Introduction Relationships can be presented through different forms: tables, equations, graphs, and words. This section explores how to compare functions presented in all four forms : Comparing Properties of Functions Given in Different Forms

Key Concepts Recall that for a quadratic function in standard form, f(x) = ax 2 + 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. Recall that linear functions have a constant slope. When x increases by 1, y increases by a constant value. A linear function will either increase, decrease, or remain constant : Comparing Properties of Functions Given in Different Forms

Key Concepts, continued 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 : Comparing Properties of Functions Given in Different Forms

Key Concepts, continued 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 : Comparing Properties of Functions Given in Different Forms

Common Errors/Misconceptions misidentifying the appropriate type of function (linear, quadratic, or exponential) thinking that an equation with a higher y-intercept has a higher maximum value : Comparing Properties of Functions Given in Different Forms

Guided Practice Example 2 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 : Comparing Properties of Functions Given in Different Forms

Guided Practice: Example 2, continued The path of Alejandro’s paper wad is modeled by the equation f(x) = –x 2 + 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 : Comparing Properties of Functions Given in Different Forms x0234 y3676

Guided Practice: Example 2, continued 1.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 : Comparing Properties of Functions Given in Different Forms

Guided Practice: Example 2, continued 2.Verify that the extremum for each function is the maximum value of each function. The value of a in 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 : Comparing Properties of Functions Given in Different Forms

Guided Practice: Example 2, continued 3.Determine the vertex of each function. Find the vertex of Alejandro’s function using : Comparing Properties of Functions Given in Different Forms Formula to find the x- coordinate of the vertex Substitute –1 for a and 2 for b. x = 1Simplify.

Guided Practice: Example 2, continued Use the given function to find The vertex is at (1, 8) : Comparing Properties of Functions Given in Different Forms f(x) = –x 2 + 2x + 7Original equation f(1) = –(1) 2 + 2(1) + 7Substitute 1 for x. f(1) = – Simplify. f(1) = 8

Guided Practice: Example 2, 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) : Comparing Properties of Functions Given in Different Forms

Guided Practice: Example 2, continued 4.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 : Comparing Properties of Functions Given in Different Forms ✔

Guided Practice: Example 2, continued : Comparing Properties of Functions Given in Different Forms

Guided Practice Example 3 Which of the following quadratic functions has a vertex with a larger y-value: f(x) = 2x 2 – 12x + 25, or g(x) as presented in the table? : Comparing Properties of Functions Given in Different Forms x–4–3–202 g(x)g(x)787–1–17

Guided Practice: Example 3, continued 1.Verify for each function whether the vertex is a minimum or maximum. For f(x), a is positive, which means that the vertex is a minimum. For g(x), as x increases, the y-values first increase and then decrease. This indicates the vertex is a maximum : Comparing Properties of Functions Given in Different Forms

Guided Practice: Example 3, continued 2.Find the vertex for each function. For f(x), complete the square of f(x) = 2x 2 – 12x : Comparing Properties of Functions Given in Different Forms f(x) 2(x 2 – 6x) + 25 Factor 2 out of each term containing an x. f(x) = 2(x 2 – 6x + 9) + 25 – 2(9)Add and subtract 2(9). f(x) = 2(x – 3) – 2(9) Rewrite the perfect square trinomial as a squared binomial. f(x) = 2(x – 3) 2 + 7Simplify.

Guided Practice: Example 3, continued From this form, we can see that the vertex of f(x) is (3, 7). For g(x), the points are symmetric about x = –3. Therefore, the line of symmetry contains (–3, 8), which must be the vertex : Comparing Properties of Functions Given in Different Forms

Guided Practice: Example 3, continued 3.Determine which vertex has a larger y-value. Compare the y-values of each function. 7 < 8 The function g(x) has a vertex with a larger y-value : Comparing Properties of Functions Given in Different Forms ✔

Guided Practice: Example 3, continued : Comparing Properties of Functions Given in Different Forms