Question

Your friend states that for functions of the form \(y=a \sin b x\) and \(y=a \cos b x\), the values of \(a\) and \(b\) affect the \(x\)-intercepts of the graph of the function. Is your friend correct? Explain.

Short answer

No, the friend is incorrect. The values of \(a\) and \(b\) do not affect the x-intercepts of the functions \(y=a \sin bx\) and \(y=a \cos bx\). Changing the amplitude and frequency does not create or eliminate x-intercepts as these occur whenever \(y = 0\), regardless of the values of \(a\) and \(b\).

Step-by-step solution

Understand the function

In a function of the form \(y = a \sin bx\) or \(y = a \cos bx\), \(a\) represents the amplitude of the function (i.e., the height of its peak), and \(b\) represents its frequency (i.e., how frequently it repeats). The x-intercepts of a function are the x-values where the y-value is 0.

Evaluate the effect of coefficients on x-intercepts

Now consider what happens if you change \(a\) or \(b\). The coefficient \(a\) only influences the height of the wave, making it taller or shorter. Therefore, this won't change its x-intercepts. However, changing \(b\) affects the wave's frequency, which means it can move the x-intercepts, but it does not create or eliminate x-intercepts. It is important to remember that x-intercepts occur when the function \(y\) is equal to 0. Thus, regardless of what values of \(a\) and \(b\) are assigned, the x-intercepts for these sine and cosine functions will always be at the same positions on the x-axis when \(y = 0\).

Conclude

So the statement of the friend is incorrect. The parameters \(a\) and \(b\) do not affect the x-intercepts of the functions \(y=a \sin bx\) and \(y=a \cos bx\) in the sense of creating or eliminating x-intercepts. Changing \(a\) only affects the amplitude of the functions and changing \(b\) only affect the frequency. Neither of these changes affect the x-intercepts as x-intercepts occur when \(y = 0\), regardless of the values of \(a\) and \(b\).

Key concepts

Amplitude of a Trigonometric Function

When discussing trigonometric functions, amplitude is a term that refers to the peak vertical distance from the midpoint of the wave to its top or bottom. In a mathematical expression like \(y = a \sin(bx)\) or \(y = a \cos(bx)\), the variable \(a\) signifies this amplitude. Imagine amplitude as how “tall” or “short” the waves of the function are. The larger the value of \(a\), the taller the peak and deeper the trough of the wave.

It is vital to note that amplitude is always a positive value, even if \(a\) is negative in the equation. This is because the amplitude is a measure of distance, which cannot be negative. While amplitude does change the overall appearance of the graph, it does not impact where the function crosses the x-axis—the x-intercepts. This is because x-intercepts occur where the function's output \(y\) is zero, a value unaffected by the amplitude \(a\).

Frequency of a Trigonometric Function

In the context of trigonometry, frequency determines how often the function repeats its pattern within a certain interval. For functions represented as \(y = a \sin(bx)\) or \(y = a \cos(bx)\), the variable \(b\) dictates the frequency. A higher value of \(b\) results in the function completing more cycles within the same length along the x-axis, effectively 'squeezing' more waves into the graph.

Altering the frequency \(b\) does change where the peaks and troughs of the wave occur, and as a result, may appear to affect the x-intercepts. However, the number of x-intercepts in one period of the wave doesn't change, and they remain at points where the function is zero. It's also noteworthy that frequency relates inversely to the period of the function—that is, the larger the frequency, the shorter the period (the width of one complete cycle of the wave).

Graphing Sine and Cosine Functions

Graphing sine and cosine functions can be a delightful visual representation of their behavior. With the standard equations \(y = a \sin(bx)\) and \(y = a \cos(bx)\), we can see the 'wave-like' nature of these functions. Their graphs feature continuous, smooth curves that oscillate above and below the x-axis.

When graphing, a full cycle of a sine or cosine wave is represented from 0 to \(2\pi\) for the standard case where \(b = 1\). Sine functions typically begin at \((0, 0)\) and move upward to their peak amplitude, whereas cosine functions start at their maximum amplitude when \(x = 0\). Both have x-intercepts where the function equals zero. These are key points that should be marked clearly on the graph. As you draw the functions, consider how changes in amplitude and frequency affect the spacing and height of the waves, though they do not fundamentally alter the starting and ending points of each cycle.

Trigonometric Function Properties

Trigonometric functions have distinct properties that define their graphical representation and overall behavior. Key properties include periodicity, symmetry, and amplitude and frequency as mentioned earlier.

Periodicity

Sine and cosine functions are periodic, meaning they repeat their values in regular intervals. The standard period for these functions is \(2\pi\), corresponding to one complete cycle of the wave.

Symmetry

Sine functions possess odd symmetry, which means they're symmetric about the origin (when graphed, they have 180-degree rotational symmetry). Cosine functions exhibit even symmetry, being symmetric about the y-axis.

The understanding of these properties is crucial for graph interpretation and knowing how the basic form of the graph will look before applying transformations such as changing amplitude and frequency.