Trigonometric Functions and Graphs - Trigonometry
Card 0 of 20
Let 
be a function defined as follows:
The 4 in the function above affects what attribute of the graph of 
?
Let
The 4 in the function above affects what attribute of the graph of
The period of the function is indicated by the coefficient in front of 
; here the period is unchanged.
The amplitude of the function is given by the coefficient in front of the 
; here the amplitude is -1.
The phase shift is given by the value being added or subtracted inside the 
function; here the shift is 
units to the right.
The only unexamined attribute of the graph is the vertical shift, so 4 is the vertical shift of the graph. A vertical shift of 4 means that the entire graph of the function will be moved up four units (in the positive y-direction).
The period of the function is indicated by the coefficient in front of
The amplitude of the function is given by the coefficient in front of the
The phase shift is given by the value being added or subtracted inside the
The only unexamined attribute of the graph is the vertical shift, so 4 is the vertical shift of the graph. A vertical shift of 4 means that the entire graph of the function will be moved up four units (in the positive y-direction).
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Let be a function defined as follows:
What is the vertical shift in this function?
Let
What is the vertical shift in this function?
The period of the function is indicated by the coefficient in front of ; here the period is unchanged.
The amplitude of the function is given by the coefficient in front of the ; here the amplitude is 3.
The phase shift is given by the value being added or subtracted inside the cosine function; here the shift is units to the right.
The only unexamined attribute of the graph is the vertical shift, so -3 is the vertical shift of the graph. A vertical shift of -3 means that the entire graph of the function will be moved down three units (in the negative y-direction).
The period of the function is indicated by the coefficient in front of
The amplitude of the function is given by the coefficient in front of the
The phase shift is given by the value being added or subtracted inside the cosine function; here the shift is
The only unexamined attribute of the graph is the vertical shift, so -3 is the vertical shift of the graph. A vertical shift of -3 means that the entire graph of the function will be moved down three units (in the negative y-direction).
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This graph shows a translated cosine function. Which of the following could be the equation of this graph?
This graph shows a translated cosine function. Which of the following could be the equation of this graph?
The correct answer is 
. There are no sign changes with vertical shifts; in other words, when the function includes 
, it directly translates to moving up three units. If you thought the answer was 
, you may have spotted the y-intercept at 
and jumped to this answer. However, recall that the y-intercept of a regular 
function is at the point 
. Beginning at 
and ending at 
corresponds to a vertical shift of 3 units.
The correct answer is 
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The graph below shows a translated sine function. Which of the following functions could be shown by this graph?
The graph below shows a translated sine function. Which of the following functions could be shown by this graph?
A normal 
graph has its y-intercept at 
. This graph has its y-intercept at 
. Therefore, the graph was shifted down three units. Therefore the function of this graph is 
.
A normal 
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Consider the function 
. What is the vertical shift of this function?
Consider the function 
The general form for the secant transformation equation is 
. 
represents the phase shift of the function. When considering 
we see that 
, so our vertical shift is 
and we would shift this function 
units up from the original secant function’s graph.
The general form for the secant transformation equation is 
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Which of the following is the graph of 
with a vertical shift of 
?
Which of the following is the graph of 
The graph of 
with a vertical shift of 
is shown below. This can also be expressed as 
.
Here is a graph that shows both 
and 
, so that you can see the "before" and "after." The original function is in blue and the translated function is in purple.
The graphs of the incorrect answer choices are 
(no vertical shift applied), 
(shifted upwards instead of downwards), 
(amplitude modified, and shifted upwards instead of downwards), and 
(shifted downwards 3 units, but this is not the correct original graph of simply 
since the amplitude was modified.)
The graph of 
Here is a graph that shows both 
The graphs of the incorrect answer choices are 
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Which of the following graphs shows one of the original six trigonometric functions with a vertical shift of 
applied?
Which of the following graphs shows one of the original six trigonometric functions with a vertical shift of 
We are looking for an answer choice that has one of the six trigonometric functions, as well as that function shifted up 3 units. The only answer choice that displays that is this graph of 
(purple) and 
(blue).
The incorrect answers depict 
and 
, 
and 
, and 
and 
.
We are looking for an answer choice that has one of the six trigonometric functions, as well as that function shifted up 3 units. The only answer choice that displays that is this graph of 
The incorrect answers depict 
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For which values of 
, where 
in the unit circle, is 
undefined?
