Finding the Distance of the Highest Point from the X-axis

The question asks for the distance of the highest point on the graph of the function \(y = \sqrt{3} \cos x + \sin x\) from the x-axis. The highest point on the graph corresponds to the maximum value that the function \(y\) can attain. The distance of this point from the x-axis is simply the absolute value of its y-coordinate.

Analyzing the Trigonometric Function

The given function is in the form \(y = a \cos x + b \sin x\), where \(a = \sqrt{3}\) and \(b = 1\). Functions of this form can be rewritten in the amplitude-phase shift form:

\[y = R \cos(x - \alpha)\]

or

\[y = R \sin(x + \alpha)\]

where \(R\) is the amplitude and is given by \(R = \sqrt{a^2 + b^2}\).

Calculating the Amplitude (Maximum Value)

Let's calculate the value of \(R\) for the given function:

\[a = \sqrt{3}, \quad b = 1\]

\[R = \sqrt{(\sqrt{3})^2 + (1)^2}\]

\[R = \sqrt{3 + 1}\]

\[R = \sqrt{4}\]

\[R = 2\]

So, the amplitude of the function \(y = \sqrt{3} \cos x + \sin x\) is 2.

Determining the Maximum Value of the Function

The transformed function can be written as \(y = 2 \cos(x - \alpha)\) or \(y = 2 \sin(x + \alpha)\). The maximum value of \(\cos \theta\) is 1, and the maximum value of \(\sin \theta\) is 1. Therefore, the maximum value of \(y\) is:

\[y_{max} = R \times (\text{maximum value of } \cos \text{ or } \sin)\]

\[y_{max} = 2 \times 1\]

\[y_{max} = 2\]

The highest point on the graph of the function \(y = \sqrt{3} \cos x + \sin x\) has a y-coordinate equal to the maximum value, which is 2.

Calculating Distance from the X-axis

The x-axis is the line \(y = 0\). The distance of a point \((x_0, y_0)\) from the x-axis is given by \(|y_0|\). In this case, the highest point has a y-coordinate of 2. The distance of this point from the x-axis is:

\[\text{Distance} = |2| = 2\]

Thus, the distance of the highest point on the graph from the x-axis is 2.

Comparing with Options

Let's compare our result with the given options:

1. \(\dfrac{1}{2}\)
2. \(\dfrac{\sqrt{3}}{2}\)
3. \(\dfrac{2}{\sqrt{3}}\)
4. 2

Our calculated distance is 2, which matches the fourth option.

Revision Table: Key Concepts for Function Analysis

Additional Information: Amplitude and Phase Shift

Any expression of the form \(a \cos x + b \sin x\) can be written as \(R \cos(x - \alpha)\) or \(R \sin(x + \beta)\), where \(R = \sqrt{a^2 + b^2}\) is the amplitude. The angle \(\alpha\) or \(\beta\) represents the phase shift. The maximum value of such a function is always \(R\), and the minimum value is \(-R\).

To find the phase shift, we use the relations:

• If converting to \(R \cos(x - \alpha)\), then \(a = R \cos \alpha\) and \(b = R \sin \alpha\). Thus, \(\tan \alpha = b/a\).
• If converting to \(R \sin(x + \beta)\), then \(a = R \sin \beta\) and \(b = R \cos \beta\). Thus, \(\tan \beta = a/b\).

In our case, \(y = \sqrt{3} \cos x + \sin x\). We found \(R=2\). We can write:

\[y = 2 \left( \dfrac{\sqrt{3}}{2} \cos x + \dfrac{1}{2} \sin x \right)\]

Using \(\cos \dfrac{\pi}{6} = \dfrac{\sqrt{3}}{2}\) and \(\sin \dfrac{\pi}{6} = \dfrac{1}{2}\):

\[y = 2 \left( \cos \dfrac{\pi}{6} \cos x + \sin \dfrac{\pi}{6} \sin x \right) = 2 \cos\left(x - \dfrac{\pi}{6}\right)\]

Using \(\sin \dfrac{\pi}{3} = \dfrac{\sqrt{3}}{2}\) and \(\cos \dfrac{\pi}{3} = \dfrac{1}{2}\):

\[y = 2 \left( \sin \dfrac{\pi}{3} \cos x + \cos \dfrac{\pi}{3} \sin x \right) = 2 \sin\left(x + \dfrac{\pi}{3}\right)\]

Both forms clearly show the amplitude is 2. The maximum value of \(\cos\left(x - \dfrac{\pi}{6}\right)\) is 1, occurring when \(x - \dfrac{\pi}{6} = 2n\pi\) for integer \(n\). The maximum value of \(2 \cos\left(x - \dfrac{\pi}{6}\right)\) is \(2 \times 1 = 2\).

Similarly, the maximum value of \(\sin\left(x + \dfrac{\pi}{3}\right)\) is 1, occurring when \(x + \dfrac{\pi}{3} = 2n\pi + \dfrac{\pi}{2}\) for integer \(n\). The maximum value of \(2 \sin\left(x + \dfrac{\pi}{3}\right)\) is \(2 \times 1 = 2\).

The highest point on the graph is where \(y\) reaches its maximum value, which is 2. The distance of this point from the x-axis (where \(y=0\)) is 2.