Skill: Solve problems using the algebra of functions. Modeling a Function: Numerical (data table) Algebraic (equation) Graphical Using Numerical Values: Look for a common difference. If the first difference is constant, the function is linear. If the second difference is constant, the function is quadratic. If the third difference is constant, the function is cubic. E : Determine the type of function using the common difference. Graph the function and write an equation for the function. 4 5 y 6 8 7 First difference 5 7 9 Second difference Second difference is constant, so it is quadratic. Graph (plot points): Because of the symmetry in a quadratic function, we can draw the entire function. Write the equation: Etending the table, we can find the y-intercept. 0, So the equation of the function is y -Intercept: the value of where the graph intersects the -ais; also called a zero or root Page of 4 Precalc Unit 0 Notes - Functions
Using an Algebraic Model: f 8 8 E : Find the zeros of the function algebraically and graph. Factor by grouping: f 8 f 8 Use the zero product property (A product of real numbers is zero if and only if at least one of the factors in the product is zero.): 8 0, 0 To sketch the graph, we can use the zeros and end behavior. Recall: In a cubic function, if a is positive, as, Zeros:,, f ; and as, f Read As: As approaches negative infinity, f of approaches negative infinity. As approaches infinity, f of approaches infinity.. Using a Graph: Watch for hidden behavior! E : Solve the equation 7 66 0 graphically. Graph on the calculator (Zoom Standard). Because this is a cubic function, we know by the end behavior that it will have one more zero. Etend the window on the -ais to 5 to 5. Window: 0,0, 0,0 Window: 5,5, 0,0 Solutions:,.5, Page of 4 Precalc Unit 0 Notes - Functions
E 4: Graph a scatter plot for gas prices and find an algebraic model. When will gas prices hit $8.00/gallon? The year is the independent variable. We will label 00 as and create a scatter plot. This appears to be an eponential model, so we will use EpReg to find an algebraic model. (Note: Store this in Y.) Use the graph of the algebraic model to estimate when gas prices will hit $8.00. y.9.59 Gas prices will hit $8.00 close to the year 0. Note: If a is a real number that solves the equation f 0, then these three statements are equivalent. a is a root (solution) of the equation f 0. a is a zero of y f. a is an -intercept of the graph of y f. Page of 4 Precalc Unit 0 Notes - Functions
You Try: Solve the equation algebraically and graphically. 9 QOD: What are two ways to solve an equation graphically? Page 4 of 4 Precalc Unit 0 Notes - Functions
Skills: Determine the domain and range of given functions and relations. Analyze the graph of a function for continuity. Describe the symmetries for the graph of a given relation. Compare values of etrema for a given relation. Find the vertical and horizontal asymptotes of the graph of a given relation. Function: a rule that assigns every element in the domain to a unique element in the range Function Notation: y f ; Read y equals f of Domain: the values of the independent variable () Range: the values of the dependent variable (y) Functions can be represented using a mapping diagram. E : Which of these mapping diagrams represents a function, where the domain represents the number of entrees ordered, and the range represents the amount paid? Taco Bell P.F. Changs 0 $60 0 $60 $00 0 $0 0 $80 Function: for every input, there is eactly one output Not a Function Think About It: What difference between the two restaurants would be the reason for one being a function and the other not? (tipping) Using a graph to determine if a relation is a function: Graphically, the vertical line test will tell if repeats. Vertical Line Test: If any vertical line intersects a graph at more than one point, then the graph is NOT a function. Page 5 of 4 Precalc Unit 0 Notes - Functions
E : Determine if the graphs are functions and eplain why or why not... Function: Vertical line does not intersect the graph at more than one point. Not a Function: Vertical line intersects the graph at more than one point. E : A function has the property that f 5? Find when the input,, equals 5: 5 Evaluate the function when : f f Determine domain and range from an equation: f for all real numbers. What is 5 7 E 4: Find the domain and range of the function. f Domain: The radicand cannot be negative, so 0. The denominator cannot equal zero, so. Range: The numerator can only be positive, and the denominator can be all real numbers. So the range is all real numbers Write the answers in interval notation. Domain: 0,, Range:, Continuity: a graph is continuous if you can sketch the graph without lifting your pencil Types of Discontinuity:. Removable (hole). Jump. Infinite Page 6 of 4 Precalc Unit 0 Notes - Functions
