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2.2 Finding Quadratic Equations In section 1.2 you learned how to generate linear equations when we had ordered pairs. We used the format y = mx + b. We’re going to do something similar for quadratics. Recall from 2.1 we talked about quadratic equations in the form: y ax 2 bx c There is another format we can use that is really helpful when we have the vertex. It is: y a( x h)2 k Where (h,k) is the vertex To complete the equation, we put in the values for “x” and “y” using a given ordered pair, substitute the “h” and “k” values, then solve for “a.” Once we get “a”, we can rewrite the equation using the values for “a”, “h”, and “k”. y a ( x h) 2 k Your text refers to this format as STANDARD FORM. I know we are going to be relying on the calculator to determine equations, but it’s important that you know how to generate a quadratic by hand. Review of Example 2.2.2 in your text. Example 2.2.2: Finding a Quadratic Equation Consider the data regarding the number of morning newspaper companies in the United States since 1940. Find an equation by hand to model the data. Then use the equation to predict the number of morning papers there will be in 2025. For simplicity we can consider 1940 to be year 0. Solution: Having graphed the data, you can see the point (20,312) is a reasonable estimate for the vertex, substituting into the equation y = a(x - h)2 + k we have: y = a(x - 20)2 + 312. The graph passes through the point (65,817). Substituting we have: 817 = a(65 - 20)2 + 312 substitute 65 for x and 817 for y 817 = 2025a + 312 simplifying (65 - 20)2 505 = 2025a subtract 312 from both sides a ≈ .249 divide both sides by 2025 The equation y = .249(x - 20)2 + 312 is a fairly good model for the data. Use it to predict for year 2025. y = .249(x - 20)2 + 312 year 2025 would mean x = 85 2 y = .249(85 - 20) + 312 replace x with 85 2 y = .249(65) + 312 parenthesis first y = .249(4225) + 312 y = 1052.025 + 312 y = 1364.025 exponents next multiplication next add Final Answer: If the pattern of growth continues in a similar fashion, there will be approximately 1364 morning newspaper companies in 2025. Finding an equation by hand using the standard form of a parabola is a confidence builder, but as we saw in the last chapter using regression is much easier and more accurate. Example 2.2.3 in your test takes the same data and uses the regression feature in your calculator to get the equation. If you read your text, you are supposed to ask me a question after example 2.2.3. ?????????? Results from 2.2.2 y 0.249( x 20)2 312 Results from 2.2.3 y 0.23x 2 8.13x 374.75 Practice 1 1. The chart shows the inches of drop in a bullet for different length shots for a particular type of Hornady bullet design. a) Use regression to find a quadratic equation to model the data, considering drop as a function of yardage. Round the numbers in your equation to 4 decimal places. b) Use your equation to predict the drop for a 215 yard shot, accurate to 1 decimal place. c) Use your equation to predict the drop for a 500 yard shot, accurate to 1 decimal place. Solutions: a) Enter into the calculator the data from the first two columns in the Hornady table, then select the quad option to generate the regression equation: y 0.0013x 2 0.3166 x 20.3480 b) Using the equation from a), c) Using the equation from a), substitute 215 for x, solve for y. substitute 500 for x, solve for y. y 0.0013 (215) 2 0.3166 215 20.3480 y 0.0013 (500)2 0.3166 500 20.3480 y 12.3715 y 187.048 y 12.4 in y 187.0 in Practice 2 2. The golden gate is a suspension bridge where the road is supported by equally spaced cables hanging from a large parabolic cable, which is itself supported by 2 large towers. a) Use the dimensions below (measured in feet) to find 3 ordered pairs that define the parabola, then regression to find the equation for the parabola, accurate to 6 decimal places. b) Use the equation to find the lengths of the 7 hanging cables between the left tower and the 40 foot center cable. Round to 2 decimal places. Solutions: a) Establish 3 ordered pairs. Enter into the calculator the data from the table, then select the quad option to generate the regression equation: y 0.000132 x 2 0.633333x 800 b The cables are 300 ft apart. Using the equation from a), substitute each increment in for “x” and solve for “y” (height of each cable). Distance From Left Tower (x) Length of Cable (y) 300 ft 600 ft 900 ft 1200 ft 1500 ft 1800 ft 2100 ft Distance Length of From Left Cable Tower 0 2400 4800 800 40 800 621.88 ft 467.52 ft 336.92 ft 230.08 ft 147.00 ft 87.68 ft 52.12 ft Homework: problems 2,6,8,9,10. Number 8 might cause you….to do some thinking. Make sure you understand the question and that your answer(s) make sense.