Functions Project Core Precalculus Extra Credit Project

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1 Name: Period: Date Due: 10/10/1 (for A das) and 10/11/1(for B das) Date Turned In: Functions Project Core Precalculus Etra Credit Project Instructions and Definitions: This project ma be used during the first nine-weeks grading period to replace our lowest test grade with the grade ou receive on this project (as long as the project grade is higher than the test grade.) Fill out all requested information for each function, and accuratel graph the function. Do not change the scale on the graphs. Each bo represents one unit. (Remember to put arrows on the ends of our graphs when a graph etends beond the coordinate plane provided.) If a function does not have one of the characteristics listed, write N/A or none, for not applicable. You ma use a graphing calculator to help ou, but be familiar with the methods for finding these characteristics without a calculator*. There will not be an projects accepted late!!!!! Parent Function: the most basic function in a famil of functions Choose from the following list of parent functions: constant, linear, absolute value, quadratic, square root, cubic, cube root, power, rational, eponential growth, eponential deca, logarithmic, or greatest integer Equation: a statement that two epressions are equal If provided with the characteristics of the function, or the graph of the function, use the information to write an equation for the function in the form f ( ) =. Domain: the set of input values, or values, of a function List the possible -values for the function in interval notation. Be sure to write the smaller number before the larger number in interval notation. You ma use R to represent the set of all real numbers. You ma use to represent the set of all integers. Remember to look for values for which the function will not be defined, and restrict the domain accordingl (e: division b zero, even roots of negative numbers, etc.) Range: the set of output values, or values, of a function List the possible -values for the function in interval notation. Be sure to write the smaller number before the larger number in interval notation. You ma use R to represent the set of all real numbers. You ma use to represent the set of all integers. Analzing the left and right hand end behaviors will help ou. Looking for maimums and minimums will also help. -intercept: the coordinates of a point where a graph intersects the -ais Substitute zero in for, then solve for to find the -coordinates of the -intercepts. Find eact answers, or round to the thousandths place. -intercept: the coordinates of a point where a graph intersects the -ais Substitute zero in for, then solve for to find the -coordinates of the -intercepts. Find eact answers, or round to the thousandths place. Relative Minimum: the coordinates of the lowest point on the graph of a function, anwhere the function starts out decreasing, and then increases Look at the graph for low points, occurring at a turning point. *Note: You need to use our calculator for this. Put function in Y =. Press nd CALC, 3 for minimum, arrow to the left of the minimum point, press enter, arrow to the right of the point, press enter twice. Record the coordinates of the minimum point to the nearest thousandth. Relative Maimum: the coordinates of the highest point on the graph of a function, anwhere the function starts out increasing, and then decreases Look at the graph for high points, occurring at a turning point. *Note: You need to use our calculator for this. Put function in Y =. Press nd CALC, 4 for maimum, arrow to the left of the maimum point, press enter, arrow to the right of the point, press enter twice. Record the coordinates of the minimum point to the nearest thousandth. Smmetr with respect to the -ais: for ever point (, ) on the graph of a function, the point (, -) is also on the graph The graph will have the -ais as a line of smmetr. Replacing with - in the equation ields an equivalent equation. Use the word -ais. Smmetr with respect to the -ais: for ever point (, ) on the graph of a function, the point (-, ) is also on the graph The graph will have the -ais as a line of smmetr. Replacing with - in the equation ields an equivalent equation. Use the word -ais. Smmetr with respect to the origin: for ever point (, ) on the graph of a function, the point (-, -) is also on the graph If ou were to put our pencil down on the origin, then turn the graph over 180, the graph would look eactl the same. Replacing with -, and with -, in the equation ields an equivalent equation. Use the word origin. Even: a function is even if its graph is smmetric with respect to the -ais Odd: a function is odd if its graph is smmetric with respect to the origin Neither: If the function is neither even nor odd, write neither. Vertical Asmptote: a vertical line that a graph approaches more and more closel For some functions, ou will need to closel eamine the graph to find the asmptotes. To find vertical asmptotes of a rational function, set the denominator of the function equal to zero and solve for. The resulting " =" equation will be the equation of the vertical asmptote(s).