Symmetry: a graph of a function can be even, odd, or neither Even Function: a function that is symmetric about the y-ais If a function is even, then f f. Eample of an even function: y Odd Function: a function that is symmetric with respect to the origin If a function is odd, then f f. Eample of an odd function: y Find f E 5: Show whether the function is odd, even or neither. y. f f f f, so the function is ODD. E 6: Show whether the function is odd, even or neither. g Find f. f f, so the function is EVEN. f f Relative Etrema: the maima or minima of a function in a local area Absolute Etrema: the maimum or minimum value of a function in its domain Increasing Function: as the -values increase, the y-values increase Decreasing Function: as the -values increase, the y-values decrease Constant Function: as the -values increase, the y-values do not change Boundedness: A function is bounded below if there is some number b that is less than or equal to every number in the range of the function. A function is bounded above if there is some number B that is greater than or equal to every number in the range of the function. A function is bounded if it is bounded both above and below. Page 7 of 4 Precalc Unit 0 Notes - Functions
Asymptotes: Horizontal Asymptote The line y b, if f approaches b as approaches or. Recall: y lim f and y lim f Vertical Asymptote The line a, if f approaches or as approaches a from either direction. Limit: the value that Notation: f lim f a approaches as approaches a given value The limit as approaches a of f. (From both sides.) Calculating a Limit (as approaches positive or negative infinity): divide every term by the largest power of E 7: Find the following for the function f. Then graph the function. Domain, range, continuity, increasing/decreasing, symmetry, boundedness, etrema, asymptotes Domain: The denominator equals zero when, so the domain is,, Range: The function can take on all real values, so the range is all Reals.. Continuity: The function has an infinite discontinuity at. Increasing/Decreasing: Always decreasing Symmetry: f, so the function is neither even nor odd. Boundedness: The function is not bounded above or below. Etrema: None Vertical Asymptote: 0 lim lim lim 0 Horizontal Asymptote: 0 lim lim lim 0 y Page 8 of 4 Precalc Unit 0 Notes - Functions
You Try: Graph the function. Identify the domain, range, continuity, increasing/decreasing, symmetry, boundedness, etrema, and asymptotes. f 9 QOD: Identify a rule for finding the horizontal asymptotes of a function. Hint: Use the degrees of the numerator and denominator. Page 9 of 4 Precalc Unit 0 Notes - Functions
Skill: Sketch the graph of a polynomial, radical, or rational function. Teacher Note: For each of the basic functions, have students identify the domain, range, increasing/decreasing, symmetry, boundedness, etrema, and asymptotes.. Identity Function y. Squaring Function y. Cubing Function y 4. Square Root Function y 5. Natural Logarithm Function y ln 6. Reciprocal Function y Page 0 of 4 Precalc Unit 0 Notes - Functions
7. Eponential Function y e 8. Sine Function y sin 9. Cosine Function y cos 0. Absolute Value Function. Greatest Integer Function int y (Step Function). Logistic Function y e Page of 4 Precalc Unit 0 Notes - Functions
E : The greatest integer function is evaluated by finding the greatest integer less than or equal to the number. Evaluate the following if a) f f. The greatest integer less than or equal to is 0. b) f The greatest integer less than or equal to is. c) f The greatest integer less than or equal to is. d) f 9. The greatest integer less than or equal to 9. is 0. Discussing the Twelve Basic Functions: Domain: Nine of the basic functions have domain the set of all real numbers. The Reciprocal Function y has a domain of all real numbers ecept 0. The Square Root Function has a domain of 0,. The Natural Logarithm Function has a domain of 0,. Continuity: Eleven of the basic functions are continuous on their entire domain. The Greatest Integer Function y has jump discontinuities at every integer value. Note: The Reciprocal Function y has an infinite discontinuity at 0. However, it is still a continuous function because 0 is not in its domain. Boundedness: Three of the basic functions are bounded. The Sine and Cosine Functions are bounded above at and below at. The Logistic Function is bounded above at and below at 0. Symmetry: Three of the basic functions are even. Four of the basic functions are odd. The Squaring Function f is even because f. The Cosine Function f cos is even because f cos cos The Absolute Value Function f is even because f. The Identity Function f is odd because f.. The Cubing Function f is odd because The Reciprocal Function The Sine Function f sin f.. f is odd because f is odd because sin f sin. Asymptotes: Two of the basic functions have vertical asymptotes at 0. Three of the basic functions have horizontal asymptotes at y 0. Page of 4 Precalc Unit 0 Notes - Functions
