The problem involves transforming the function $y = x + 10$ through two translations. Let me explain the process step by step:

We are given two translation vectors:

1. $T_1 = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$
2. $T_2 = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$

Step 1: Apply the first translation $T_1$

The translation vector $T_1 = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ means that we shift the graph of the function by 1 unit in the x-direction and 2 units in the y-direction.

The original function is: $y = x + 10$

After applying the first translation, the new coordinates will be: $x' = x + 1$ $y' = y + 2$ Substituting $y = x + 10$ into this, we get: $y' = (x + 1) + 10 + 2 = x + 11 + 2 = x + 13$ So, after the first translation, the new function is: $y = x + 13$

Step 2: Apply the second translation $T_2$

Now, apply the second translation vector $T_2 = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$, which translates the graph by 3 units in the x-direction and 4 units in the y-direction.

Thus: $x'' = x + 3$ $y'' = y + 4$

Starting with the function $y = x + 13$ from the previous step, substitute and perform the translation: $y'' = (x + 3) + 13 + 4 = x + 16 + 4 = x + 17$

Final result:

After both translations, the final equation of the transformed function is: $y = x + 17$

Would you like more details on this process or have any questions?

Related Questions:

1. What is the geometric meaning of translating a function by a vector?
2. How would the translation affect a quadratic function like $y = x^2 + 5$?
3. Can multiple translations be combined into a single translation?
4. How would the function change if the translation vectors were reversed?
5. What if we apply a reflection instead of a translation?

Tip:

Translations only affect the position of the graph, not its shape. The slope remains the same when dealing with linear equations.