IMP 3 Function POW # Linear, Quadratic and Cubic Functions with some extension to higher degree polynomials Directions: ) Graphing: Use a graphing calculator to do all the graphing. This will save you precious time and energy! Graph the parent function in y. Graph the transformed function one at a time in y. This will help you see the changes in the functions. Graph the transformed functions in order. ou do not have to draw any of your graphs. However, you may wish to sketch a few for future use. Look for patterns (similarities/differences in and between sets) that will help you answer the questions.* ) Completing the POW: our results/answers to any italicized questions or directions must be written in your POW. The results should be in the order of the directions/questions. The results/answers should be written in complete sentences. Any tables may be completed on this sheet. 3) Submitting the POW: A cover sheet (see below for example) should be the first page. Next, attach your results/answers pages in order! Clearly label each section. Example Linear Functions Then, attach the POW downloads so that you include all completed tables. Next, attach the Required Worksheet. Attach the rubric as the last page. * If you already know the patterns or figure out the pattern before you have graphed all the transformed functions, you may go directly to answering the questions. Cover Page Example: IMP 3 Function POW # Linear, Quadratic and Cubic Functions Hypatia al Khwarizmi* Period 6 Due Date: --358 *Do you know who Hypatia was? Do you know who al Khwarizmi was? What s special about the date in the example?
Linear functions: Parent function: y = x Transformed function: y = ax + b (often learned as y = mx + b) Set A: Set B: Set C: Set D: y = x y = -x y = x + y = x - y = 5x y = -5x y = x + 5 y = x - 5 y = x y = - x y = x + y = x - y =.5 x y = -.5x y =.5 x. If a is positive will the graph be increasing or decreasing? Why?. If a is negative, will the graph be increasing or decreasing? Why? 3. If a > (If a is a large negative OR positive number), how will the graph change? 4. If a < (If a is a small negative OR positive number), how will the graph change? 5. If b is positive, how is the graph transformed? 6. If b is negative, how will the graph be transformed? 7. Describe how you would recognize a linear graph. 8. Complete the tables below: y = 3x What is the value of a? - y = 5x + 4 What is the a-value? - 9. How does the a-value relate to the difference between the y-values (as x increases by )?. Why does this relationship define a linear function?. How do we refer to the a-value when we discuss linear functions? What word do we use?. What is the formula for finding this relationship? 3. If the x-values in a table did NOT increase by, how could we tell if a function was linear or not? Use your procedure to show that the table below represents a linear function. 5 3 3 7 9 8 46
4. How would you recognize a linear relationship in a word problem? 5. How would you identify the b-value from a table when x= is given? 6. How could you calculate the b-value algebraically when x = is not given in the table? 7. What is another term for the b-value? ****************************************************************************** Quadratic Functions: Part - Parent function: y = x Transformed standard form function: y = a x + bx + c y = x y = x + x y = x + y = x + x + 3 y = -x y = x + 5x y = x + 5 y = x + 5x + 3 y = -x y = x - x y = x - y = x + x - 3 y = x y = x - 5x y = x - 5 y = x + 5 x -3 y =.5x y = x +.5x y = x + x + 5 y =.5x y = x + x -5 y = x - x + 5. If a is positive, what is true about the shape of the transformed graph?. If a is negative, what is true about the shape of the transformed graph? 3. If a >, what is true about the shape of the transformed graph? 4. If a <, what is true about the shape of the transformed graph? 5. If c is positive, how is the graph transformed? 6. If c is negative, how is the graph transformed? HONORS: 7. How do a and b interact to transform the graph? Be detailed. Part Parent function: y = x Transformed vertex form function: y = a(x h) + k = (x 5) y = x 5 y = (x 5) 5 y = 4(x 5) = (x + ) y = x + 3 y = (x + ) + 3 y = (x 5) = (x 3) y = x + 6 y = (x-3) 3 + 6 y = ½ (x 5) = -3(x 5) 8. How do the h and k values affect the graph of y = x? Be specific, discussing positive and negative values.
