Introduction The tourism industry thrives on being able to provide travelers with an amazing travel experience. Specifically, in areas known for having tropical weather, tour planners want to maximize profit each month by identifying the warmest and coolest months, and then plan tours accordingly. Tour planners might use quadratic models to determine when profits are increasing or decreasing, when they maximized, and/or how profits change in the earlier months versus the later months by looking at the key features of the quadratic functions : Interpreting Key Features of Quadratic Functions

Introduction, continued In this lesson, you will learn about the key features of a quadratic function and how to use graphs, tables, and verbal descriptions to identify and apply the key features : Interpreting Key Features of Quadratic Functions

Key Concepts The key features of a quadratic function are distinguishing characteristics used to describe, draw, and compare quadratic functions. These key features include the x-intercepts, y-intercept, where the function is increasing and decreasing, where the function is positive and negative, relative minimums and maximums, symmetries, and end behavior of the function : Interpreting Key Features of Quadratic Functions

Key Concepts, continued : Interpreting Key Features of Quadratic Functions The x-intercepts of a quadratic function occur when the parabola intersects the x-axis. In the graph at right, the x-intercepts occur when x = 2 and when x = –2.

Key Concepts, continued The equation of the x-axis is y = 0; therefore, the x-intercepts can also be found in a table by identifying when the y-value is 0. The table of values at right corresponds to the parabola we just saw. Notice that the same x- intercepts can be found where the table shows y is equal to : Interpreting Key Features of Quadratic Functions xy –412 –20 0–

Key Concepts, continued The ordered pair that corresponds to an x-intercept is always of the form (x, 0). The x-intercepts are also the solutions of a quadratic function. The y-intercepts of a quadratic function occur when the parabola intersects the y-axis. In the next graph, the y-intercept occurs when y = – : Interpreting Key Features of Quadratic Functions

Key Concepts, continued : Interpreting Key Features of Quadratic Functions

Key Concepts, continued The equation of the y-axis is x = 0; therefore, the y-intercept can also be found in a table by identifying when the x-value is 0. Notice in the table of values (at right) that corresponds to the parabola we just saw, the same y-intercept can be found where x is : Interpreting Key Features of Quadratic Functions xy –412 –20 0–

Key Concepts, continued The ordered pair that corresponds to a y-intercept is always of the form (0, y). Recall that the vertex is the point on a parabola where the graph changes direction. The maximum or minimum of the function occurs at the vertex of the parabola : Interpreting Key Features of Quadratic Functions

Key Concepts, continued The vertex is also the point where the parabola changes from increasing to decreasing. Increasing refers to the interval of a function for which the output values are becoming larger as the input values are becoming larger. Decreasing refers to the interval of a function for which the output values are becoming smaller as the input values are becoming larger. Recall that parabolas are symmetric to a line that extends through the vertex, called the axis of symmetry : Interpreting Key Features of Quadratic Functions

Key Concepts, continued Any point to the right or left of the parabola is equidistant to another point on the other side of the parabola. A parabola only increases or decreases as x becomes larger or smaller. Read the graph from left to right to determine when the function is increasing or decreasing. Trace the path of the graph with a pencil tip. If your pencil tip goes down as you move toward increasing values of x, then f(x) is decreasing : Interpreting Key Features of Quadratic Functions

Key Concepts, continued If your pencil tip goes up as you move toward increasing values of x, then f(x) is increasing. For a quadratic, if the graph has a minimum value, then the quadratic will start by decreasing toward the vertex, and then it will increase. If the graph has a maximum value, then the quadratic will start by increasing toward the vertex, and then it will decrease. The vertex is called an extremum. Extrema are the maxima or minima of a function : Interpreting Key Features of Quadratic Functions

Key Concepts, continued The concavity of a parabola is the property of being arched upward or downward. A quadratic with positive concavity will increase on either side of the vertex, meaning that the vertex is the minimum or lowest point of the curve. A quadratic with negative concavity will decrease on either side of the vertex, meaning that the vertex is the maximum or highest point of the curve : Interpreting Key Features of Quadratic Functions

Key Concepts, continued A quadratic that has a minimum value is concave up because the graph of the function is bent upward. A quadratic that has a maximum value is concave down because the graph of the function is bent downward. The graphs that follow demonstrate examples of parabolas as they decrease and then increase, and vice versa. Trace the path of each parabola from left to right with your pencil to see the difference : Interpreting Key Features of Quadratic Functions

Key Concepts, continued : Interpreting Key Features of Quadratic Functions Decreasing then Increasing Vertex: (0, –4); minimum x < 0 = decreasing x > 0 = increasing Direction: concave up

Key Concepts, continued : Interpreting Key Features of Quadratic Functions Increasing then Decreasing Vertex: (0, 4); maximum x < 0 = increasing x > 0 = decreasing Direction: concave down

Key Concepts, continued : Interpreting Key Features of Quadratic Functions The inflection point of a graph is a point on a curve at which the sign of the curvature (i.e., the concavity) changes. In the graph on the following slide, the curvature starts out as concave down, but then switches to concave up at (–1, 1). The point (–1, 1) is the point of inflection. The vertex of a quadratic function is also the point of inflection.

