UNIT 2 QUADRATIC FUNCTIONS AND MODELING Lesson 2: Interpreting Quadratic Functions Instruction
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1 Prerequisite Skills This lesson requires the use of the following skills: knowing the standard form of quadratic functions using graphing technolog to model quadratic functions Introduction The tourism industr thrives on being able to provide travelers with an amazing travel eperience. Specificall, in areas known for having tropical weather, tour planners want to maimize profit each month b identifing the warmest and coolest months, and then plan tours accordingl. Tour planners might use quadratic models to determine when profits are increasing or decreasing, when the maimized, and/or how profits change in the earlier months versus the later months b looking at the ke features of the quadratic functions. In this lesson, ou will review the definitions of ke features of a quadratic function and how to use graphs, tables, and verbal descriptions to identif and appl the ke features. Ke Concepts The ke features of a quadratic function are distinguishing characteristics used to describe, draw, and compare quadratic functions. Ke features include the -intercepts, -intercept, minimums and maimums, and smmetries, as well as where the function is increasing and decreasing, where the function is positive and negative, and the end behavior of the function. Recall each of the forms of quadratic functions, outlined as follows. Standard Form The standard form, or general form, of a quadratic function is written as f() = a + b + c, where a is the coefficient of the quadratic term, b is the coefficient of the linear term, and c is the constant term. The -intercept is the value of c. The verte of the function can be found b first determining the value of, and then finding the corresponding -value. Verte Form The verte form of a quadratic function is written as f() = a( h) + k. The verte is (h, k). The ais of smmetr is identified from verte form as = h. U-59
2 Factored Form The factored form, or intercept form, of a quadratic function is written as f() = a( p) ( q). The -intercepts of the function are p and q. The -intercepts of a quadratic function occur when the parabola intersects the -ais. In the graph that follows, the -intercepts occur when = and when = intercepts The equation of the -ais is = 0; therefore, the -intercepts can also be found in a table b identifing when the -value is 0. The table of values below corresponds to the parabola illustrated above. Notice that the same -intercepts can be found where the table shows is equal to The ordered pair that corresponds to an -intercept is alwas of the form (, 0). The -intercepts are also the solutions of a quadratic function. U-60
3 The -intercepts of a quadratic function occur when the parabola intersects the -ais. In the net graph, the -intercept occurs when = intercept The equation of the -ais is = 0; therefore, the -intercept can also be found in a table b identifing when the -value is 0. Notice in the table of values that corresponds to the parabola above, the same -intercept can be found where is The ordered pair that corresponds to a -intercept is alwas of the form (0, ). Recall that the verte is the maimum or minimum of the function. The verte is also the point where the parabola changes from increasing to decreasing. U-61
4 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 smmetric to a line that etends through the verte, called the ais of smmetr. An point to the right or left of the parabola is equidistant to another point on the other side of the parabola. A parabola onl increases or decreases as 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 our pencil tip goes down as ou move toward increasing values of, then f() is decreasing. If our pencil tip goes up as ou move toward increasing values of, then f() is increasing. For a quadratic, if the graph has a minimum value, then the quadratic will start b decreasing toward the verte, and then it will increase. If the graph has a maimum value, then the quadratic will start b increasing toward the verte, and then it will decrease. The verte is called an etremum. Etrema are the maima or minima of a function. The concavit of a parabola is the propert of being arched upward or downward. A quadratic with positive concavit will increase on either side of the verte, meaning that the verte is the minimum or lowest point of the curve. A quadratic with negative concavit will decrease on either side of the verte, meaning that the verte is the maimum or highest point of the curve. A quadratic that has a minimum value is concave up because the graph of the function is bent upward. A quadratic that has a maimum value is concave down because the graph of the function is bent downward. The graphs that follow demonstrate eamples of parabolas as the decrease and then increase, and vice versa. Trace the path of each parabola from left to right with our pencil to see the difference. U-6
5 Decreasing then Increasing Verte: (0, ); minimum < 0 = decreasing > 0 = increasing Direction: concave up Increasing then Decreasing Verte: (0, ); maimum < 0 = increasing > 0 = decreasing Direction: concave down 0 0 The inflection point of a graph is a point on a curve at which the sign of the curvature (i.e., the concavit) changes. In the following graph, 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 verte of a quadratic function is also the point of inflection End behavior is the behavior of the graph as approaches positive or negative infinit U-63
6 If the highest eponent of a function is even, and the coefficient of the same term is positive, then the function is approaching positive infinit as approaches both positive and negative infinit. If the highest eponent of a function is even, but the coefficient of the same term is negative, then the function is approaching negative infinit as approaches both positive and negative infinit. Even and Positive f() = Highest eponent: Coefficient of : positive As approaches positive infinit, f() approaches positive infinit. As approaches negative infinit, f() approaches positive infinit. f() = + Highest eponent: Even and Negative Coefficient of : negative As approaches positive infinit, f() approaches negative infinit. As approaches negative infinit, f() approaches negative infinit. 0 0 Functions can be defined as odd or even based on the output ielded when evaluating the function for. For an odd function, f( ) = f(). That is, if ou evaluate a function for, the resulting function is the opposite of the original function. For an even function, f( ) = f(). That is, if ou evaluate a function for, the resulting function is the same as the original function. U-6
