Problem 38 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Describe the transformation of the graph of \(f\) represented by the function \(g\). \(f(x)=\sin x, g(x)=3 \sin \left(x+\frac{\pi}{4}\right)-2\)

Step-by-Step Solution

Answer

The graph of \(f(x) = \sin x\) is transformed into \(g(x) = 3 \sin \left(x+\frac{\pi}{4}\right)-2\) by stretching vertically by a factor of 3, shifting left by \(\frac{\pi}{4}\) units, and downward by 2 units. There is no horizontal scaling.

Step 1: Identify the Vertical Scaling Factor

The coefficient of the sine function in the equation for \(g(x)\) is 3. This means that the graph of \(f(x)\) is stretched vertically by a factor of 3. This stretch does not move the graph up or down; it simply makes the waves three times as tall.

Step 2: Identify the Horizontal Shift

The term present inside the sine function \(x+\frac{\pi}{4}\) stands for shifting the graph to the left by \(\frac{\pi}{4}\) units. To see this, note that the graph of \(f(x) = \sin x\) gets shifted to the left whenever we add a positive constant inside the sine function. So, in this case, \(\frac{\pi}{4}\) inside the sine function shifts the graph left by \(\frac{\pi}{4}\) units.

Step 3: Identify the Vertical Shift

The term \(-2\) outside the sine function represents a vertical shift down by 2 units. This shift can be applied after considering the vertical scaling (in step 1) and the horizontal shift (in step 2).

Step 4: Identify the Horizontal Scaling

Since there is no coefficient inside the sine function other than 1 in \(x+\frac{\pi}{4}\), the graph of \(f(x)\) is not stretched or compressed horizontally. So, there is no horizontal scaling in this case.

Key Concepts

Vertical Scaling

Vertical scaling is a transformation that alters the height of the graph. In the function \( g(x) = 3 \sin \left( x + \frac{\pi}{4} \right) - 2 \), the number 3 before the sine function represents the vertical scaling factor. This means that for each point on the graph of the original function \( f(x) = \sin x \), its distance from the x-axis is multiplied by 3.
This transformation affects the amplitude of the trigonometric function.
When the amplitude is altered, the peaks and troughs of the sine wave become more pronounced:

• If the scaling factor is greater than 1, the graph is stretched upwards and downwards.
• If the scaling factor is between 0 and 1, the graph is squished towards the x-axis.
• A negative scaling factor would also reflect the graph over the x-axis, but in our case, it's positive.

Horizontal Shift

Horizontal shift, also known as phase shift in the context of trigonometric functions, involves moving the graph left or right along the x-axis. Examining our function \( g(x) = 3 \sin \left( x + \frac{\pi}{4} \right) - 2 \), we see the \( x + \frac{\pi}{4} \) term.
Adding \( \frac{\pi}{4} \) indicates a horizontal shift to the left by \( \frac{\pi}{4} \) units.
Understanding horizontal shifts can be simplified:

• Adding a positive value inside the function moves the graph to the left.
• Subtracting a value moves it to the right.

Remember, the shift does not alter the shape of the graph, just its position along the x-axis.

Vertical Shift

Vertical shift changes the position of the graph up or down along the y-axis. In the function we are analyzing, \( g(x) = 3 \sin \left( x + \frac{\pi}{4} \right) - 2 \), the "-2" represents a vertical shift downward by 2 units.
Vertical shifts are straightforward:

• Add a constant outside the function to shift up.
• Subtract a constant to shift down.

The entire wave of the sine function is shifted, affecting both the peaks and troughs equally, without changing the amplitude or the wave's period.

Trigonometric Functions

Trigonometric functions like sine and cosine are fundamental in understanding periodic phenomena, due to their wave-like nature. The sine function in particular, with its equation \( f(x) = \sin x \), naturally oscillates between 1 and -1, creating a smooth, continuous wave.
Transformations, like scaling or shifting, are often applied to alter its periodic appearance without changing its intrinsic frequency.
This flexibility explains why trigonometric functions are used extensively in physics and engineering to model everything from sound waves to electrical currents:

• Sine waves can be modified to fit various amplitudes and positions using transformations.
• Understanding how these transformations affect the graph is essential in mastering trigonometric functions for practical uses.

These transformations include vertical and horizontal scaling, shifts, and even reflections, giving students powerful tools to manipulate and interpret these functions effectively in real-world applications.