The Trig Graphs Build-A-Graph Day (5.5)

POD #1 Complete these limit statements. For limit statements, go to the graphs, not the unit circle. For which of these does direction not matter? In other words, which ones have the same limit at the given value of x, from either side?

POD #1 Complete these limit statements. For limit statements, go to the graphs, not the unit circle. For which of these does direction not matter? Sine and tangent

POD #2– if we have time By tables, on calculators, graph one of these functions. You can do it on the calculator. Compare it to its parent function. Then one person from each table go to the board to sketch the graph and its parent. Then we’ll compare them to their parents.

What we’ve done… … is transform the graphs of two basic trig functions, sine and cosine. Ultimately, we’re going to graph and each change in the equation from the parent corresponds to a change in the graph from the parent. As you might expect, the change in the graph is opposite the change in the equation.

Amplitude Amplitude refers to what aspect of the graph? How do we determine amplitude from a graph? Which graphs besides these have amplitude? From the equation, amplitude is determined by |a|. Amplitude is a scale change of which variable? How is this “opposite” from the equation? What happens when a is negative?

Period Period refers to what aspect of the graph? How do we determine period from a graph? From the equation, period is determined by 2π/|b|. Period is a scale change of which variable? How is this “opposite” from the equation? What happens when b is negative?

Vertical shift Vertical shift refers to what aspect of the graph? How do we determine vertical shift from a graph? From the equation, vertical shift is determined by d. Vertical shift is a translation of which variable? How is this “opposite” from the equation? What happens when d is negative?

Phase shift Horizontal (phase) shift refers to what aspect of the graph? From the equation, phase shift is determined by –c/b. THIS PHASE SHIFT IS SEEN IN MOVEMENT FROM THE Y-AXIS. Phase shift is a translation of which variable? What happens when c/b changes sign?

Cycles These are VERY helpful when graphing trig functions. You can determine the interval for a full cycle of a trig function by solving for x: 0 ≤ bx+c ≤ 2π This will give us an interval starting when sin x = 0 and cos x = 1, the same values we have on the y-axis for these parent functions.

Use it Find the amplitude, period, and vertical and phase shifts for. What is an interval for a complete cycle?

Use it Find the amplitude, period, and vertical and phase shifts for What is the interval for a complete cycle? Amplitude: 2 *How does the Period: πinterval for the Vertical shift: 1cycle compare Phase shift: -π/4with the values for Cycle: [-π/4,3π4]*period and phase shift? (Foot stomp)

Use it Using this information, graph the function.

Use it Using this information, graph the function. In this graph, the x interval is π/4, and the y interval is 1. I also graphed y = 1 to show the vertical shift. Note how the cycle starts at –π/4 (where sine is 0) and runs a length of π.

Use it Whaddya think? Even, odd, or neither?

Use it Find the equation from the graph. (We simply work backwards.) In this graph both the x and y intervals are 1. Which parent function would you use? How could you use the other one?

Use it Find the equation from the graph. (We simply work backwards.) In this graph both the x and y intervals are 1. What are the amplitude and period? What are the shifts?

Use it Find the equation from the graph. (We simply work backwards.) In this graph both the x and y intervals are 1. What are the amplitude and period? Amplitude is 10/2 and period is 4. Vertical shift is 3.

Use it So, a = 5, b = π/2, and d = 3. If we used cosine, we’d have no phase shift, c = 0.

Use it If we used sine, we would have a phase shift. We can determine the phase shift by seeing where the graph crosses the “center,” y = 3.

Use it The interval “starts” at x = -1, so we have a phase shift of -1. This means –c/b = -1. Using our value for b, we get c = π/2. Test: 0 ≤ π/2 x + π/2 ≤ 2π - π/2 ≤ π/2 x ≤ 3π/2 -1 ≤ x ≤ 3 It checks.