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Use quadratic functions to model data. Use quadratic models to analyze and predict.

Objectives. Use quadratic functions to model data. Use quadratic models to analyze and predict. Vocabulary. quadratic model quadratic regression.

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Use quadratic functions to model data. Use quadratic models to analyze and predict.

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  1. Objectives Use quadratic functions to model data. Use quadratic models to analyze and predict.

  2. Vocabulary quadratic model quadratic regression

  3. Recall that you can use differences to analyze patterns in data. For a set of ordered parts with equally spaced x-values, a quadratic function has constant nonzero second differences, as shown below.

  4. Example 1A: Identifying Quadratic Data Determine whether the data set could represent a quadratic function. Explain. Find the first and second differences. Equally spaced x-values Quadratic function: second differences are constant for equally spaced x-values 1st 2 6 10 14 2nd 4 4 4

  5. Example 1B Determine whether the data set could represent a quadratic function. Explain. Find the first and second differences. Equally spaced x-values Not a quadratic function: first differences are constant so the function is linear. 1st 2 222 2nd 0 0 0

  6. 1 2 3 Example 2: Writing a Quadratic Function from Data Write a quadratic function that fits the points (1, –5), (3, 5) and (4, 16). Use each point to write a system of equations to find a, b, and c in f(x) = ax2 + bx + c.

  7. Subtract equation by equation to get . 9a + 3b + c = 5 16a + 4b + c = 16 a + b + c = –5 a + b + c = –5 8a + 2b + 0c = 10 15a + 3b + 0c = 21 1 1 1 3 2 4 5 1 2 5 3 4 Example 2 Continued Subtract equation by equation to get .

  8. 5 4 4 5 Example 2 Continued Solve equation and equation for a and b using elimination. 30a + 6b = 42 2(15a + 3b = 21) Multiply by 2. –3(8a + 2b = 10) – 24a – 6b = –30 Multiply by –3. 6a + 0b = 12 Subtract. a = 2 Solve for a.

  9. 4 5 Example 2 Continued Substitute 2 for a into equation or equation to get b. 15(2) +3b = 21 8(2) +2b = 10 2b = –6 3b = –9 b = –3 b = –3

  10. 1 f(x) = ax2 + bx + c f(x)= 2x2 – 3x – 4 Example 2 Continued Substitute a = 2 and b = –3 into equation to solve for c. (2) +(–3) + c = –5 –1 + c = –5 c = –4 Write the function using a = 2, b = –3 and c = –4.

  11. A quadratic model is a quadratic function that represents a real data set. Models are useful for making estimates.

  12. Lesson Quiz: Part I Determine whether each data set could represent a quadratic function. 1. not quadratic quadratic 2. 3. Write a quadratic function that fits the points (2, 0), (3, –2), and (5, –12). f(x) = –x2 + 3x – 2

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