174 CHAPTER 3 Linear and Quadratic Functions Fitting a Quadratic Function to Data Photosynthetic efficiency is the percentage of light energy that is converted into chemical energy by a plant. The chemical energy is used by a plant to maintain the plant’s activities. A sample of grass was used to determine the efficiency of the grass species. Table 8 shows the results of the study at various temperatures, measured in degrees Celsius. The efficiency is measured as a percent. EXAMPLE 4 Temperature, x Efficiency, y Temperature, x Efficiency, y −1.5 33 15 91 0 46 17 89 2.5 55 20 77 5 80 22 72 7 87 25 54 10 93 27 46 12 95 Source: D. Brown and P. Rothery, Models in Biology: Mathematics, Statistics, and Computing. Table 8 Figure 30 y 5 ax2 1 bx 1 c, a , 0 (b) y 5 ax2 1 bx 1 c, a . 0 (a) CAREER TIP Fitting a curve to data, sometimes called regression, is frequently used to find possible patterns in observed data. One goal of pattern recognition is to make reliable predictions. The power of this type of modeling has led to many jobs and careers in analytics and data science. j Use these facts to find the value of a in y ax .2 = y ax a a 526 2100 526 2100 2 2 2 = = ⋅ = = = x y 2100, 526 The equation of the parabola is y x 526 21002 2 = When x 1000, = the height of the cable is y 526 2100 1000 119.3 feet 2 2 = ⋅ ≈ The cable is 119.3 feet above the road at a distance of 1000 feet from the center of the bridge. Now Work PROBLEM 11 2 Build Quadratic Models from Data In Section 3.2, we found the line of best fit for data that appeared to be linearly related. But data may also follow a nonlinear relation. Figures 30(a) and (b) show scatter plots of data that follow a quadratic relation. (a) Draw a scatter plot of the data, treating temperature as the independent variable. Comment on the type of relation that exists between temperature and efficiency.
RkJQdWJsaXNoZXIy NjM5ODQ=