7 If evaluating the function for 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 when the quadratic includes both a linear and a constant term. Common Errors/Misconceptions incorrectl identifing when a function is increasing or decreasing making sign errors when determining if a function is odd, even, or neither U-65
8 Guided Practice Eample 1 A local store s monthl revenue from T-shirt sales is modeled b the function f() = Use the equation and graph to answer the following questions: At what prices is the revenue increasing? Decreasing? What is the maimum revenue? What prices ield no revenue? Is the function even, odd, or neither? 100 Monthl T-shirt revenue (dollars) T-shirt price (dollars) 1. Determine when the function is increasing and decreasing. Use our pencil to determine when the function is increasing and decreasing. Moving from left to right, trace our pencil along the function. The function increases until it reaches the verte, then decreases. The revenue is increasing when the price per shirt is less than $15 or when < 15. The verte of this function has an -value of 15. The revenue is decreasing when the price per shirt is more than $15 or when > 15. U-66
9 . Determine the maimum revenue. Use the verte of the function to determine the maimum revenue. The T-shirt price that maimizes revenue is = 15. The maimum is the corresponding -value. Since it is difficult to estimate accuratel from this graph, substitute into the function to solve. f() = Original function f(15) = 5(15) + 150(15) 7 Substitute 15 for. f(15) = 1118 The maimum revenue is $1,118. Simplif. 3. Determine the prices that ield no revenue. Identif the -intercepts. The -intercepts are 0 and 30, so the store has no revenue when the shirts cost $0 and $30.. Determine if the function is even, odd, or neither. Evaluate the function for. f() = Original function f( ) = 5( ) + 150( ) 7 Substitute for. f( ) = Simplif. Since f( ) is neither the original function nor the opposite of the original function, the function is not even or odd; it is neither. 5. Use the graph of the function to verif that the function is neither odd nor even. Since the function is not smmetric over the -ais or the origin, the function is neither even nor odd. U-67
10 Eample A function has a minimum value of 5 and -intercepts of 8 and. What is the value of that minimizes the function? For what values of is the function increasing? Decreasing? 1. Determine the -value that minimizes the function. Quadratics are smmetric functions about the verte and the ais of smmetr, the line that divides the parabola in half and etends through the verte. The -value that minimizes the function is the midpoint of the two -intercepts. Find the midpoint of the two points b taking the average of the two -coordinates. = 8 + = The value of that minimizes the function is.. Determine when the function is increasing and decreasing. Use the verte to determine when the function is increasing and when it is decreasing. The minimum value is 5 and the verte of the function is (, 5). From left to right, the function decreases as it approaches the minimum and then increases. The function is increasing when > and decreasing when <. U-68
11 Eample 3 The table below shows the predicted temperatures for a summer da in Woodland, California. At what times is the temperature increasing? Decreasing? Time Temperature (ºF) 8 a.m a.m. 6 1 p.m. 7 p.m. 78 p.m p.m Use the table to determine approimate intervals of increasing and decreasing temperatures. Eamine what is happening to the temperatures as the da progresses from morning to evening. The values are increasing when 1 is positive, or when the subsequent output value is larger than the preceding value. In this case, the temperature starts at 5º at 8 a.m. and increases to 81º at p.m. The values are decreasing when 1 is negative or when the subsequent output value is less than the preceding value. At 81º, the temperature decreases to 76º. The temperatures in Woodland on this summer da appear to be increasing from about 8 a.m. to p.m. The temperatures are decreasing from p.m. to 6 p.m. U-69
12 . Use graphing technolog to verif the information that is assumed from the table. On a TI-83/8: Step 1: Press [STAT]. Step : Press [ENTER] to select Edit. Step 3: Enter -values into L1. Enter times based on a -hour clock for times after 1 p.m. For eample, 1 p.m. should be entered as hour 13. Step : Enter -values into L. Step 5: Press [nd][y=]. Step 6: Press [ENTER] twice to turn on the Stat Plot. Step 7: Press [ZOOM][9] to select ZoomStat and show the scatter plot. Step 8: Press [STAT]. Step 9: Arrow to the right to select Calc. Step 10: Press [5] to select QuadReg. Step 11: Enter [L1][,][L], Y 1. To enter Y 1, press [VARS] and arrow over to the right to Y-VARS. Select 1: Function. Select 1: Y 1. Step 1: Press [ENTER] to see the graph of the data and the quadratic equation. On a TI-Nspire: Step 1: Press the [home] ke and select the Lists & Spreadsheet icon. Step : Name Column A time and Column B temperature. Step 3: Enter -values under Column A. Enter times based on a -hour clock for times after 1 p.m. For eample, 1 p.m. should be entered as hour 13. Step : Enter -values under Column B. Step 5: Select Menu, then 3: Data, and then 6: Quick Graph. Step 6: Press [enter]. Step 7: Move the cursor to the -ais and choose time. (continued) U-70
13 Step 8: Move the mouse to the -ais and choose temperature. Step 9: Select Menu, then : Analze, then 6: Regression, and then : Show Quadratic. Step 10: Move the cursor over the equation and press the center ke in the navigation pad to drag the equation for viewing, if necessar Temperature (ºF) Time (-hour clock) 3. State our conclusion. The highest temperature (the maimum) in the table will occur at the point of inflection, or in this case, the time at which the temperature goes from increasing to decreasing. The highest temperature is 81º, and this occurs at p.m. The maimum temperature appears to happen at hour 16, according to the quadratic model, or at around p.m. The high is slightl less than 81º, the predicted temperature for that hour. U-71
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