For which values of
Recall that 
. Since the ratio of any two real numbers is undefined when the denominator is equal to 
, 
must be undefined for those values of 
where 
. Restricting our attention to those values of 
between 
and 
, 
when 
or 
. Hence, 
is undefined when 
or 
.
Recall that
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What is the domain of f(x) = sin x?
What is the domain of f(x) = sin x?
The domain of a function is the range of all possible inputs, or x-values, that yield a real value for f(x). Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Sine is defined as the ratio between the side length opposite to the angle in question and the hypotenuse (SOH, or sin x = opposite/hypotenuse). In any triangle created by the angle x and the x-axis, the hypotenuse is a nonzero number. As a result, the denominator of the fraction created by the definition sin x = opposite/hypotenuse is not equal to zero for any angle value x. Therefore, the domain of f(x) = sin x is all real numbers.
The domain of a function is the range of all possible inputs, or x-values, that yield a real value for f(x). Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Sine is defined as the ratio between the side length opposite to the angle in question and the hypotenuse (SOH, or sin x = opposite/hypotenuse). In any triangle created by the angle x and the x-axis, the hypotenuse is a nonzero number. As a result, the denominator of the fraction created by the definition sin x = opposite/hypotenuse is not equal to zero for any angle value x. Therefore, the domain of f(x) = sin x is all real numbers.
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What is the domain of f(x) = cos x?
What is the domain of f(x) = cos x?
The domain of a function is the range of all possible inputs, or x-values, that yield a real value for f(x). Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Cosine is defined as the ratio between the side length opposite to the angle in question and the hypotenuse (CAH, or cos x = adjacent/hypotenuse). In any triangle created by the angle x and the x-axis, the hypotenuse is a nonzero number. As a result, the denominator of the fraction created by the definition cos x = adjacent/hypotenuse is not equal to zero for any angle value x. Therefore, the domain of f(x) = cos x is all real numbers.
The domain of a function is the range of all possible inputs, or x-values, that yield a real value for f(x). Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Cosine is defined as the ratio between the side length opposite to the angle in question and the hypotenuse (CAH, or cos x = adjacent/hypotenuse). In any triangle created by the angle x and the x-axis, the hypotenuse is a nonzero number. As a result, the denominator of the fraction created by the definition cos x = adjacent/hypotenuse is not equal to zero for any angle value x. Therefore, the domain of f(x) = cos x is all real numbers.
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Which of the following trigonometric functions is undefined?
Which of the following trigonometric functions is undefined?
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Secant is the reciprocal of cosine, so the secant of any angle x for which cos x = 0 must be undefined, since it would have a denominator equal to 0. The value of cos (pi/2) is 0, so the secant of (pi)/2 must be undefined.
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Secant is the reciprocal of cosine, so the secant of any angle x for which cos x = 0 must be undefined, since it would have a denominator equal to 0. The value of cos (pi/2) is 0, so the secant of (pi)/2 must be undefined.
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Which of the following trigonometric functions is undefined?
Which of the following trigonometric functions is undefined?
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Cotangent is the reciprocal of tangent, so the cotangent of any angle x for which tan x = 0 must be undefined, since it would have a denominator equal to 0. The value of tan (pi) is 0, so the cotangent of (pi) must be undefined.
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Cotangent is the reciprocal of tangent, so the cotangent of any angle x for which tan x = 0 must be undefined, since it would have a denominator equal to 0. The value of tan (pi) is 0, so the cotangent of (pi) must be undefined.
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Which of the following trigonometric functions is undefined?
Which of the following trigonometric functions is undefined?
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Secant is the reciprocal of cosine, so the secant of any angle x for which cos x = 0 must be undefined, since it would have a denominator equal to 0. The value of cos 3(pi/2) is 0, so the secant of 3(pi)/2 must be undefined.
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Secant is the reciprocal of cosine, so the secant of any angle x for which cos x = 0 must be undefined, since it would have a denominator equal to 0. The value of cos 3(pi/2) is 0, so the secant of 3(pi)/2 must be undefined.
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Which of the following trigonometric functions is undefined?
Which of the following trigonometric functions is undefined?
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Cotangent is the reciprocal of tangent, so the cotangent of any angle x for which tan x = 0 must be undefined, since it would have a denominator equal to 0. The value of tan (0) is 0, so the cotangent of (0) must be undefined.
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Cotangent is the reciprocal of tangent, so the cotangent of any angle x for which tan x = 0 must be undefined, since it would have a denominator equal to 0. The value of tan (0) is 0, so the cotangent of (0) must be undefined.
