New SAT Math Workbook

Petersonscom additional geometry topics data analysis

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Unformatted text preview: ve). To answer these questions, you do not need to know the equations that define such graphs; simply apply your knowledge of the xy-coordinate system and, for some questions, function notation (see Chapter 15). Example: The figure above shows the graph of a certain equation in the xy-plane. The graph is a circle with center O and circumference 6π. At how many different values of y does x = –7.5 ? (A) 0 (B) 1 (C) 2 (D) 4 (E) Infinitely many Solution: The correct answer is (C). First, find the circle’s radius from its circumference: C = 6π = 2πr; r = 3. Since the circle’s center (0) lies at (–5,–6), the minimum value in the domain of x is –8. In other words, the left-most point along the circle’s circumference is at (–8,–6), 3 units to the left of O. Thus, the graph of x = –7.5, which is a vertical line passing through (–7.5, 0), intersects the circle at exactly two points. That is, when x = –7.5, there are two different corresponding values of y. Other questions on the new SAT will involve transformations of linear and quadratic functions and the effect of transformations on the graphs of such functions. The function f(x) is transformed by substituting an expression containing the variable x for x in the function — for example: If f(x) = 2x, then f(x + 1) = 2(x + 1), or 2x + 2 Transforming a function alters the graph of the function in the xy-plane. The effect of a transformation might be any of the following: * To move, or translate, the graph (either vertically, horizontally, or both) to another position in the plane * To alter the slope of a line (in the case of a linear function) * To alter the shape of a curve (in the case of a quadratic function) 288 Chapter 16 For example, if f(x) = x, then f(x + 1) = x + 1. In the xy-plane, the graph of f(x) = x (or y = x), is a line with slope 1 passing through the origin (0,0). The effect of transforming f(x) to f(x + 1) on the graph of f(x) is the translation of the line one unit upward. (The y-intercept...
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