UNIT 2 QUADRATIC FUNCTIONS AND MODELING Lesson 2: Interpreting Quadratic Functions. Instruction. Guided Practice Example 1

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1 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 graph 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? y 1200 Monthly T-shirt revenue (dollars) x T-shirt price (dollars) 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 < 15. 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 > 15. U2-66

2 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. f(x) = 5x x 7 Original function f(15) = 5(15) (15) 7 Substitute 15 for x. f(15) = 1118 The maximum revenue is $1,118. Simplify. 3. Determine the prices that yield no revenue. Identify the x-intercepts. The x-intercepts are 0 and 30, so the store has no revenue when the shirts cost $0 and $ Determine if the function is even, odd, or neither. Evaluate the function for x. f(x) = 5x x 7 Original function f( x) = 5( x) ( x) 7 Substitute x for x. f( x) = 5x 2 150x 7 Simplify. 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. 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. U2-67

3 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? 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. Find the midpoint of the two points by taking the average of the two x-coordinates. x = = 2 2 The value of x that minimizes the function is 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 < 2. U2-68

4 Example 3 The table below shows the predicted temperatures for a summer day in Woodland, California. At what times is the temperature increasing? Decreasing? Time Temperature (ºF) 8 a.m a.m p.m p.m p.m p.m Use the table to determine approximate intervals of increasing and decreasing temperatures. Examine what is happening to the temperatures as the day progresses from morning to evening. The values are increasing when y 2 y 1 is positive, or when the subsequent output value is larger than the preceding value. In this case, the temperature starts at 52º at 8 a.m. and increases to 81º at 4 p.m. The values are decreasing when y 2 y 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 day appear to be increasing from about 8 a.m. to 4 p.m. The temperatures are decreasing from 4 p.m. to 6 p.m. U2-69

5 2. Use graphing technology to verify the information that is assumed from the table. On a TI-83/84: Step 1: Press [STAT]. Step 2: Press [ENTER] to select Edit. Step 3: Enter x-values into L1. Enter times based on a 24-hour clock for times after 12 p.m. For example, 1 p.m. should be entered as hour 13. Step 4: Enter y-values into L2. Step 5: Press [2nd][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][,][L2], 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 12: Press [ENTER] to see the graph of the data and the quadratic equation. On a TI-Nspire: Step 1: Press the [home] key and select the Lists & Spreadsheet icon. Step 2: Name Column A time and Column B temperature. Step 3: Enter x-values under Column A. Enter times based on a 24-hour clock for times after 12 p.m. For example, 1 p.m. should be entered as hour 13. Step 4: Enter y-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 x-axis and choose time. (continued) U2-70

6 Step 8: Move the mouse to the y-axis and choose temperature. Step 9: Select Menu, then 4: Analyze, then 6: Regression, and then 4: Show Quadratic. Step 10: Move the cursor over the equation and press the center key in the navigation pad to drag the equation for viewing, if necessary. 80 y 60 Temperature (ºF) Time (24-hour clock) x 3. State your conclusion. The highest temperature (the maximum) 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 4 p.m. The maximum temperature appears to happen at hour 16, according to the quadratic model, or at around 4 p.m. The high is slightly less than 81º, the predicted temperature for that hour. U2-71