Trigonometric graphs for SQA National 5 Maths
This page covers the following topics:
1. Graphs of trigonometric functions
2. Transformations of trigonometric graphs
The graphs for sine and cosine are almost identical, with the only difference being that the cosine graph is the sine graph shifted by 90° to the right. The graphs for sine and cosine both have an amplitude of 1, meaning that the maximum value that the two trigonometric functions can take is 1. They are both 360° periodic, meaning that the graph will be repeated after every 360°.
The cosine graph is symmetric over the y-axis, since when it is flipped across it, the two halves of the graph match up. On the other hand, the graph of sine is symmetric about the origin, since the graph is flipped over both the x-axis and the y-axis. This means that for every x value with a corresponding value of y, −x matches to −y.
Tangent graph is significantly different to both sine and cosine graphs. The graph of tangent is 180° periodic. The graph passes through the origin and tends to infinity both in the positive and the negative direction, and thus its amplitude is undefined. Since tangent is undefined for odd multiples of 90°, at these values of x, the graph of tangent has asymptotes.
If y = f(x) is a trigonometric function, then y = f(ax) is a horizontal stretch of scale factor 1/a, whereas y = af(x) is a vertical stretch with a scale factor a. y = f(−x) is a reflection in the y-axis, whereas y = −f(x) is a reflection in the x-axis. y = f(x) + c is a translation of c units up, whereas y = f(x + c) is a translation of c units to the left.
State the maximum amplitude reached by the graph of y = 7sinx + 2.
The graph of y = 7sinx + 2 is a vertical stretch of the y = sinx graph of scale factor 7 and a translation of 2 units up. Since the amplitude of the y = sinx graph is 1, the amplitude of y = 7sinx + 2 is 7 × 1 + 2 = 9.
What is the first positive solution of the equation 2cos(3x) = 0?
The first positive solution of y = cosx is x = 90°. Since y = 2cos(3x) is a vertical stretch of scale factor 2 and a horizontal stretch of scale factor 1/3 of the y = cosx graph, the first solution of 2cos(3x) = 0 is x = 30°.
Explain why the graph of tangent has asymptotes.
Tangent is undefined for odd multiples of 90°, therefore for those x values, the graph will not have a point, thus it has asymptotes there.
Tangent is undefined at odd multiples of 90°.
State the roots of the equation cosx + 3 = 0.
y = cosx + 3 is a translation of the graph of y = cosx 3 units up. This graph does not cross the x-axis, therefore y = cosx + 3 does not have any roots.
Sketch the graph of y = sin(−2x).
If y = f(x) is a trigonometric function, then y = f(ax) is a horizontal stretch of scale factor 1/a, y = f(−x) is a reflection in the y-axis. Thus, the sine graph is compressed by 2 times and is reflected in the y-axis.
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