9. How does the a-value affect the graph? If you can reference a previous answer, please feel free to do so. (i.e.: See answer #75 ) Part 3 -. Complete the tables for each function below. Then, find the first and second differences of the y-values. y = x y = 3x + 4 - - - -. Describe how you would recognize a quadratic relationship in a word problem.. What is one of the most common geometric applications for quadratic functions? ************************************************************************ Cubic Functions: Part - Parent Function: y = x 3 Transformed standard form Function: y = ax 3 + bx + cx + d y = x 3 y = x 3 + y = x 3 + x y = x 3 + x y = 5x 3 y = x 3 + 5 y = x 3 x y = x 3 x y =.5x 3 y = x 3 4 y = x 3 + 3x y = x 3 + 3x y = -x 3 y = x 3 y = x 3 3x y = x 3 3x y = -5x 3 y = x 3 +.5x y = x 3 + 5 x. If a is positive, what is true about the shape of the transformed graph?. If a is negative, what is true about the shape of the transformed graph? 3. If a >, what is true about the shape of the transformed graph? 4. If a <, what is true about the shape of the transformed graph? 5. If d is positive, how is the graph transformed? 6. If d is negative, how is the graph transformed? HONORS: 7. Now, experiment with several functions in which a, b, c and d are all changed at the same time. What do you notice? ou might look carefully at the point of inflection.
Part Parent Function: y = x 3 Transformed function in a different form: y = a(x h) 3 + k y = (x 5)3 y = x 3 5 y = (x 5) 3 5 y = 4(x 5) 3 y= (x + ) 3 y = x 3 + 3 y = (x + ) 3 + 3 y = (x 5) 3 y= (x 3) 3 y = x 3 + 6 y = (x-3) 3 + 6 y = ½ (x 5) 3 y = -3(x 5) 3 8. How do the h and k values affect the graph of y = x? Be specific, discussing positive and negative values. ou might look carefully at the point of inflection. 9. How does the a-value affect the graph? If you can reference a previous answer, please feel free to do so. (i.e.: See answer #75 ) Part 3 -. Complete the x-y table below for at least two of the functions above.. Find the first, second and third differences of the y-values.. How would the third differences help you identify a cubic function? - - 3. What common geometric application uses cubic functions? 4. How could we use our calculators to find the x-values where the graph of a cubic crosses the x-axis? 5. How could we extend the zero product rule to find the zeros of the equation below? Please state the zeros. =(3x + 8)(x 9)(x 6) Extension for everyone: 6. How can you calculate the y-intercept of any function? HONORS The degree of a function is the value of its largest exponent. 7. Using your results from the past three explorations, how could you determine if you have a 5 th degree function? 8. An nth degree function? 9. How could we extend the zero product rule to a 5 th degree function? A 6 th degree function? an nth degree function?
. End Behavior of Polynomials a. Graph the following functions: y = x 4 y = -x 4 y = x 5 y = -x 5 y = x 6 y = -x 6 y = x 7 y = -x 7 y =.5x 4 + x 3 3x + y = -x 5 x 4 + 5x 3 + x - 3 (Graph more functions with higher degrees if you need more examples.) b. Based on your work throughout this POW, complete the table below by cutting and pasting (or re-drawing) the four possible polynomial function shapes. The ( ) mean that we are concerned only with the ends of the shape, not the wiggle in the middle. Degree is odd Leading Coefficient is Positive Leading coefficient is Negative Degree is even End Behavior Shapes: A. B..... C. D.....
Vocabulary terms: Point of inflection The point on any graph where the curve changes from concave up to concave down. Concave up: Concave down Point of inflection First, second and third differences of the y-values The first differences are the differences between the actual y-values (when x increases by one). The second differences are the differences between the differences. Etc. Example: (The parentheses show the calculation!) st Difference nd Difference 3 rd Difference 3 (4-) 4 9 (-3) (6-4) 7 (36 9) 6 36 48 8 3 64 44 9 43 4 56 576 768 5 4 Increasing function - If a function is Increasing, the y-values get bigger as the x- values get bigger. The graph goes up from left to right. Decreasing function - If a function is decreasing, the y-values get smaller as the x-values get bigger. The graph goes down from left to right. End Behavior The shape (or behavior) of a graph/function when the x-values are really big positive numbers or really big negative numbers.