2 Horizontal Asmptote: a horizontal line that a graph approaches more and more closel For some functions, ou will need to closel eamine the graph to find the asmptotes. To find horizontal asmptotes of a rational function, use the following rules: 1) If the degree of the numerator is less than the degree of the denominator, the line = 0 is a horizontal asmptote. a ) If the degree of the numerator is equal to the degree of the denominator, the equation = is the equation of the horizontal asmptote, where a is the leading b coefficient of the numerator and b is the leading coefficient of the denominator 3) If the degree of the numerator is greater than the degree of the denominator, the graph has no horizontal asmptotes. Continuous: a function whose graph is unbroken over its domain If the graph can be drawn without picking up our pencil, it is continuous. Discontinuous: a function whose graph is broken over its domain If the graph cannot be drawn without picking up our pencil, it is discontinuous. Inverse of a Function: functions f and g are inverses provided that f ( g( ) ) line = If the inverse is not a function, write N/A. To find the inverse of a function follow these steps: 1) Replace f ( ) with. ) Switch and. 3) Solve for. 4) Replace with '( ) f. = and ( ( )) g f = ; additionall, the graphs are smmetric of each other across the Increasing: a function is increasing over an interval if the -values increase as the -values increase from left to right on the graph, "the graph is rising from left to right" List the possible -values over which the graph is increasing in interval notation. Use parentheses at the turning points, thereb not including them in the interval. Be sure to write the smaller number before the larger number in interval notation. You ma use R to represent all real numbers. Decreasing: a function is decreasing over an interval if the -values decrease as the -values increase from left to right on the graph, "the graph is falling from left to right" List the possible -values over which the graph is increasing in interval notation. Use parentheses at the turning points, thereb not including them in the interval. Be sure to write the smaller number before the larger number in interval notation. You ma use R to represent all real numbers. Constant: a function is constant over an interval if the -values remain the same value as the -values increase from left to right on the graph, "the graph is constant, or flat, from left to right" List the possible -values over which the graph is increasing in interval notation. Use parentheses at the turning points, thereb not including them in the interval. Be sure to write the smaller number before the larger number in interval notation. You ma use R to represent all real numbers. Transformation: a change in a graph's size, shape, position, or orientation Choose from the following list of transformations. (Eamples are listed using the parent function ( ) For an real number c, greater than zero, Vertical Shift Upward: f ( ) + c f ( ) = + Vertical Shift Downward: f ( ) c f ( ) = Horizontal Shift to the Right: f ( c) f ( ) = ( ) Horizontal Shift to the Left: f ( + c) f ( ) = ( + ) Reflection across the -ais: f ( ) f ( ) = Reflection across the -ais: f ( ) f ( ) = ( ) Vertical Stretch: cf ( ), where c is greater than 1 f ( ) = 1 f 1 = f = Vertical Shrink: cf ( ), where c is between 0 and 1 f ( ) = Horizontal Stretch: f ( c ), where c is between 0 and 1 ( ) Horizontal Shrink: f ( c ), where c is greater than 1 ( ) ( ) f = ):

3 f = + 7 1) Parent Function Name: Equation: ( ) ( ) 3 -intercept(s): -intercept(s): Relative Minimum(s): Relative Maimum(s): Smmetr: Even / Odd / Neither: Vertical Asmptote(s): Horizontal Asmptote(s): ) Parent Function Name: Equation: f ( ) 1 = -intercept(s): -intercept(s): Relative Minimum(s): Relative Maimum(s): Smmetr: Even / Odd / Neither: Vertical Asmptote(s): Horizontal Asmptote(s): 3) Parent Function Name: Equation: -intercept(s): -intercept(s): Relative Minimum(s): Relative Maimum(s): Smmetr: Even / Odd / Neither: Vertical Asmptote(s): Horizontal Asmptote(s):

4 4) Parent Function Name: _cube root Equation: Domain: R Range: R -intercept(s): (0, 0) -intercept(s): (0, 0) Relative Minimum(s): N/A Relative Maimum(s): N/A Smmetr: origin Even / Odd / Neither: odd Vertical Asmptote(s): N/A Horizontal Asmptote(s): _N/A Continuous / Discontinuous: _continuous Inverse: _ ( ) 1 3 f = Increasing: N/A Decreasing: R Constant: N/A reflection across the -ais 5) Parent Function Name: linear Equation: Domain: R Range: R -intercept(s): 3,0 -intercept(s): _(0, -3) 7 Relative Minimum(s): N/A Relative Maimum(s): N/A Smmetr: N/A Even / Odd / Neither: Neither Vertical Asmptote(s): N/A Horizontal Asmptote(s): N/A Continuous / Discontinuous: _continuous Inverse: f ( ) = 7 Increasing: R Decreasing: N/A Constant: N/A vertical stretch b 7, vertical shift down 3 units 6) Parent Function Name: Equation: ( ) 1 f = log 3 -intercept(s): -intercept(s): Relative Minimum(s): Relative Maimum(s): Smmetr: Even / Odd / Neither: Vertical Asmptote(s): Horizontal Asmptote(s):

5 7) Parent Function Name: Equation: f ( ) 1 = + 3 -intercept(s): -intercept(s): Relative Minimum(s): Relative Maimum(s): Smmetr: Even / Odd / Neither: Vertical Asmptote(s): Horizontal Asmptote(s): 8) Parent Function Name: Equation: -intercept(s): -intercept(s): Relative Minimum(s): Relative Maimum(s): Smmetr: Even / Odd / Neither: Vertical Asmptote(s): Horizontal Asmptote(s): 9) Parent Function Name: Equation: -intercept(s): -intercept(s): Relative Minimum(s): Relative Maimum(s): Smmetr: Even / Odd / Neither: Left-Hand Behavior: Right-Hand Behavior: Vertical Asmptote(s): Horizontal Asmptote(s):

6 10) Parent Function Name: Equation: ( ) 1 f = -intercept(s): -intercept(s): Relative Minimum(s): Relative Maimum(s): Smmetr: Even / Odd / Neither: Vertical Asmptote(s): Horizontal Asmptote(s): 11) Parent Function Name: square root Equation: Domain: [ 5, ) Range: [ 0, ) -intercept(s): (5, 0) -intercept(s): N/A Relative Minimum(s): N/A Relative Maimum(s): N/A Smmetr: N/A Even / Odd / Neither: neither Vertical Asmptote(s): N/A Horizontal Asmptote(s): N/A 1 Continuous / Discontinuous: _continuous Inverse: f ( ) = Increasing: ( 5, ) Decreasing: N/A Constant: N/A vertical stretch b 3, horizontal shift 5 units right