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Chapter 10: Problem 27

Solve the given problems. Find the function and graph it for a function of the form \(y=a \sin x\) that passes through \((\pi / 2,-2).\)

Short Answer

Expert verified
The function is \(y = -2 \sin x\), and it reflects \(y = \sin x\) across the x-axis with an amplitude of 2.

Step by step solution

01

Understand the Form of the Function

We are given the function form \(y = a \sin x\). To solve the problem, we need to determine the value of \(a\) such that the graph passes through the point \((\pi / 2, -2)\).
02

Plug the Given Point into the Function

Substitute the point \((\pi / 2, -2)\) into the function \(y = a \sin x\). This means we replace \(y\) with \(-2\) and \(x\) with \(\pi / 2\), giving us: \(-2 = a \sin (\pi / 2)\).
03

Simplify the Equation

Recall that \(\sin (\pi / 2) = 1\). Substituting this into the equation from Step 2, we have \(-2 = a \cdot 1\), simplifying to \(a = -2\).
04

Write the Specific Function

Now that we have determined that \(a = -2\), we can write the specific function as \(y = -2 \sin x\).
05

Consider Graphing the Function

To graph \(y = -2 \sin x\), start by plotting the basic points where \(y = \sin x\) is relevant, such as \(x = 0\), \(x = \pi / 2\), \(x = \pi\), etc. Multiply these sine values by \(-2\) and plot them accordingly. The graph will have an amplitude of 2 and reflect across the x-axis due to the negative coefficient.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Graphing
Graphing a function helps visualize its behavior and properties. It gives insight into how the function changes over different values of its variable. Here, we examine the function of the form \(y = a \sin x\). To graph such functions, steps typically include identifying key points across one complete wave (from \(0\) to \(2\pi\) for sine).

  • Start by marking the x-values where sine has known values: \( x = 0, \pi/2, \pi, 3\pi/2, \text{and } 2\pi\).
  • Calculate the corresponding y-values using the determined sine values and the coefficient \(a\).
These points help in sketching the shape of the sine wave, which generally appears smooth and periodic.
Sine Function
The Sine function, written as \(\sin(x)\), is a basic trigonometric function tapping into the natural circular properties. It is periodic, meaning it repeats its shape over regular intervals, specifically every \(2\pi\) radians.

  • The standard sine function \(y = \sin x\) oscillates between \(-1\) and \(1\).
  • Key points include \(\sin 0 = 0\), \(\sin \frac{\pi}{2} = 1\), \(\sin \pi = 0\), and so forth, defining its wave-pattern.
Understanding these points is crucial because any changes in the equation, like multiplying by a coefficient, alter the wave's appearance without changing its fundamental periodic nature.
Amplitude and Reflection
Amplitude in trigonometric functions pertains to the height from the base to the peak. For \(y = a \sin x\), the amplitude is the absolute value of \(a\). It determines the stretch or compression of the wave.

  • If \(a = 1\), the wave peaks at \(1\) and troughs at \(-1\).
  • If \( |a| > 1\), the wave is taller; \( |a| < 1\), the wave is shorter.
Reflection occurs when the coefficient \(a\) is negative, causing the entire wave to flip over the x-axis. For the function \(y = -2 \sin x\), the amplitude is 2 due to \(\mid a \mid = 2\), and the graph is reflected because of the negative sign, meaning it inverts its peaks and troughs.
Equation Solving
Equation solving often involves substituting specific values to uncover unknown coefficients or solutions. In solving for \(a\) in \(y = a \sin x\), introduce known points to rearrange and solve the function.

  • Substitute \(x\) and \(y\) from the given point \((\pi/2, -2)\).
  • Understand that \(\sin (\pi/2) = 1\), allowing simplification of the equation.
By substituting these reliable values, the equation becomes easily solvable for \(a\). Thus, confirming that the function passing through the point must be \(y = -2 \sin x\), determined from the algebraic process.

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Most popular questions from this chapter

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