Graphing functions by plotting points. Knowing the values of the sine function for the special angles.

Transcription

1 Spaghetti Sine Graphs Summary In this lesson, students use uncooked spaghetti and string to measure heights on the unit circle and create the graph of the y = sin(x). This is a great lesson to help students understand why the graph of the sine is a wave. Utah State Core Standard Graph sine and cosine functions. Calculate the exact values of the sine, cosine, and tangent functions for the special angles of the unit circle. Desired Results Benchmark/Enduring Understanding Students will understand the important relationship between the graph of y = sin(x) and the unit circle definition of the sine function. Essential Questions What does the graph of y = sin(x) look like? Is the unit circle a graph of y = sin(x)? How does the unit circle relate to the graph of y = sin(x)? Assessment Evidence Skills Graphing functions by plotting points. Knowing the values of the sine function for the special angles. The questions at the end of the lesson assess student understanding of the graph of y = sin(x). This lesson is an introduction to graphing sine functions. After completing a unit on graphing trig functions, students should be able to sketch the graph of sine and cosine functions using transformation. Instructional Activities Launch: Post the essential questions. Ask students to vote on whether or not the unit circle is a graph of y = sin(x). Discuss their responses. Explore: Students can work in pairs to complete the worksheet. Summarize: Discuss student responses to the questions at the end of the lesson. Materials Needed Uncooked spaghetti String Scissors Legal paper Copies of worksheet Tape

2 Spaghetti Sine Name: Period: Materials -spaghetti -tape -4 pieces of legal size paper taped together -yarn or string (approx 4 ) -circle partitioned off (see 3 rd page) Instructions 1. Tape two pieces of paper together lengthwise. Fold the paper hotdog-style & draw a line along the fold. Then draw a perpendicular line to the line you just drew approximately one inch from the edge of the paper.. Wrap the yarn around the circle so that it is the length of the circumference of the circle. 3. Use a marker to mark each angle on the yarn/string. 4. Place the yarn/string on the horizontal line on your paper. Transfer the marks from the yarn/string onto your paper. Now label each tick mark you just wrote on your paper so that they correspond to the angles on the circle.,,, Break one piece of spaghetti so that it is the vertical distance from the initial angle 0 to the point 6 on the circle. 6. Tape this piece above the string at the mark labeled Continue doing this for all the points on the circle. 8. Write the y value of sine at each radian measure. 9. Repeat this for the Cosine exercise with a different color of yarn and the horizontal distances.

3 QUESTIONS Spaghetti Sine/Cosine Graphs Name Date Period 1. Looking at the graph, what is the range of the graph (Interval notation)?. What do you think will occur after? 3. What do you think will occur before 0? 4. Using Questions and 3, what is the domain of this portion of the graph included on your paper (interval notation)? 5. How is the sine function related, or similar to, the cosine function? 6. Using what you ve learned about the transformation of graphs in prior context, what do you think the graph of f(x) = sin θ + will look like?. Graph the function accurately below. Label the x-axis for the quadrantal angles between 0 and π. y x 7. Using what you ve learned about the transformation of graphs in prior context, what do you predict the graph of f(x) = sin (θ + π/) will look like? Graph the function accurately below. Label the x-axis for the quadrantal angles between -π and π.

4 8. Using what you ve learned about the transformation of graphs in prior context, what do you think the graph of f(x) = sin (-θ + π/) will look like? Graph the function accurately below. Label the x-axis for the quadrantal angles between -π and π. 9. Go to the computer lab and graph the following trig functions. Connect your calculator to the computer and go to TI Connect to print each one. Then paste them in the space provided. For each graph make sure to graph either y = sin x or y = cos x in y 1 first so you can see the transformation. Change the size of your graph to by before you print it. Procedure: i.) Make sure your mode is in radians. ii.) Change your window so that x min & x max are between -π and π and the x-scale is π/. iii.) Change your window so that y min & x max are between -5 and 5 and the y-scale is 1. a.) y = -sin x b.) y = -cos x paste graph here paste graph here What did you discover about this transformation?

5 b.) y =3sin x b.) y = 3 1 cos x paste graph here paste graph here What did you discover about this transformation? 10. Summary: Define what you discovered about the transformations of the sine and cosine graphs. What shifts the graphs up/down? What shifts the graphs right/left? What reflects the graph over the x-axis? What happens when the angle is negative? 11. Graph the following trig functions BY HAND. Accurately label the x- and y-axis for each grid line. You must first graph either y = sin x or y = cos x and then the final function IN A DIFFERENT COLOR. Do not graph between grid lines or outside of the graph. a.) 3 y sin x b.) y cos x 4

6 c.) y sin x 1 d.) y cos x e.) y sin x f.) y cos x

7 5 g.) y sin x h.) y cos x 4 i.) 1 y sin x 3 j.) y cos x

8 3 k.) y sin 3 x l.) y cos x 1 5 m.) y 3sin x 4 n.) y cos x 1 3 4