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Which of the following trigonometric functions is undefined?
Which of the following trigonometric functions is undefined?
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Cosecant is the reciprocal of sine, so the cosecant of any angle x for which sin x = 0 must be undefined, since it would have a denominator equal to 0. The value of sin (0) is 0, so the cosecant of 0 must be undefined.
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Cosecant is the reciprocal of sine, so the cosecant of any angle x for which sin x = 0 must be undefined, since it would have a denominator equal to 0. The value of sin (0) is 0, so the cosecant of 0 must be undefined.
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Which of the following trigonometric functions is undefined?
Which of the following trigonometric functions is undefined?
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Tangent is defined as the ratio between the side length opposite to the angle in question and the side length adjacent to it (TOA, or tan x = opposite/adjacent). In a triangle created by the angle x and the x-axis, the adjacent side length lies along the x-axis; however, when the angle x lies on the y-axis, no length can be drawn along the x-axis to represent the angle. As a result, the denominator of the fraction created by the definition tan x = opposite/adjacent is equal to zero for any angle along the y-axis (90 or 270 degrees, or pi/2 or 3pi/2 in radians.) Therefore, tan 3(pi)/2 is undefined.
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Tangent is defined as the ratio between the side length opposite to the angle in question and the side length adjacent to it (TOA, or tan x = opposite/adjacent). In a triangle created by the angle x and the x-axis, the adjacent side length lies along the x-axis; however, when the angle x lies on the y-axis, no length can be drawn along the x-axis to represent the angle. As a result, the denominator of the fraction created by the definition tan x = opposite/adjacent is equal to zero for any angle along the y-axis (90 or 270 degrees, or pi/2 or 3pi/2 in radians.) Therefore, tan 3(pi)/2 is undefined.
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Which of the following trigonometric functions is undefined?
Which of the following trigonometric functions is undefined?
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Cosecant is the reciprocal of sine, so the cosecant of any angle x for which sin x = 0 must be undefined, since it would have a denominator equal to 0. The value of sin (pi) is 0, so the cosecant of pi must be undefined.
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Cosecant is the reciprocal of sine, so the cosecant of any angle x for which sin x = 0 must be undefined, since it would have a denominator equal to 0. The value of sin (pi) is 0, so the cosecant of pi must be undefined.
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Which of the following trigonometric functions is undefined?
Which of the following trigonometric functions is undefined?
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Tangent is defined as the ratio between the side length opposite to the angle in question and the side length adjacent to it (TOA, or tan x = opposite/adjacent). In a triangle created by the angle x and the x-axis, the adjacent side length lies along the x-axis; however, when the angle x lies on the y-axis, no length can be drawn along the x-axis to represent the angle. As a result, the denominator of the fraction created by the definition tan x = opposite/adjacent is equal to zero for any angle along the y-axis (90 or 270 degrees, or pi/2 or 3pi/2 in radians.) Therefore, tan (pi)/2 is undefined.
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Tangent is defined as the ratio between the side length opposite to the angle in question and the side length adjacent to it (TOA, or tan x = opposite/adjacent). In a triangle created by the angle x and the x-axis, the adjacent side length lies along the x-axis; however, when the angle x lies on the y-axis, no length can be drawn along the x-axis to represent the angle. As a result, the denominator of the fraction created by the definition tan x = opposite/adjacent is equal to zero for any angle along the y-axis (90 or 270 degrees, or pi/2 or 3pi/2 in radians.) Therefore, tan (pi)/2 is undefined.
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This is the graph of what function of x?
This is the graph of what function of x?
is shown in red, and 
is shown in blue.
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Give the equation of the following graph.
Give the equation of the following graph.
Looking at our graph, we can tell that the period is 
. Using the formula
where 
is the coefficient of 
and 
is the period, we can calculate
This eliminates one answer choice. We then retrun to our graph and see that the amplitude is 3. Remembering that the amplitude is the number in front of the function, we can eliminate two more choices.
We then examine our graph and realize it contains the point 
. Plugging 0 into our two remaining choices, we can determine which one gives us 4 for a result.
Looking at our graph, we can tell that the period is
This eliminates one answer choice. We then retrun to our graph and see that the amplitude is 3. Remembering that the amplitude is the number in front of the function, we can eliminate two more choices.
We then examine our graph and realize it contains the point
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