Section 5.3 Trigonometric Graphs Chapter 5 – Trigonometric Functions: Unit Circle Approach Section 5.3 Trigonometric Graphs 5.3 - Trigonometric Graphs

5.3 - Trigonometric Graphs Graph y = sinx We graph a function by plotting points that satisfy that function in the coordinate system. We will graph the sine function on the period [0,2] 5.3 - Trigonometric Graphs

5.3 - Trigonometric Graphs This is called the primary cycle of the sine function. 5.3 - Trigonometric Graphs

5.3 - Trigonometric Graphs Remember that the sine function will repeat indefinitely because it is periodic so we have the graph: 5.3 - Trigonometric Graphs

5.3 - Trigonometric Graphs Properties Domain Range Period Even/Odd x-int Min/Max 5.3 - Trigonometric Graphs

5.3 - Trigonometric Graphs y = A sin (Bx - C) + D We need to know what A, B, C, and D mean on the sine graph. 5.3 - Trigonometric Graphs

5.3 - Trigonometric Graphs Amplitude Amplitude, |A|, adjusts the height of the graph and determines the range of the graph. The range of the function is [-|A|, |A|]. If |A| ≥ 1, then the graph stretches. If |A| ≤ 1, then the graph shrinks. If A < 0, then the graph reflects over the x-axis. 5.3 - Trigonometric Graphs

5.3 - Trigonometric Graphs Period The period is (2)/B 5.3 - Trigonometric Graphs

5.3 - Trigonometric Graphs Phase Shift The phase shift, C/B, will help us find the starting point and ending point of the primary cycle. If C/B < 0, then the graph shifts left. If C/B > 0, then the graph shifts right. 5.3 - Trigonometric Graphs

5.3 - Trigonometric Graphs Vertical Shift Vertical shift, D, shifts the graph vertically. If D > 0, then the graph shifts up D units. If D < 0, then the graph shifts down D units. Max or min y-value is A+D or –A+D. The smaller value is the min and the large value is the max. The graph will be centered vertically on the line y = D. 5.3 - Trigonometric Graphs

Graphing the Sine Function Identify the A, B, C, and D values Identify the amplitude, period, and phase shift. Find the values for the five key points – the three intersection points to y=D, the max, and the min. Connect the 5 key points with a smooth curve and graph the primary cycle. Extend the graph if needed. 5.3 - Trigonometric Graphs

Graphing the Negative Sine Function When graphing the negative sine function, start off by graphing the positive sine function. Then reflect the graph over the y = D line. NOTE: This will change your minimum and maximums. 5.3 - Trigonometric Graphs

5.3 - Trigonometric Graphs Examples Determine the Sine equation. 5.3 - Trigonometric Graphs

5.3 - Trigonometric Graphs Examples Determine the amplitude, period, phase shift and vertical shift of the function below. Graph the primary cycle. 5.3 - Trigonometric Graphs

5.3 - Trigonometric Graphs Graph y = cosx We graph a function by plotting points that satisfy that function in the coordinate system. We will graph the cosine function on the period [0,2] 5.3 - Trigonometric Graphs

5.3 - Trigonometric Graphs The primary cycle is in dark blue. 5.3 - Trigonometric Graphs

5.3 - Trigonometric Graphs Properties Domain Range Period Even/Odd x-int Min/Max 5.3 - Trigonometric Graphs

5.3 - Trigonometric Graphs y = Acos(Bx - C) + D Notice that we have the same values as the sine function so we will follow a similar method to graph the cosine function. 5.3 - Trigonometric Graphs

Graphing the Cosine Function Identify the A, B, C, and D values Identify the amplitude, period, and phase shift. Find the values for the five key points –the two maxs, the min, and the two intersection points to y = D, Connect the 5 key points with a smooth curve and graph the primary cycle. Extend the graph if needed. 5.3 - Trigonometric Graphs

Graphing the Negative Cosine Function When graphing the negative cosine function, start off by graphing the positive cosine function. Then reflect the graph over the y = D line. NOTE: This will change your minimum and maximums. 5.3 - Trigonometric Graphs

5.3 - Trigonometric Graphs Examples Determine the cosine equation. 5.3 - Trigonometric Graphs

5.3 - Trigonometric Graphs Examples Determine the amplitude, period, phase shift and vertical shift of the function below. Graph the primary cycle. 5.3 - Trigonometric Graphs