Question

Find DRG of following functions (mention graphical operations (CST1) and draw graph of each step)

(a) $f(x) = sin^2x$

(b) $f(x) = sin(x - 1). sin(x + 1)$

Answer 1

The DRG operations for (a) are: 1. Start with $y=sin x$. 2. Replace $y$ by $y^2$ (squaring every ordinate). Resulting graph: $y=sin^2x$ with period $\pi$ and range $[0,1]$. The DRG operations for (b) are: 1. Use the identity to convert $f(x)=sin(x-1)sin(x+1)$ into $f(x)=\frac{cos2-cos2x}{2}$. 2. Start with $y=cos2x$ (period $\pi$). 3. Multiply by $-\frac{1}{2}$ and add the constant $\frac{cos2}{2}$. Resulting graph is a cosine function modified by these transformations.

Answer 2

(a) $f(x)=\sin^2x$

Step 1. Start with the basic function:
Graph $y=\sin x$.
Note: The sine function has period $2\pi$ and oscillates between $-1$ and $1$.

Step 2. Apply the square transformation:
Replace each $y$ value by $y^2$.

• Operation: $\sin x \to (\sin x)^2$.
• Effect: Negative values become positive. The new graph oscillates between $0$ and $1$ and its period becomes $\pi$ (because squaring “hides” the sign change over half the period).

Alternate Viewpoint (Using an Identity):
Rewrite

$$\sin^2x=\frac{1-\cos2x}{2}.$$

This shows the operations as:

1. Horizontal Compression: $\cos x \to \cos 2x$ (period becomes $\pi$).
2. Reflection & Scaling: Multiply by $-\frac{1}{2}$.
3. Vertical Shift: Add $\frac{1}{2}$.

(b) $f(x)=\sin(x-1)\cdot \sin(x+1)$

Step 1. Use a Product-to-Sum Identity:
Recall that:

$$\sin A\,\sin B = \frac{\cos(A-B)-\cos(A+B)}{2}.$$

Let $A=x-1$ and $B=x+1$. Then:

$$A - B = (x-1)-(x+1) = -2\quad \text{and}\quad A+B= 2x.$$

Thus,

$$\sin(x-1)\,\sin(x+1) = \frac{\cos(-2)-\cos2x}{2}.$$

Since $\cos(-2)=\cos2$,

$$f(x)=\frac{\cos2-\cos2x}{2}.$$

Step 2. Identify the Graphical Operations:
Write the function as:

$$f(x) = \frac{\cos2}{2} - \frac{1}{2}\cos2x.$$

The transformation steps are:

1. Basic Function: Begin with $y=\cos2x$ (a cosine graph horizontally compressed; period $\pi$).
2. Vertical Scaling & Reflection: Multiply by $-\frac{1}{2}$ to get $y=-\frac{1}{2}\cos2x$.
3. Vertical Translation: Shift upward by $\frac{\cos2}{2}$ (note that $\cos2$ is simply a constant, approximately $\cos2 \approx -0.416$, so the shift is by roughly $-0.208$; the exact value is maintained symbolically).