Question

Graph the function. \(g(x)=-\cos x+3\)

Short answer

The graph of the function \(g(x)=-\cos x+3\) starts at (0,3), goes down to reach a minimum at \((\pi/2,2)\), rises back to (π,3), reaches a maximum at \((3\pi/2,4)\) and then comes back down to \((2\pi,3)\). It keeps repeating this cycle as x increases or decreases.

Step-by-step solution

Understand basic cosine function

The first step is to understand the basic cosine function. The graph of the cosine function, \(y=\cos x\), starts at (0,1) and oscillates between -1 and 1 as x increases or decreases. It completes one full cycle between \(0\) and \(2\pi\), with an amplitude (the maximum distance of the function from its midline) of 1.

Account for Reflection

A negative sign in front of the cosine function, as seen in \(-\cos x\), reflects the graph over the x-axis. This means where the graph of \(y=\cos x\) would be above the x-axis, the graph of \(y=-\cos x\) would be below, and vice versa. All the peaks become troughs and all troughs become peaks.

Account for Vertical Shift

The \(+3\) in \(g(x)=-\cos x+3\) translates the graph vertically upwards by 3 units. This means every y-coordinate on the graph of \(y=-\cos x\) is increased by 3 to get the corresponding y-coordinate on the graph of \(g(x)=-\cos x+3\). The amplitude remains the same, but the midline of the graph shifts up to \(y=3\). The graph still oscillates between the lines \(y=2\) and \(y=4\).

Graph the Function

To graph \(g(x)=-\cos x+3\), start by graphing \(y=-\cos x\), keeping points like (0,0), \((\pi/2,-1)\), \((\pi,0)\), \((3\pi/2,1)\) and \((2\pi,0)\). Then, shift every point up by three units to arrive at the graph of the given function. For instance, \(x=0\) now corresponds to \(y=3\) instead of \(y=0\).

Key concepts

Cosine Function

Understanding the cosine function is fundamental when dealing with trigonometric graphing. The cosine function, denoted as \(y = \cos x\), is an oscillating function that represents the x-coordinate of a point on the unit circle as it moves around the circle with respect to an angle \(x\). The function is periodic, meaning it repeats values in a predictable pattern, with a period of \(2\pi\) radians, corresponding to a full revolution around the circle.

When graphing the basic cosine function, you will notice the graph starts at the peak value of 1 when \(x=0\), and oscillates to the minimum value of -1, completing what we call a 'wave' over the interval from 0 to \(2\pi\). These 'waves' continue infinitely in both directions along the x-axis. The peaks and troughs of the wave are pivotal points in understanding the transformations like amplitude changes and vertical shifts, as they will move accordingly.

Amplitude

The amplitude of a trigonometric function, such as the cosine function, is a measure of its vertical stretch, illustrating how far the peaks and troughs are from its midline. For the standard cosine function, the amplitude is 1 since the function’s values range from 1 to -1. However, if the cosine function is multiplied by a factor, like \(a\) in \(y = a \cos x\), that factor represents the amplitude, and the graph will stretch or compress accordingly.

For a negative value, such as \(-\cos x\), while the amplitude remains the same (the absolute value of the coefficient, which is 1), the function reflects across the x-axis due to the negative sign. This means that the graph’s peaks and troughs invert positions, but the distance from the midline to the peaks (or troughs) remains unchanged. It's crucial to remember that amplitude is always a positive value, representing a distance without direction.

Vertical Shift

A vertical shift in graphing trigonometric functions occurs when a constant is added or subtracted from the function. This shifting does not alter the period or amplitude of the function but simply moves the entire wave up or down on the graph. In the function \(g(x)=-\cos x+3\), the '+3' signifies an upward vertical shift of 3 units.

This transformation changes the midline, which is the horizontal line equidistant from the peaks and troughs of the function, from \(y=0\) (for the basic cosine function) to \(y=3\). As a result, every point on the graph of the function shifts vertically, and the function now oscillates between the new bounds set by this vertical shift, in this case, from \(y=2\) to \(y=4\). It is also essential to bear in mind that the vertical shift affects where the function crosses the y-axis; this crossing point moves up or down together with the entire graph.