LEARNING GOALS – LESSON 5-2 DAY 1 Warm Up Give the coordinate of the vertex of each function. 1. f(x) = (x – 2)2 + 3 2. f(x) = 2(x + 1)2 – 4 3. Give the domain and range of the following function. {(–2, 4), (0, 6), (2, 8), (4, 10)} LEARNING GOALS – LESSON 5-2 DAY 1 5.2.1 Identify the axis of symmetry of a quadratic graph. 5.2.2 From it’s Standard Form equation, graph a quadratic function and determine if the graph of a quadratic function opens up or down. 5.2.3 Find the axis of symmetry, max/min values, vertex, y- intercept and the domain and range of a quadratic function. 5.2.4 Identify and use maximums and minimums to solve problems. The axis of symmetry is ALWAYS a ____________ ________ of the form ____ = ____ The example to the right shows that parabolas are symmetric curves. The __________ ______ _____________ is the line of symmetry of a parabola that divides the parabola into two congruent halves.

Example 1: Identifying the Axis of Symmetry A. Identify the axis of symmetry for the graph of: B. Identify the axis of symmetry for the graph of: Another form of quadratic functions is the __________________of a quadratic function is f(x)= ax2 + bx + c, where a ≠ 0. f(x)= a(x – h)2 + k Multiply to expand (x – h)2. Distribute a. Simplify and group terms.

Example 2: Graphing Quadratic Functions in Standard Form a is positive a is negative Example 2: Graphing Quadratic Functions in Standard Form Graph the function f(x) = 2x2 – 4x + 5 using its what you know about it properties. a. Determine whether the graph opens upward or downward. b. Find the axis of symmetry. c. Find the vertex. d. Find the y-intercept. e. Do you need another point? If so make a table of values to determine another point.

Example 2: Graphing Quadratic Functions in Standard Form Consider the function f(x) = –x2 – 2x + 3. a. Determine whether the graph opens upward or downward. b. Find the axis of symmetry. c. Find the vertex. d. Find the y-intercept. Graph: f(x)= –2x2 – 4x

LEARNING GOALS – LESSON 5-2 DAY 2 5.2.3 Find the max/min values the domain and range of a quadratic function. 5.2.4 Identify and use maximums and minimums to solve problems. Domain is the set of possible _____ - values. This means x inputs that you can “legally” plug in to the function. Range is the set of possible _____ - values. This means possible y outputs that could come out of the function. Minimum and Maximum Values Minimums Maximums When a parabola opens _____________, the _____-value of the vertex is the MINIMUM VALUE. When a parabola opens _____________, the _____-value of the vertex is the MAXIMUM VALUE. y = x2 – 2 y = - x2 + 2 Domain: Range: Graph:

Example 3: Finding Minimum or Maximum Values Find the minimum/maximum value of f(x) = –3x2 + 2x – 4. Then state the domain and range. Step 1 Determine whether the function has minimum or maximum value. Step 2 Find the x-value of the vertex. Step 3 Then find the y-value of the vertex. B. Find the minimum/maximum value of f(x) = x2 – 6x + 3. Then state the domain and range. C. Find the minimum or maximum value of g(x) = –2x2 – 4. Then state the domain and range. Example 4: Agricultural Application A. The average height h in centimeters of a certain type of grain can be modeled by the function h(r) = 0.024r2 – 1.28r + 33.6 where r is the distance in centimeters between the rows in which the grain is planted. What is the minimum average height of the grain, and what is the row spacing that results in this height?

B. The highway mileage m in miles per gallon for a compact car is approximately by m(s) = –0.025s2 + 2.45s – 30, where s is the speed in miles per hour. What is the maximum mileage for this compact car to the nearest tenth of a mile per gallon? What speed results in this mileage? Consider the function f(x)= -2x2 + 8x – 6. 1. Determine whether the graph opens upward or downward. 2. Find the axis of symmetry. 3. Find the vertex. 4. Identify the maximum or minimum value of the function. 5. Find the y-intercept. 6. Graph the function. 7. Find the domain and range of the function.