Question

If \(\mathbf{r}(t)\) is a vector-valued function, is the graph of the vectorvalued function \(\mathbf{u}(t)=\mathbf{r}(t-2)\) a horizontal translation of the graph of \(\mathbf{r}(t) ?\) Explain your reasoning.

Short answer

Yes, \(\mathbf{u}(t)=\mathbf{r}(t-2)\) is a horizontal translation of the graph of \(\mathbf{r}(t)\), moved 2 units to the right on the horizontal axis.

Step-by-step solution

Understand the Concept of Translation in a Function

In general, for any function How a function behaves or shifts upon addition or subtraction of constants in its variables (input) is based on a simple rule. For instance, \(f(x-h)\) they represent a shift of the graph \(h\) units to the right along the horizontal axis.

Relate to Vector-Valued Function

In our context, \(\mathbf{r}(t)\) is a vector-valued function. By applying the same concept as explained above, by introducing \(t-2\) instead of \(t\) i.e., \(\mathbf{u}(t)=\mathbf{r}(t-2)\), we get a graph that is a horizontal shift of \(\mathbf{r}(t)\) 2 units to the right.

The Resulting Translation Effect

Therefore, based on the nature of function shifts, the graph of the vector valued function \(\mathbf{u}(t)=\mathbf{r}(t-2)\) is indeed a horizontal translation of the graph of \(\mathbf{r}(t)\), moved 2 units to the right on the horizontal axis.