NAME DATE PERIOD 1-5 Study Guide and Intervention

1 Lesson 1-5 name date PERIOD Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. PDF PassChapter 1 27 Glencoe Precalculus1-5 Parent Functions A parent function is the simplest of the functions in a FunctionFormNotesconstant functionf(x) = cgraph is a horizontal lineidentity functionf(x) = xpoints on graph have coordinates (a, a)quadratic functionf(x) = x2graph is U-shapedcubic functionf(x) = x3graph is symmetric about the originsquare root functionf(x) = x graph is in first quadrantreciprocal functionf(x) = 1 x graph has two branchesabsolute value functionf(x)

2 = | x |graph is V-shapedgreatest integer functionf(x) = x defined as the greatest integer less than or equal to x; type of step function Describe the following characteristics of the graph of the parent function f(x) = x3 : domain, range, intercepts, symmetry, continuity, end behavior, and intervals on which the graph is increasing/decreasing. The graph confirms that D = {x | x } and R = {y | y }.The graph intersects the origin, so the x-intercept is 0 and the y-intercept is 0. It is symmetric about the origin and it is an odd function: f(-x) = -f(x). The graph is continuous because it can be traced without lifting the pencil off the paper.

3 As x decreases, y approaches negative infinity, and as x increases, y approaches positive infinity. lim x - f(x) = - and lim x f(x) = The graph is always increasing, so it is increasing for (- , ).ExerciseDescribe the following characteristics of the graph of the parent function f(x) = x2 : domain, range, intercepts, symmetry, continuity, end behavior, and intervals on which the graph is increasing/decreasing. Study Guide and InterventionParent Functions and TransformationsExampleyx0f(x)= 2710/5/09 10:28:21 PM10/5/09 10:28:21 PMCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, date PERIOD PDF PassChapter 1 28 Glencoe Precalculus1-5 Transformations of Parent Functions Parent functions can be transformed to create other members in a family of graphs.

4 Translationsg(x) = f(x) + k is the graph of f(x) units up when k > units down when k < (x) = f(x - h) is the graph of f(x) units right when h > units left when h < (x) = -f(x) is the graph of f(x)..reflected in the (x) = f(-x) is the graph of f(x)..reflected in the (x) = a f(x) is the graph of f(x)..expanded vertically if a > vertically if 0 < a < (x) = f(ax) is the graph of f(x)..compressed horizontally if a > horizontally if 0 < a < 1. Identify the parent function f(x) of g(x) = -x - 1, and describe how the graphs of g(x) and f(x) are related. Then graph f(x) and g(x) on the same axes. The graph of g(x) is the graph of the square root function f(x) = x reflected in the y-axis and then translated one unit down.

5 ExercisesIdentify the parent function f(x) of g(x), and describe how the graphs of g(x) and f(x) are related. Then graph f(x) and g(x) on the same axes. 1. g(x) = x+ 4 2. g(x) = 2x2- 4 Study Guide and Intervention (continued)Parent Functions and TransformationsExampleyx0g(x)= -x-1f(x)= xyx048 4 848 8 4yx84 4 848 8 2810/5/09 10:28:28 PM10/5/09 10:28:28 PMCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, 1-5 name date PERIOD Pdf PassChapter 1 29 Glencoe Precalculus1-5 1.

6 Use the graph of f(x) = x to graph g(x) = x + 3 + 1. yx0 2. Use the graph of f(x) = x to graph g(x) = -|2x|. yx0 3. Describe how the graph of f(x) = x2 and g(x) are related. Then write an equation for g(x). 4. Identify the parent function f(x) of g(x) = 2|x + 2| - 3. Describe how the graphs of g(x) and f(x) are related. Then graph f(x) and g(x) on the same axes. 5. Graph f(x) = yx0 6. Use the graph of f(x) = x3 to graph g(x) = (x + 1)3 . yx0 PracticeParent Functions and Transformations yx0 yx2468 4 6 82468 8 6 4 -1 if x -31 + x if -2 < x 2. x if 4 x 293/22/09 5:51:57 PM3/22/09 5:51:57 PMCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, date PERIOD Pdf PassChapter 1 30 Glencoe Precalculus1-5 1.

7 AREA The width w of a rectangular plot of land with fixed area A is modeled by the function w( ) =A , where is the If the area is 1000 square feet, describe the transformations of the parent function f(x) =1 xused to graph w(x). b. Describe a function of the length that could be used to find a minimum perimeter for a given areac. Is the function you found in part ba transformation of f(x)? Find the minimum perimeter for an area of 1000 square feet. 2. GOLFThe path of the flight of a golf ball can be modeled by h(x) =- 1 10x2+2x, where h(x) is the distance above the ground in yards and x is the horizontal distance from the tee in Describe the transformation of the parent function f(x) =x2 used to graph h(x).

8 B. Suppose the same shot was made from a tee located 10 yards behind the original tee. Rewrite h(x) to reflect this TAXES Graph the tax rates for the different incomes by using a step HORIZONThe function f(x) = can be used to approximate the distance to the apparent horizon, or how far a person can see on a clear day, where f(x) is the distance in miles and x is the person s elevation in feet. a. How does the graph of f(x) compare to the graph of its parent function?b. The function g(x) = x is also used to approximate the distance to the horizon. How does the graph of g(x) compare to the graph of its parent function?Word Problem PracticeParent Functions and Transformations Tax Rate (%)2010304050 Taxable Income(thousands)30 60 90 120 150 180 210 240 270 300 Source: Information Please Almanac Income Tax Rates for a Couple Filing JointlyLimits of Taxable Income ($)Tax Rate (%)0 to 41,200 1541,201 to 99,600 2899,601 to 151,750 31151,751 to 271,050 36271,051 and up 303/22/09 5:52:06 PM3/22/09 5:52.

9 06 PMCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, date PERIOD Pdf PassChapter 1 32 Glencoe PrecalculusOperations with Functions Two functions can be added, subtracted, multiplied, or divided to form a new function. For the new function, the domain consists of the intersection of the domains of the two functions, excluding values that make a denominator equal to zero. Given f(x) = x2 - x - 6 and g(x) = x + 2, find each function and its domain. a.

10 (f + g)(x) (f + g)x = f(x) + g(x) = x2 - x - 6 + x + 2 = x2 - 4 The domains of f and g are both (- , ), so the domain of (f + g) is (- , ). b. ( f g ) (x) ( f g ) x = f(x) g(x) = x2 - x - 6 x + 2 = (x - 3)(x + 2) x + 2 = x - 3 The domains of f and g are both (- , ), but x = -2 yields a zero in the denominator of ( f g ) . So, the domain is {x | x -2, x }. Given f(x) = x2 - 3 and g(x) = 1 x , find each function and its domain. a. (f - g)(x) (f - g)x = f(x) - g(x)= x2 - 3 - 1 x The domain of f is (- , ) and the domain of g is ( , 0) (0, ), so the domain of (f - g) is ( , 0) (0, ).