Mathematics Reviewer | 4Q | Graphs and Properties of
Basic Trigonometric Functions: Transformations
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Practice Exercise
Graphs and Properties of y = asinb(x+c) +d
we consider transformations on trigonometric functions of the
form:
y = asinb(x+c) +d
and
y = acosb(x+c) +d
Match the given function to one of the graphs on the second
column.
for real numbers a,b, c, and d. Let us look at each parameter and
see the behavior of the graph
Case 3: y = sin(x+c)
The horizontal shift of the graph is defined by the number −c.
If the horizontal shift is negative, then the graph is shifted to the
left, and if it is positive, then the graph is moved to the right.
Case 1: y = asinx
For a ≠ 0, notice that as a varies, the amount of vertical stretching
of the sine function changes
If a < 0, the graph of y = asinx is reflected with respect to x−axis.
|a| is referred to as the amplitude and the y− values of the form
y = asinx will always satisfy
−|a| ≤ asinx ≤ |a|
Lesson Summary
Note-Taking Space
Case 4: y = sinx+d.
Note that y = sin(x+d) is not equivalent to y = sinx+d
Notice that if d > 0, then the graph is shifted d units up
while if d < 0, then the graph is shifted d units down
Case 2: y = sin(bx)
The horizontal compression is determined by the value of b
If |b| > 1, the graph of y = sinbx and y = cosbx is
compressed horizontally by a factor of b.
In other words, d dictates the vertical shift of the graph of the
form y = asinb(x+c) +d
This is analogous to the case of y = acosb(x+c) +d
If 0 < |b| < 1, the graph is stretched horizontally by a
factor of b.
The period of the sine and cosine functions is defined by the
formula
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Mathematics Reviewer | 4Q | Inverse Trigonometric
Functions
by ® efptozlped
Finding the inverse of a trigonometric function:
change f(x) for y switch x’s and y’s solve for y write using function
notation:
f(x) = sin (x)
y = sin (x)
x = sin (y)
y = arcsin (x)
f -1 (x) = arcsin (x)
f -1 (x) = sin -1 (x)
Inverse Trigonometric Function Notation:
Inverse sine
arcsinx or sin -1 x
Inverse cosine
arccosx or cos -1 x
Inverse tangent
arctanx or tan -1 x
Notice that if we restrict the domain to
3. The Inverse Tangent Function
f(x) = tan x must also be restricted to find its inverse.
, the graph will look like the one below, which is already
one-to-one. At the same time, all the values [-1. 1] of the sine
function can be found within this interval.
Tan x has an inverse function on this interval.
Notice that if we restrict the domain to
2. The Inverse Cosine Function
f(x) = cos x must be restricted to find its inverse since it is also not
one-to-one and does not pass the horizontal line test.
, the graph will look like the one below, which is already
one-to-one. At the same time, all the values [-1. 1] of the sine
function can be found within this interval.
1. The Inverse Sine Function
The figure below shows the graph of the sine function (y = sin x)
with a horizontal line.
Recall that for a function to have an inverse, it must be a
one-to-one function and pass the Horizontal Line Test.
Cos x has an inverse function on this interval.
Notice that if we restrict the domain to
, the graph will look like the one below, which is already
one-to-one. At the same time, all the values [-1. 1] of the sine
function can be found within this interval.
The graph of y = f(x) = sin(x) is a sinusoidal wave.
No periodic function is one-to-one because each y value (from its
range) corresponds to multiple x values (from its domain) as it
goes through an infinite number of periods.
Finding the Exact Values of sin−1 x, cos−1 x, tan−1 x
Finding the inverse of a trigonometric function algebraically
Given
y = sin (x)
½ = sin (π/6)
(π/6) = sin (½) ​
(π/6) = arcsin (½)
(π/6) = sin -1 (½)
​
(switch x and y values and solve for y)
But does it mean that the sine function, or any trigonometric
function for that matter, does not have an inverse?
The Inverse Sine Function
f(x) = sin x is not one-to-one & does not pass the Horizontal Line
Test. Hence, it can only have an inverse if we restrict its domain.
Sin x has an inverse function on this interval.
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Graphs and Properties of Basic Inverse Trigonometric Functions
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Note-Taking Space
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Note-Taking Space
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Properties of Basic Inverse Trigonometric Functions
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