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Regents Review #2 Functions f(x) = 2x – 5 g(x) = (½) x y = ¾ x f(x) = 1.5(4) x Linear & Exponential

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Functions What is a function? A relation in which every x-value(input) is assigned to exactly one y-value (output) Which relation represents a Function? xy xy FunctionNot a Function 2 6 7

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Functions We can recognize functions using the vertical line test Vertical Line Test: If a graph intersects a vertical line in more than one place, the graph is not a function Which graph represents a function? Function Not a function

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Functions Functions can be written using function notation f(x) is read f of x Example: f(x) = 2x – 3 is the same as y = 2x – 3 x: input f(x): output Evaluating Functions: Find f(-10) f(-10) = 2(-10) – 3 f(-10) = -20 – 3 f(-10) = -23 (-10, -23)

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Linear Functions Linear Functions y = mx +b The easiest ways to graph a linear function are… 1)Table of Values 2) Slope-Intercept Method

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Linear Functions Table of Values Method Graph 2x – 4y = 12 y = ½ x – 3 xy x – 4y = 12 Domain: {x|x is all Real #s} Range: {y|y is all Real #s} Positive Slope

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Linear Functions Slope-Intercept Method y = mx + b m = slope b = y –intercept (0,b) Graph 6x + 3y = 9 y = -2x + 3 m = b = 3 (0, 3) 6x + 3y = 9 Negative Slope Domain: {x|x is all Real #s} Range: {y|y is all Real #s}

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Linear Functions Horizontal Lines y = b where b represents the y-intercept y = 4 (zero slope) Vertical Lines x = a where a represents the x-intercept x = 4 (undefined slope) y = 4 x = 4 Domain: all real #s Range: y|y = 4Domain: x|x = 4 Range: all real #s

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Linear Functions Writing the Equation of a Line Write the equation of a line that runs through the points (-3,1) and (0,-1) Find the slope (m) (-3,1) (0,-1) Find the y-intercept (b) y = mx + b Pt.(-3,1) Write the equation in y = mx + b y = x – 1 m = -2/3 1 = (-2/3)(-3) + b 1 = 2 + b -1 = b b = -1

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Linear Functions Write the equation of a line that is parallel to y – 2x = 4 and runs through the point (-2,4) Find the slope Parallel lines have the same slope y – 2x = 4 y = 2x + 4 m = 2 Find the y-intercept y = mx + b Pt.(-2,4) 4 = 2(-2) + b 4 = -4 + b 8 = b b = 8 Write the equation in y = mx + b y = 2x + 8

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Linear Functions The graph shows yearly cost based on the number of golf games played at a private club. Write an equation that represents the relationship shown. y-int: (0, 90) $90 initial fee slope (rate of change): $30 per game y = 30x + 90 x: # of golf games y: total cost (2,150) (3,180)

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Linear Functions Max purchased a box of green tea mints. The nutrition label on the box stated that a serving of three mints contains a total of 10 calories. Graph the function, C, where C(x) represents the number of calories in x mints. Mints x Calories C(x)

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Modeling Data with Functions Scatter Plots: A graph of plotted points that show the relationship between two sets of data. Correlation Coefficient (r): A number in between -1 and 1 that describes the strength of the data. 2 nd 0 (CATALOG) Scroll down to DIAGNOSTICS ON ENTER, ENTER Calculator :

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Modeling Data with Functions The local ice cream shop keeps track of how much ice cream they sell as compared to the noon temperature on that day. Here are their figures for the last 12 days: Ice Cream Sales vs Temperature Temperature °C Ice Cream Sales 14.2°$ °$ °$ °$ °$ °$ °$ °$ °$ °$ °$ °$408 Regression Line Trend Line Line of Best Fit Least Squares Line

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Modeling Data with Functions Interpolation is where we find a value inside our set of data points. Here we use interpolation to estimate the sales at 21 °C. Extrapolation is where we find a value outside our set of data points. Here we use extrapolation to estimate the sales at 29 °C (which is higher than any value we have).

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Modeling Data with Functions Write the regression equation (y = ax + b) for the raw score based on the hours tutored. Round all values to the nearest hundredth. Equation: y = 6.32x x: # of hours tutored y: raw test score Using the regression equation, predict the score of a student who was tutored for 3 hours. y = 6.32(3) y = Predicted Raw Test Score: 41 1) STAT Edit (#1) 2) Enter data into L 1 and L 2 3) STAT CALC LinReg(ax + b)

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Modeling Data with Functions Calculating Residuals: A residual is calculated by finding the difference between the actual data value and the predicted value (Actual – Predicted). y = 6.32(3) y = Resdiual: 35 – = A – P = – 6.4 The actual score is about 6 points below what I would expect after 3 hours of tutoring.

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Exponential Functions There are two types of Exponential Functions 1)Exponential Growth y = ab x where b > 1 2)Exponential Decay y = ab x where 0 < b < 1 Rate of Change is NOT Constant. An average rate of change can be calculated over a specified interval (see study guide for example).

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Exponential Functions xy = 1/ = 1/ = 1/ = = = = 8 Domain: All real Numbers {x|x is all Real Numbers} Range: All real numbers greater than 0 {y|y > 0} The function is increasing (x and y both increase)

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Exponential Functions xy -3(½) -3 = 8 -2(½) -2 = 4 (½) -1 = 2 0(½) 0 = 1 1(½) 1 = 1/2 2(½) 2 = 1/4 3(½) 3 = 1/8 Domain: All real Numbers {x|x is all Real Numbers} Range: All real numbers greater than 0 {y|y > 0} The function is decreasing (x increases and y decreases)

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Exponential Functions What happens to f(x) = 2 x when…. 4 is added multiplied by -1 f(x) = 2 x + 4 f(x) = -2 x Moves f(x) = 2 x up 4 units Reflects f(x) = 2 x in the x-axis

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Exponential Functions Exponential Growth Model y = a(1 + r) t The cost of maintenance on an automobile increases each year by 8%. If Alberto paid $400 this year for maintenance for his car, what will the cost be (to the nearest dollar) seven years from now? y = a(1 + r) t y = 400(1 +.08) 7 y = 400(1.08) 7 y = … The cost will be $ a: initial value r: growth rate t: time 1 + r: growth factor

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Exponential Functions Exponential Decay Model y = a(1 – r) t A used car was purchased in July 1999 for $12,900. If the car loses 14% of its value each year, what was the value of the car (to the nearest penny) in July 2003? y = a(1 – r) t y = 12,900(1 –.14) 4 y = 12,900(.86) 4 y = … The cost of the car was $ a: initial value r: decay rate t: time 1 – r: decay factor

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Sequences Use these formulas to define sequences and find the nth term of any sequence. Arithmetic: a n = a 1 + d(n – 1) Geometric: a n = a 1 r n – 1 A sequence is an ordered list of numbers. a 1 : first term in the sequence d: common difference ( + ) r: common ratio (x)

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Sequences The first row of the theater has 15 seats in it. Each subsequent row has 3 more seats than the previous row. Write an explicit formula to find the number of seats in the nth row. Arithmetic: Find the number of seats in the tenth row.

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Sequences Brian has 2 parents, 4 grandparents, 8 great-grandparents and so on. Write an explicit formula for the number of ancestors Brian has in a generation if he goes back to the nth generation. Geometric: Find the number of ancestors in the 7 th generation.

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Now its your turn to review on your own! Using the information presented today and the study guide posted on halgebra.org, complete the practice problem set. Regents Review #3 Friday, May 16 th BE THERE!

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