*Ray is a licensed engineer in the Philippines. He loves to write about mathematics and civil engineering.*

*Secant* is an even function that serves as the reciprocal of the cosine function. It is a trigonometric function with a more extensive graph in the negative direction rather than a smaller one. As the fractions in the cosine function get smaller or closer to zero, their reciprocals in the secant function get broader and longer in the opposite direction.** **

A secant function has the same period as its reciprocal. The cosine function has a period of 2π, so the secant function has a period of 2π. It is discontinuous at values for x, for which its reciprocal is zero. For secant functions, say y = sec (x), the values for which the function is not defined are x = ± [(2n +1)π] / 2 and the asymptotes are at x = ± [(2n +1)π] / 2. Near those values of x for which each function is insignificant, the values of the trigonometric functions are unbounded. Hence, the amplitude is insignificant for these functions.

The graph of a secant can be sketched by first sketching the graph of y = cos (x) and then estimating the reciprocal of cos (x). Take a look at the y = sec (x) and y = cos (x) shown below. As observed, the u-shapes of the secant graph touch the graph of cos (x) at maximum and minimum points.

## Secant Graph Properties

For the secant function y =** **α sec βx,

**Period**. The period is the absolute value of |2π / β|. For y = sec (x), the period is 2π.**Amplitude**. Secant functions have no defined amplitude. The secant graphs go on unending or infinite in vertical directions. However, for some cases, given the equation y = α sec (βx - c) - d, the secant graph will have an amplitude of |α|. It means that one should multiply all points of the vertical axis (y-coordinates) in the graph by the value of α. However, still, the graph goes on forever in the vertical direction.**Domain**. The domain of y = sec (x) is all real numbers x, except x = ± [(2n +1)π] / 2. In other words, the domain of the secant function excludes π/2 +*k*n, where k is applicable for all integers. Ordered pairs of the form (x, sec (x)) make up the secant function. Since sec (x) is the reciprocal of cos (x), the domain of the secant function is the set of real numbers, except those for which cos (x) = 0. Hence, the domain contains all elements x ∈**ℝ**such that: x ≠ π/2 + nπ satisfies. Since |cos x| < 1 for all s ∈**ℝ**, its reciprocal satisfies |sec x| > 1, which determines the range of the secant function.**Range.**The range of y = sec (x) is all real numbers y, except -1 < y < 1. The range of y = sec (x) is equal to (-∞, -1] U [1, ∞).**Continuity**. The function y = sec (x) is discontinuous at x = ± [(2n +1)π] / 2.**Vertical Asymptotes**. The vertical asymptotes of y = sec (x) occur at π/2 and repeat every π units. The x-intercepts of y = cos (x) are the asymptotes for y = sec (x). The vertical asymptotes of y = sec (x) is given by the equation x = π/2 + πn. In other resources, it is easy to identify the vertical asymptotes for the secant graph by solving the inequality -π/2 < βx + c < π/2.**Maximum and Minimum Values.**The maximum values of y = cos (x) are minimum values of y = sec(x). On the other hand, the minimum values of y = cos (x) are the maximum values of the negative sections of y = sec (x).**X-intercepts**. The trig secant graph has no x-intercepts.**Symmetry.**Secant graphs are always symmetrical about the y-axis.

**Note: **For the secant function with more terms, apply phase shifting. Say, y = α sec (βx - c) - d, the period of the function is still 2π / β, the phase shift/horizontal shift is c/β, and the vertical shift is -d.

## How to Graph a Secant Function

In drawing the secant curve on graphing paper, take note that secant graphs with no shifts automatically have x and y coordinates equal to (0,1), (1,-1), and (2,1). However, there are few things needed to consider for cases with variations to draw the secant graph fully. Just follow the steps shown below.

**Identify the period of the given secant equation**. As mentioned earlier, the period of any secant function is 2π / β, given an equation y = α sec (βx - c) - d. Therefore, the period of the secant graph 2π / β is the length of one cycle of the secant curve.**Locate the asymptotes and other parameters of the secant graph.**Since secant is the reciprocal of the cosine function, any point on the cosine graph where the value is zero generates a vertical asymptote on the secant graph. It is helpful since it defines the secant graph early. Take note that the parent graph y = cos (x) has 0 values at angles 3π/2 and π/2.**Identify if there are horizontal and vertical shifts.**For y = α sec (βx - c) - d, the horizontal shift is c/β to the right, and the vertical shift is -d in the downward direction.**Identify the amplitude.**The amplitude of the secant equation y = α sec (βx - c) - d equals |α|. Multiply all values or points with |α|.**Tabulate values of the cosine and secant functions.**Apply this step if values along the curves of the secant graph are needed. Substitute the radians into the given equation, of course, excluding the asymptotes obtained from the first step.

To be able to graph a secant function, see the examples given below.

## Example 1: Sketching the Secant Graph With Horizontal Shift

Sketch the graph of the secant equation y = sec (x - π/4).

**Solution**

First, identify the parameters before sketching the trigonometric secant graph.

