Venus (named Aphrodite in ancient Greece), after she is born full-grown, standing on a large shell that washes up on the shore of the Mediterranean island of either Cyprus or Cythera, guided by the wind, blown by Zephyrus, who holds the nymph Chloris. A goddess of spring, sometimes identified as the Roman garden goddess Pomona, pre- pares to cover Venus with a floral robe, a suggestion of her modesty. Venus was frequently depicted in antiquity, and Botticelli echoes the modest pose of the Aphrodite of Knidos by Praxiteles of Athens (see Fig. 15.1), a copy of which he could have seen in Rome. Botticelli was less interested in the spatial developments of the Renaissance and more focused on graceful, lyrical figures set against a flat background to create a perfect harmony of parts, epitomizing Platonic ideas of love and beauty. While Florence was the center of the artistic inno- vations that characterize the Renaissance, many other city-states began to use art to assert political power and cultural authority, resulting in rich artistic communities across all of Europe.