Question

A signal \(x(t)\) is shown below.

The plot of \(y(t)=1-x(4-t)\) is shown in ______.

• A.

  

• B.

  

• C.

  

• D.

  

Hint:

To sketch \(y(t)=1-x(4-t)\), first time-reverse \(x(t)\) to get \(x(-t)\), shift it right by \(4\) to get \(x(4-t)\), then flip the amplitude and add \(1\).

Answer 1

The Correct Option is B

Step 1: Rewrite the rule as a fold and a shift.
The rule is $y(t)=1-x(4-t)$. Since $4-t=-(t-4)$, we get $x(4-t)$ by first folding $x(t)$ about the vertical axis to get $x(-t)$, then sliding that folded graph $4$ units to the right. The outer minus sign and the added $1$ then flip the curve top to bottom and lift it by one unit.

Step 2: List the corner points of $x(t)$ with their fold rule.
A corner that sits at time $s$ on the original graph lands at time $t=4-s$ on the new graph, since folding then shifting by $4$ is the same as the single map $s\to 4-s$. We read off each corner of $x(t)$ from the given plot: $(-3,0)$, the jump at $s=-1$ from $-2$ up to $1$, the flat run to $(3,1)$, and the ramp down to the jump at $s=5$ from $-2$ down to $0$.

Step 3: Map every corner using $t=4-s$.
$s=-3,\ x=0 \Rightarrow t=7,\ y=1-0=1$.
$s\to -1^{-},\ x\to -2 \Rightarrow t\to 5^{+},\ y\to 1-(-2)=3$.
$s\to -1^{+},\ x=1 \Rightarrow t\to 5^{-},\ y=1-1=0$.
$s=3,\ x=1 \Rightarrow t=1,\ y=1-1=0$.
$s\to 5^{-},\ x\to -2 \Rightarrow t\to -1^{+},\ y\to 1-(-2)=3$.
$s\to 5^{+},\ x=0 \Rightarrow t\to -1^{-},\ y=1-0=1$.

Step 4: Read the new graph off this table of mapped corners.
Sorting the mapped points by $t$: flat at $1$ up to $t=-1$, a jump up to $3$ right at $t=-1$, a straight line down to $0$ at $t=1$, flat at $0$ up to $t=5$, a jump up to $3$ right at $t=5$, a straight line down to $1$ at $t=7$, then flat at $1$ after $t=7$.

Step 5: Compare with the choices.
This two-spike shape, with jumps up to $3$ at $t=-1$ and $t=5$ and straight falls to $0$ (at $t=1$) and to $1$ (at $t=7$), is exactly what option (B) draws, whose marked axis values $-1,0,1,5,6,7$ and $1,2,3$ line up with these numbers. The other three sketches place their corners at different times or reach a different peak height, so they do not fit the mapping above.
\[ \boxed{\text{Option (B)}} \]