The Reciprocal Function y and the Natural Logarithm Function y ln have vertical asymptotes at 0. The Reciprocal Function y, the Eponential Function y e, and the Logistic Function y have horizontal asymptotes at y 0. e Piecewise Function: A function whose rule includes more than one formula. The formula for each piece of the function is applied to certain values of the domain, as specified in the definition of the function., 0 E : Graph the piecewise function. f, 0 Graph each piece. The bolded portion is the first piece. Note: This is the Absolute Value Function! cos 0 E : Graph the piecewise function. 0 Graph each piece. The bolded portion is the first piece. Note: There is a jump discontinuity at 0. You Try: Graph the piecewise function. QOD: Which of the twelve basic functions are identical ecept for a horizontal shift? Page of 4 Precalc Unit 0 Notes - Functions
Skills: Solve problems using the algebra of functions. Find the composition of two or more functions. Function Operations Sum f g f g Difference f g f g Product fg f g Quotient f f g g g, 0 The domains of the new functions consist of all the numbers that belong to both of the domains of all of the original functions. E : Find the following for f and g a) f g. f g f g 4 b) f g f f g, g Note: Domain can also be written as,,. c) f g f g f g 4 E : Find the rule and domain if f 9 and g a) f g Domain of f: f g 9. 9 0 9 or, or, Domain of g: 0 Domain of f g :, b) g g g g Domain: 0, Page 4 of 4 Precalc Unit 0 Notes - Functions
g. E : Find the rule and domain if f and fg 9 4 9 9 a) fg or 0, Domain of f: 0 0 0 0 Critical Points:,0, Domain of g: Sign Chart:,0, Domain of fg:, + + b) ff ff 9 Recall: Domain of ff: 0, c) f / g Domain of f / g : f / g 0, Composition of Functions: The composition f of g uses the notations f g f g f g This is read f of g of. In the composition f of g, the domain of f intersects the range of g. The domain of the composition functions consists of all -values in the domain of g that are also g -values in the domain of f. E 4: Find the rule and domain for f and g a) f g f g = Domain: 5 0, 5 5, b) g f Domain: 0 0, 5. f g 5 5 5 g f g f Page 5 of 4 Precalc Unit 0 Notes - Functions
Decomposing Functions E 5: Find The inside function, f and g if f g 7. g is. So Solution: f 7 and g f g g 7. Implicitly-Defined Function: independent and dependent variables are on one side of the equation Implicit: y 6 Eplicit: y 6 Solve for y: E 6: Find two functions defined implicitly by the given relation. y 5 y 5 y y 5 or 5 y 5 Relation: set of ordered pairs Note: Functions are special types of relations. E 7: Describe the graph of the relation y 9. The graph of this relation is a circle with center 0,0 and radius. A circle is not a function because the equation relates two different y-values with the same -value. For eample, if 0, y or y. You Try: Find f and g if g f. Is your answer unique? QOD: Describe how to find the domain of a composite function. Page 6 of 4 Precalc Unit 0 Notes - Functions
Skills: Solve problems using parametric equations. Find the inverse of a given function. Compare the domain and range of a given function with those of its inverse. Parametric Equations: a pair of continuous functions that define the and y coordinates of points in a plane in terms of a third variable, t, called the parameter. E : Find y, determined by the parameters t for the function defined by the equations t y t,. Find a direct relationship between & y and indicate if the relation is a function. Then graph the curve. Create a table for t,, & y. t t y t 7 4 0 0 4 7 Solve for t and substitute to find a direct relationship between and y. t t Graph by plotting the points in the table. y t y y 4 Application Parametric Equations E : A stuntwoman drives a car off a 50 m cliff at 5 m/s. The path of the car is modeled by the equations t y t 5 & 50 4.9. How long does it take to hit the ground and how far from the base of the cliff is the impact? The car hits the ground when y 0. 0 50 4.9t t 0.04 t.94sec The distance from the base of the cliff is. t 5 5.94 79.85m Page 7 of 4 Precalc Unit 0 Notes - Functions
Parametric Equations on the Graphing Calculator E : Graph the situation in the previous eample on the calculator. Change to Parametric Mode. Type each equation into the Y= menu and choose an appropriate window. Teacher Note: Emphasize to students that this is the path of the car. Have students discuss the best viewing window for the graph. Inverse: An inverse relation contains all points ba, for the relation with all points ab,. Notation: If g is the inverse of Caution: f, then g f. f is NOT f to the power. It is the INVERSE of f. Inverses are reflections over the line y. The inverse of a relation will be a function if the original relation passes the Horizontal Line Test (a horizontal line will not pass through more than one point at a time). The composition of inverse functions equals the identity function, y. So, if g f, then f g g f. The domain of f is the range of f. The range of f is the domain of f. Find the composite functions f g, E 4: Confirm that f and g are inverses. f g and g f. f g g f One-to One Function: a function whose inverse is a function; must pass both the vertical and horizontal line tests Steps for Finding an Inverse Relation ) Switch the and y in the relation. ) Solve for y. Page 8 of 4 Precalc Unit 0 Notes - Functions
