Problem 40
Question
Sketch the graph of each of the given expressions. $$g(x)=\arcsin x+\pi$$
Step-by-Step Solution
Verified Answer
The graph of \(y = \arcsin x + \pi\) is the same as the graph of \(y = \arcsin x\) but shifted upwards by \(\pi\) units.
1Step 1: Understand the Basic Graph of the Arcsine Function
It is important to know that the domain of the arcsine function is between -1 and 1, and the range is between -\(\pi/2\) and \(\pi/2\). The function crosses the origin, i.e., (0,0) is a point on the graph. The shape of the graph in this domain is a rising curve.
2Step 2: Understand the Effect of Adding a Constant
Adding a constant to a function results in a vertical shift of the graph. In this case, \(g(x) = \arcsin x + \pi\) would shift the entire graph of \(y = \arcsin x\) upwards by \(\pi\) units.
3Step 3: Sketch the Graph
Firstly, draw the basic graph of \(y = \arcsin x\) which is a curve that passes through the points (-1, -\(\pi/2\)), (0,0), and (1, \(\pi/2\)) with its domain from -1 to 1 and range from -\(\pi/2\) to \(\pi/2\). Then, shift this curve upwards by \(\pi\) units to get the graph of \(y = \arcsin x + \pi\).
Key Concepts
Domain and Range of the Arcsine FunctionVertical Shift in GraphsSketching Function Graphs
Domain and Range of the Arcsine Function
When delving into the arcsine function, or any trigonometric function, two fundamental concepts to grasp are the domain and range. The domain refers to the set of all possible input values for which the function is defined, while the range represents the possible output values the function can take.
For the arcsine function, denoted as \(arcsin x\), the domain is restricted to the closed interval \[ -1, 1 \]. This limitation is due to the nature of the sine function that arcsine is the inverse of. Since the sine of an angle cannot exceed 1 or be less than -1, the arcsine can only accept values within this range.
The range of arcsine function is between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \), which corresponds to the possible angles in radians that can produce the sine values within \( -1 \) and \( 1 \). This means that when you input any number between -1 and 1 into \(arcsin x\), the output will be an angle between \( -90\degree \) (or \( -\frac{\pi}{2} \) radians) and \( 90\degree \) (or \( \frac{\pi}{2} \) radians).
For the arcsine function, denoted as \(arcsin x\), the domain is restricted to the closed interval \[ -1, 1 \]. This limitation is due to the nature of the sine function that arcsine is the inverse of. Since the sine of an angle cannot exceed 1 or be less than -1, the arcsine can only accept values within this range.
The range of arcsine function is between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \), which corresponds to the possible angles in radians that can produce the sine values within \( -1 \) and \( 1 \). This means that when you input any number between -1 and 1 into \(arcsin x\), the output will be an angle between \( -90\degree \) (or \( -\frac{\pi}{2} \) radians) and \( 90\degree \) (or \( \frac{\pi}{2} \) radians).
Vertical Shift in Graphs
A vertical shift in the context of function graphs is a transformation that moves the entire graph up or down on the coordinate plane. It occurs when a constant is added to or subtracted from the function's formula.
For instance, if we consider our function \(g(x) = arcsin x + \pi\), the addition of \(\pi\) to \(arcsin x\) means we take the existing graph of the arcsine function and move it up vertically by \(\pi\) units. It's important to note that this transformation does not affect the shape of the graph, nor does it change the domain. It solely influences the range, which, in our example, after the vertical shift of \(\pi\), changes from \( -\frac{\pi}{2} \) to \( \frac{3\pi}{2} \).
A straightforward way to implement a vertical shift when sketching is to add the constant to each y-coordinate of existing points on the graph. This process results in a new graph that remains identical in form but is relocated higher or lower based on the constant added.
For instance, if we consider our function \(g(x) = arcsin x + \pi\), the addition of \(\pi\) to \(arcsin x\) means we take the existing graph of the arcsine function and move it up vertically by \(\pi\) units. It's important to note that this transformation does not affect the shape of the graph, nor does it change the domain. It solely influences the range, which, in our example, after the vertical shift of \(\pi\), changes from \( -\frac{\pi}{2} \) to \( \frac{3\pi}{2} \).
A straightforward way to implement a vertical shift when sketching is to add the constant to each y-coordinate of existing points on the graph. This process results in a new graph that remains identical in form but is relocated higher or lower based on the constant added.
Sketching Function Graphs
The art of sketching function graphs is a valuable skill, enabling us to visualize relationships between variables. When sketching a graph, start with the function's basic shape, keeping in mind key characteristics such as intercepts, slope, curvature, and asymptotes, if they exist.
In our specific example of \(g(x) = arcsin x + \pi\), begin by plotting the core function \(y = arcsin x\) between its domain limits of -1 and 1. The points to note are \( (-1, -\frac{\pi}{2}), (0,0),\) and \( (1, \frac{\pi}{2}) \), as these are the intercepts and extremities of the arcsine graph. You would then establish a smooth, continuous curve connecting these points.
Afterwards, each point on the arcsine curve is shifted up by \(\pi\) units for the vertical transformation due to the constant added to the function. There's no need to recalculate the entire graph. It's as simple as moving the key points and redrawing the curve. The result is a graph that maintains the original shape but has been adjusted vertically in alignment with the transformation applied.
In our specific example of \(g(x) = arcsin x + \pi\), begin by plotting the core function \(y = arcsin x\) between its domain limits of -1 and 1. The points to note are \( (-1, -\frac{\pi}{2}), (0,0),\) and \( (1, \frac{\pi}{2}) \), as these are the intercepts and extremities of the arcsine graph. You would then establish a smooth, continuous curve connecting these points.
Afterwards, each point on the arcsine curve is shifted up by \(\pi\) units for the vertical transformation due to the constant added to the function. There's no need to recalculate the entire graph. It's as simple as moving the key points and redrawing the curve. The result is a graph that maintains the original shape but has been adjusted vertically in alignment with the transformation applied.
Other exercises in this chapter
Problem 39
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