Problem 122 - Precalculus

Question

Find an expression involving \(C\) and \(A\) that describes the values of \(C\) for which the graph of \(y=A \cos B x+C\) does not cross the \(x\) -axis. (Assume that \(A > 0 .\) )

Step-by-Step Solution

Answer

The values of \(C\) for which the graph does not cross the x-axis are \(C < -A\) or \(C > A\).

Step 1: Understand the Condition for Crossing the x-axis

The graph of the function crosses the x-axis whenever the function's value is equal to zero. Thus, the condition for the cosine function to cross the x-axis is when \(y = A \cos Bx + C = 0\), which simplifies to \(A \cos Bx = -C\).

Step 2: Determine the Range of the Cosine Function

Recall that the range of the cosine function is \([-1, 1]\). Therefore, \(\cos Bx\) can take any value within this interval.

Step 3: Apply the Range to the General Condition

Given that \(-1 \leq \cos Bx \leq 1\), we have \(-A \leq A \cos Bx \leq A\). This means that \(A \cos Bx\) includes all values from \(-A\) to \(A\).

Step 4: Ensure the Graph Does Not Cross the x-axis

For the graph to not cross the x-axis, \(-C\) must be lesser than the smallest possible value of \(A \cos Bx\) or greater than the largest possible value of \(A \cos Bx\). Hence, either \(-C < -A\) leading to \(C > A\) or \(-C > A\) which simplifies to \(C < -A\).

Step 5: Write the Final Condition for Values of C

By combining the conditions from Step 4, we find that for the graph to not intersect the x-axis, the values of \(C\) must satisfy \(C < -A\) or \(C > A\).

Key Concepts

Cosine Function

The cosine function, represented as \( \cos(x) \), is a fundamental trigonometric function. It is defined for all real numbers and, unlike sine and tangent functions, the cosine function starts at its maximum value of 1 when \( x = 0 \). One of the key features of the cosine function is its periodicity, repeating its values in a regular cycle every \( 2\pi \) radians.

Some essential properties of the cosine function include:

• **Range**: The cosine function always takes values between -1 and 1, i.e., \( -1 \leq \cos(x) \leq 1 \).
• **Even Function**: The cosine function is symmetric about the y-axis, meaning \( \cos(-x) = \cos(x) \).
• **Periodicity**: The cosine function has a period of \( 2\pi \), meaning \( \cos(x) = \cos(x + 2\pi k) \) for any integer \( k \).

These properties make the cosine function a building block for more complex trigonometric expressions. In the context of the exercise, understanding how the cosine function behaves helps in determining when the function \( y = A \cos Bx + C \) will or will not cross the x-axis.

Graph Behavior

The behavior of the graph of a trigonometric function like \( y = A \cos Bx + C \) is determined by its components. Each of these components affects the graph in a unique way:

• **Amplitude \( A \):** This parameter stretches or compresses the graph vertically. The amplitude is the height from the centerline of the graph to its peak. For \( A \cos Bx \), the graph's maximum value is \( A \) and the minimum is \( -A \).
• **Frequency \( B \):** This affects how frequently the graph cycles through its period within a given interval. A higher \( B \) means more oscillations over the same interval.
• **Vertical Shift \( C \):** This shifts the entire graph up or down along the y-axis by a constant amount, \( C \). If \( C > 0 \), the graph moves up; if \( C < 0 \), it moves down. The vertical shift is key when determining if and where the graph intersects the x-axis.

In the exercise, these components help us determine the conditions for the graph to avoid crossing the x-axis. If the entire graph is shifted above or below the x-axis sufficiently, it never intersects, leading to specific conditions for \( C \).

X-axis Intersections

Understanding x-axis intersections involves recognizing when the function value equals zero. For the function \( y = A \cos Bx + C \), the graph crosses the x-axis wherever \( y = 0 \). Setting the equation to zero provides:
\[ A \cos Bx + C = 0 \]
which simplifies to \( A \cos Bx = -C \).

To find out when this equation is unsolvable (thus the graph does not cross the x-axis), we investigate the range of \( A \cos Bx \), which is bounded by \(-A\) and \(A\). The graph avoids crossing the x-axis if:

• \( C > A \), meaning \( -C < -A \). This situation pulls the graph entirely below the x-axis.
• \( C < -A \), meaning \( -C > A \). Here, the graph is entirely above the x-axis.

When either of these conditions hold, \( y = 0 \) has no solutions within the bounded range of \( A \cos Bx \), ensuring no x-axis intersections occur. This understanding helps solve the problem of determining the specific range of \( C \) easily.