E 5: Find the inverse relation and state the domain and range. Verify your answer graphically and algebraically. f y The function can be written y. Switch and y. y Solve for y. y y y y y y y y f Domain of f:,, Range of f:,, Domain of f :,, Range of Verify graphically that this is the inverse (show that f :,, f is the reflection of f over the line y ). Verify algebraically using composite functions. 6 6 6 f f 6 6 6 f f You Try: For the parametric equations, find the points determined by integer values of t, t. Find a direct algebraic relationship between and y. Then graph the relationship in the y-plane. t, y t QOD: Eplain why the domain and range are switched in the inverse of a function. Page 9 of 4 Precalc Unit 0 Notes - Functions
Skill: Construct the graph of a function under a given translation, dilation, or reflection. Rigid Transformation: leaves the size and shape of a graph unchanged, such as translations and reflections Non-Rigid Transformation: distorts the shape of a graph, such as horizontal and vertical stretches and shrinks Vertical Translation: a shift of the graph up or down on the coordinate plane E : Graph y and y on the same coordinate plane. Describe the transformation. Vertical Shift UP units. Vertical Translation: y f +/ c is a translation up/down c units. E : Graph y transformation. and y 4 on the same coordinate plane. Describe the Horizontal Shift LEFT 4 units. Horizontal Translation: y f / c is a translation left/right c units. Reflection: a flip of a graph over a line y f is a reflection of y f y f is a reflection of y f Teacher Note: Have students plot points to see this relationship. across the -ais. across the y-ais. Page 0 of 4 Precalc Unit 0 Notes - Functions
y on E : Describe the transformations of y Reflection across the -ais. y ( ) Translate RIGHT unit. y Translate DOWN units. y. Then sketch the graph. Horizontal Stretches or Shrinks y f c y f c Vertical Stretches or Shrinks is a horizontal stretch by a factor of c of f if c is a horizontal shrink by a factor of c of f if c y c f is a vertical stretch by a factor of c of y c f is a vertical shrink by a factor of c of f if c f if c E 4: Graph the following on the same coordinate plane and describe the transformation of y. y, y, y y, y y is a vertical stretch of y by a factor of. y, y y is a vertical shrink of y by a factor of. Page of 4 Precalc Unit 0 Notes - Functions
E 5: Graph the following on the same coordinate plane and describe the transformation of y. y, y, y y, y y is a horizontal shrink of y by a factor of. Note: y y, y y is a horizontal stretch of y by a factor of. Note: A horizontal stretch causes a vertical shrink and a horizontal shrink causes a vertical stretch. E 6: The function shown in the graph is. f. Sketch the graph of y f Vertical Shrink by a factor of ½. Reflect over the -ais. Shift right units and up unit. You Try: Use the graph of y Solution:. Write the equation of the graph that results after the following transformations. Then apply the transformations in the opposite order and find the equation of the graph that results. Shift left units, reflect over the -ais, shift up 4 units. QOD: Does a vertical stretch or shrink affect the graph s -intercepts? Eplain why or why not. Page of 4 Precalc Unit 0 Notes - Functions
Skill: Set up functions to model real-world problems. Modeling a Function Using Data Eploration Activity: Create a table that relates the number of diagonals to the number of sides of a polygon. Use the graphing calculator to graph a scatter plot and find at least two regression equations and their r and r values. Note by the second differences that this is quadratic! n d 0 4 5 5 6 9 7 4 8 0 9 7 0 5 Teacher Note: If a student does not see r and r values on their calculator, use the command DiagnosticsOn in the Catalog. Also After the command QuadReg, type in Y (in the VARS menu) to have the function automatically appear in Y to graph. Correlation Coefficient (r) & Coefficient of Determination (r ): The closer the absolute value is to, the better the curve fits the data. E : Find a regression equation and its r and r values. Then predict the population in 00. Let 0 be the year 000 and make a scatter plot. y 0.084.4 y 0 0.084 0.4.0 million Page of 4 Precalc Unit 0 Notes - Functions
Modeling Functions with Equations E : A beaker contains 400 ml of a 5% benzene solution. How much 40% benzene solution must be added to produce a 5% benzene solution? Write the equation that models the problem: 0.5400 0.4 0.5400 Solve the equation: 60 0.4 40 0.5 0.05 80 600mL E : An open bo is formed by cutting squares from the corners of a 0 ft by 4 ft rectangular piece of cardboard and folding up the flaps. What is the maimum volume of the bo? Let be the length of the sides of the squares cut from the corners. Write the volume V of the bo as a function of. 4 0 Graph V on the calculator and find the maimum value. V lwh The maimum volume is approimately 0 ft. You Try: How many rotations per second does a 4 in. diameter tire make on a car going 5 miles per hour? QOD: Describe how to determine which regression equation to use on the graphing calculator to represent a set of data. Page 4 of 4 Precalc Unit 0 Notes - Functions