Key Concepts, continued : Interpreting Key Features of Quadratic Functions

Key Concepts, continued End behavior is the behavior of the graph as x becomes larger or smaller. If the highest exponent of a function is even, and the coefficient of the same term is positive, then the function is approaching positive infinity as x approaches both positive and negative infinity. If the highest exponent of a function is even, but the coefficient of the same term is negative, then the function is approaching negative infinity as x approaches both positive and negative infinity : Interpreting Key Features of Quadratic Functions

Key Concepts, continued : Interpreting Key Features of Quadratic Functions Even and Positive f(x) = x 2 – 4 Highest exponent: 2 Coefficient of x 2 : positive As x approaches positive infinity, f(x) approaches positive infinity. As x approaches negative infinity, f(x) approaches positive infinity.

Key Concepts, continued : Interpreting Key Features of Quadratic Functions Even and Negative f(x) = –x Highest exponent: 2 Coefficient of x 2 : negative As x approaches positive infinity, f(x) approaches negative infinity. As x approaches negative infinity, f(x) approaches negative infinity.

Key Concepts, continued Functions can be defined as odd or even based on the output yielded when evaluating the function for –x. For an odd function, f(–x) = –f(x). That is, if you evaluate a function for –x, the resulting function is the opposite of the original function. For an even function, f(–x) = f(x). That is, if you evaluate a function for –x, the resulting function is the same as the original function : Interpreting Key Features of Quadratic Functions

Key Concepts, continued If evaluating the function for –x does not result in the opposite of the original function or the original function, then the function is neither odd nor even. Though all quadratics have an even power, not all quadratics are even functions. It is important to evaluate the function for –x when the quadratic includes both a linear and a constant term : Interpreting Key Features of Quadratic Functions

Common Errors/Misconceptions incorrectly identifying when a function is increasing or decreasing making sign errors when determining if a function is odd, even, or neither : Interpreting Key Features of Quadratic Functions

Guided Practice Example 1 A local store’s monthly revenue from T-shirt sales is modeled by the function f(x) = –5x x – 7. Use the equation and the graph on the next slide to answer the following questions: At what prices is the revenue increasing? Decreasing? What is the maximum revenue? What prices yield no revenue? Is the function even, odd, or neither? : Interpreting Key Features of Quadratic Functions

Guided Practice: Example 1, continued : Interpreting Key Features of Quadratic Functions

Guided Practice: Example 1, continued 1.Determine when the function is increasing and decreasing. Use your pencil to determine when the function is increasing and decreasing. Moving from left to right, trace your pencil along the function. The function increases until it reaches the vertex, then decreases. The revenue is increasing when the price per shirt is less than $15 or when x < : Interpreting Key Features of Quadratic Functions

Guided Practice: Example 1, continued The vertex of this function has an x-value of 15. The revenue is decreasing when the price per shirt is more than $15 or when x > : Interpreting Key Features of Quadratic Functions

Guided Practice: Example 1, continued 2.Determine the maximum revenue. Use the vertex of the function to determine the maximum revenue. The T-shirt price that maximizes revenue is x = 15. The maximum is the corresponding y-value. Since it is difficult to estimate accurately from this graph, substitute x into the function to solve : Interpreting Key Features of Quadratic Functions

Guided Practice: Example 1, continued The maximum revenue is $1, : Interpreting Key Features of Quadratic Functions f(x) = –5x x – 7Original function f(15) = –5(15) (15) – 7Substitute 15 for x. f(15) = 1118Simplify.

Guided Practice: Example 1, continued 3.Determine the prices that don’t yield revenue. Identify the x-intercepts. The x-intercepts are 0 and 30, so the store has no revenue when the shirts cost $0 and $ : Interpreting Key Features of Quadratic Functions

Guided Practice: Example 1, continued 4.Determine if the function is even, odd, or neither. Evaluate the function for –x. Since f(–x) is neither the original function nor the opposite of the original function, the function is not even or odd; it is neither : Interpreting Key Features of Quadratic Functions f(x) = –5x x – 7Original function f(–x) = –5(–x) (–x) – 7Substitute –x for x. f(x) = –5x 2 – 150x – 7Simplify.

Guided Practice: Example 1, continued 5.Use the graph of the function to verify that the function is neither odd nor even. Since the function is not symmetric over the y-axis or the origin, the function is neither even nor odd : Interpreting Key Features of Quadratic Functions ✔

Guided Practice: Example 1, continued : Interpreting Key Features of Quadratic Functions

Guided Practice Example 2 A function has a minimum value of –5 and x-intercepts of –8 and 4. What is the value of x that minimizes the function? For what values of x is the function increasing? Decreasing? : Interpreting Key Features of Quadratic Functions

Guided Practice: Example 2, continued 1.Determine the x-value that minimizes the function. Quadratics are symmetric functions about the vertex and the axis of symmetry, the line that divides the parabola in half and extends through the vertex. The x-value that minimizes the function is the midpoint of the two x-intercepts : Interpreting Key Features of Quadratic Functions

Guided Practice: Example 2, continued Find the midpoint of the two points by taking the average of the two x-coordinates. The value of x that minimizes the function is – : Interpreting Key Features of Quadratic Functions

Guided Practice: Example 2, continued 2.Determine when the function is increasing and decreasing. Use the vertex to determine when the function is increasing and when it is decreasing. The minimum value is –5 and the vertex of the function is (–2, –5). From left to right, the function decreases as it approaches the minimum and then increases. The function is increasing when x > –2 and decreasing when x < – : Interpreting Key Features of Quadratic Functions ✔

Guided Practice: Example 2, continued : Interpreting Key Features of Quadratic Functions