Comparing Properties of Functions Given in Different Forms

1. Comparing Properties of Functions Given in Different Forms Adapted from Walch Education

2. Key Concepts For a quadratic function in standard form, f(x) = ax2 + bx + c, the value of a determines whether the quadratic has a maximum or a minimum. • If a is negative, the quadratic function will have a maximum. • If a is positive, the quadratic function will have a minimum. • Linear functions have a constant slope. 5.6.3: Comparing Properties of Functions Given in Different Forms

3. Key Concepts • The change in y when x increases by 1 is called a first difference. • Recall that exponential functions either increase or decrease. • There is a constant multiple in the rate of change between the y-values. • A growing exponential function will eventually exceed a linear or quadratic function. • A quadratic function changes from increasing to decreasing, or vice versa. 5.6.3: Comparing Properties of Functions Given in Different Forms

4. Key Concepts • There is neither a constant rate of change nor a constant multiple in a quadratic function. • In a quadratic model, the change in the first differences is constant. • The change in first differences is called a second difference. • Compare the y-values of the vertices of quadratic functions to determine the greatest maximum or least minimum values between two or more quadratic functions. 5.6.3: Comparing Properties of Functions Given in Different Forms

5. Practice # 1 Three students are shooting wads of paper with a rubber band, aiming for a trash can in the front of the room. The height of each student’s paper wad in feet is given as a function of the time in seconds. Which student’s paper wad flies the highest? The functions for each of the three students are described on the next slide. 5.6.3: Comparing Properties of Functions Given in Different Forms

6. Continued, • The path of Alejandro’s paper wad is modeled by the equation f(x) = –x2+ 2x + 7. • Melissa’s paper wad is estimated to reach the heights shown in the table below. • After 3 seconds, Connor’s paper wad achieves a maximum height of 6.5 feet above the floor. 5.6.3: Comparing Properties of Functions Given in Different Forms

7. Determine if each function represents a quadratic • Alejandro’s path is represented by a quadratic function written in standard form. • Melissa’s table has symmetry about x = 3, which implies a quadratic relationship. • In general, a projectile follows a parabolic path, gaining height until reaching a maximum, then descending. We can assume Connor’s paper wad will follow a parabolic path. 5.6.3: Comparing Properties of Functions Given in Different Forms

8. Verify that the extremum for each function is the maximum value of each function • The value of ain the equation for Alejandro’s paper wad is a negative value. Therefore, the function has a maximum value. • The table for Melissa’s paper wad increases then decreases, so its path will have a maximum value. • The maximum height of the path of Connor’s paper wad is given. 5.6.3: Comparing Properties of Functions Given in Different Forms

9. Determine the vertex of each function Find the vertex of Alejandro’s function using 5.6.3: Comparing Properties of Functions Given in Different Forms

10. Continued, • Use the given function to find The vertex is at (1, 8). 5.6.3: Comparing Properties of Functions Given in Different Forms

11. Continued, • Find the vertex for Melissa’s paper wad using symmetry. Her paper wad’s function is symmetric about the vertex at (3, 7). • The vertex for Connor’s paper wad is given as (3, 6.5). 5.6.3: Comparing Properties of Functions Given in Different Forms

12. Use the vertices to determine whose paper wad goes the highest • Compare the y-values of the vertices. 8 > 7 > 6.5 • Alejandro’s paper wad flies the highest, at 8 feet. 5.6.3: Comparing Properties of Functions Given in Different Forms

13. Thanks for Watching! Ms. Dambreville