7.6 Modeling Data: Exponential, Logarithmic, and Quadratic Functions

Scatter Plots & Regression Lines Scatter Plot—data presented as a set of points Regression Line—the line that best fits those points Each point represents a country

Modeling with Exponential Function Exponential Function— y = b x or f(x) = b x where b is a positive constant other than 1 (b > 0 and b 1) and x is a real number. E.g. f(x) = 3 x g(x) = 5 x

Graphing an exponential function Graph: f(x) = 2 x xf(x) = 2x(x, y) -3f(-2) = 2 -3 = 1/8(-3, 1/8) -2f(-2) = 2 -2 = ¼(-2, ¼) f(-1) = 2 -1 = ½(-1. ½) 0f(0) = 2 0 = 1(0, 1) 1f(1) = 2 1 = 2(1, 2) 2f(2) = 2 2 = 4(2, 4) 3f(3) = 2 3 = 8(3, 8)

Graphing a exponential function Graph: f(x) = 2 x x(x, y) -3(-3, 1/8) -2(-2, ¼) (-1. ½) 0(0, 1) 1(1, 2) 2(2, 4) 3(3, 8)

Comparing Linear and Exponential Models The graphs show the world populations for seven selected years from 1950 through One is a bar graph and the other is scatter plot.

Comparing Linear and Exponential Models Inputting the data into a program, the following models are produced. Linear model: y = ax + b Exponential model: y = ab x

Comparing Linear and Exponential Models 1.Express each model in function notation, with numbers rounded to 3 decimal places.  Linear model: f(x) = 0.074x  Exponential model: g(x) = 2.566(1.017) x

Comparing Linear and Exponential Models 2.How well do the functions model the world population in 2008?  Linear model: f(x) = 0.074x f(59) = 0.074(59) f(59) ≈ 6.7  Exponential model: g(x) = 2.566(1.017)x g(59) = (1.017)59zzzz g(59) ≈ 6.9

Comparing Linear and Exponential Models 3.By one projection, world population is expected to reach 8 billion in the year Which function serves as a better model for this prediction? x = 77 (2026 – 1949) f(x) = 0.074x f(77) =0.074(77) ≈8.0 g(x) = 2.566(1.017) x g(77) = 2.566(1.017)77 ≈ 9.4 It seems that linear functions serves as a better model for the projected population 8 billion in 2026.

Logarithmic Functions Definition Given: b y = x, then y = log b x is an equivalent statement. f(x) = log b x is the logarithmic function with base b. E.g. 10 y = x is equivalent to y = log 10 x. Note: log of a number is the exponent to base b.

Graphing Logarithmic Function Graph: y = log 2 x. Because y = log 2 x means 2 y = x, we can use the exponential form of the equation. x = 2 y y(x,y)(x,y) 2 -2 = ¼−2−2(¼,−2) 2 -1 = ½−1−1(½,−1) 2 0 = 10(1,0) 2 1 = 21(2,1) 2 2 = 42(4,2) 2 3 = 83(8,3)

Temperature in Enclosed Vehicle When the outside air temperature is anywhere from 72° to 96°F, the temperature in an enclosed vehicle climbs by 43°in the first hour. The bar graph and scatter plot are given below

Temperature (cont.) After entering data in a computer program, it displays a logarithmic model y = a b ln x, where ln x is called the natural logarithm. a.Express the model in function notation, with numbers rounded to one decimal place. f(x) = ln x b.Use the function to find temperature increase, to the nearest degree, after 50 minutes. f(x) = − ln x f(50) = − ln 50 f(50) ≈ 41

Modeling with Quadratic Functions Quadratic function: y = ax 2 + bx + c or f(x) = ax 2 + bx + c Graph of a quadratic function is a parabola Vertex of a parabola: the lowest (or the highest) point in the graph.

Vertex of Parabola

Graphing Quadratic Functions

Graphing Parabola

3.Find x-intercepts. Let y = 0. y = x2 – 2x – 3 0 = x2 – 2x – 3 0 = (x – 3)(x + 1) (x – 3) = 0 → x = 3 (x + 1) = 0 → x = -1 Thus, graph passes through (3, 0) and (-1, 0)

Graphing Parabola 4.Find the y-intercept Let x = 0 in the equation. Y = x 2 – 2x – 3 y = 02 – 2(0) – 3 = -3 Thus, the parabola passes through (0, -3) 5.Sketch the graph with vertex, x-intercepts, and y-intercept.