**Range**: Identify the range of the given secant equation.

Range: (-∞, -1) U (1, +∞)

**Period**: Solve for the period of y = sec (2x - π/3) using the formula p = 2π/β. Since the resulting period is π, this means that the secant graph is

2π/β = 2π/2

2π/2 = π

**Horizontal Shifts**: Since the term -π/4 is present in the equation, then the secant graph shifts horizontally. It means that the graph will move horizontally to the right direction π/4 units.

y = sec (x - π/4)

Horizontal shift = - π/4

**Vertical Shifts**: None

**Vertical Asymptotes**: Solve for the vertical asymptotes of the equation using the given inequality below.

-π/2 < x - π/4 < π/2

x - π/4 < π/2

x < π/2 + π/4

x < 3π/4

-π/2 < x - π/4

-π/2 + π/4 < x

-π/4 < x

Finally, sketch the graph by locating the vertical asymptotes, the horizontal shift, and the period. Also, check the tabulated values of the given function to locate specific points as shown in the table shown.

## Example 2: Secant Graph With Amplitude

Sketch the graph of the secant equation y = 2 sec (x - π/4).

**Solution**

First, identify the parameters before sketching the trigonometric secant graph.

**Range**: Identify the range of the given secant equation.

Range: (-∞, -1) U (1, +∞)

**Period**: Solve for the period of y = sec (2x - π/3) using the formula p = 2π/β. Since the resulting period is π, this means that the secant graph is

2π/β = 2π/2

2π/1 = π

**Horizontal Shifts**: Since the term -π/4 is present in the equation, then the secant graph shifts horizontally. It means that the graph will move horizontally to the right direction π/4 units.

y = sec (x - π/4)

Horizontal shift = - π/4

**Vertical Shifts**: None

**Vertical Asymptotes**: Solve for the vertical asymptotes of the equation using the given inequality below.

-π/2 < x - π/4 < π/2

x - π/4 < π/2

x < π/2 + π/4

x < 3π/4

-π/2 < x - π/4

-π/2 + π/4 < x

-π/4 < x

**Amplitude: **Multiply the y-coordinates of the graph to execute the amplitude of the secant graph.

Finally, sketch the graph by locating the vertical asymptotes, the horizontal shift, and the period.

## Example 3: Secant Graph With Vertical Shift

Identify the parameters of the secant equation y = sec (x) - 3 and sketch it.

**Solution**

Identify the parameters before sketching the graph of the secant function.

**Range**: Identify the range of the given secant equation.

Range: (-∞, -1) U (1, +∞)

**Period**: Solve for the period of y = sec (x) - 3 using the formula p = 2π/β. Since the resulting period is π, this means that the secant graph is

2π/β = 2π/1

2π/1 = 2π

**Vertical Shifts**: There is a vertical shift of 3 units downward since it is a negative shift.

**Vertical Asymptotes**: The vertical asymptotes of the equation using the inequality formula is -π/2 < x < π/2. It means that the unique feature about this secant graph is its vertical shift and no adjustments for its vertical asymptotes.

Lastly, sketch the graph by locating the vertical asymptotes, the vertical shift, and the period. Please see the tabulated values for reference.

## Example 4: Secant Graph With Horizontal and Vertical Shifts

Sketch the graph y = sec (x - 2) + 3.

**Solution**

**Period**: Solve for the period of y = sec (x -2) + 3 using the formula p = 2π/β.

2π/β = 2π/1

2π/β = 2π

**Horizontal Shifts**: Since the term -2 is present in the equation, then the secant graph shifts horizontally. It means that the graph will move horizontally to the right direction two units.

y = α sec (βx - c)

Horizontal shift = c/β

Horizontal shift = - 2/1

Horizontal shift = - 2

**Vertical Shifts**: There is a vertical shift of 3 units upward since it is a positive movement.

**Vertical Asymptotes**: Solve for the vertical asymptotes of the given equation using the given below.

-π/2 < x - 2 < π/2

x - 2 < π/2

x < π/2 + 2

-π/2 < x - 2

2 - π/2 < x

Lastly, sketch the graph by locating the vertical asymptotes, horizontal and vertical shifts, and the period. Please see the tabulated values for reference.

## Example 5: Sketching a Secant Graph With Horizontal Shift

Sketch the graph of y = sec (2x - π/3).

**Solution**

Identify all parameters in sketching the graph of the given secant function.

**Range**: First is to identify all the parameters needed in graphing the given secant function. Identify the range of the given secant equation.

Range: (-∞, -1) U (1, +∞)

**Period**: Solve for the period of y = sec (2x - π/3) using the formula p = 2π/β.

2π/β = 2π/2 = π

**Horizontal Shifts**: Since the term -π/3 is present in the equation, then the secant graph shifts horizontally. However, first, rewrite the equation eliminating the coefficient of the variable x. It means that the graph will move horizontally to the right direction π/6 units.

y = α sec (βx - c)

Horizontal shift = c/β

y = sec (2x - π/3)

y = sec [2 (x - π/6)]

Horizontal shift = - π/6

**Vertical Asymptotes**: Solve for the vertical asymptotes of the given equation using the given below.

2x - π/3 = π/2 + kπ

2x = π/2 + π/3 + kπ

2x = 5π/6 + kπ

x = 5π/12 + kπ/2

At k = 0

x = 5π/12 + (0)π/2

x = 5π/12

Finally, sketch the secant function y = sec (2x - π/3). Start with π/6 units to the right and ends at 7π/6, which is π/6 + π.

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*This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.*

**© 2021 Ray**