The question from the image is:

Diketahui fungsi $f(x) = x^2 + 3x - 4$. Jika dilakukan translasi terhadap grafik fungsi tersebut dengan vektor $T(2,1)$, tentukan persamaan fungsi bayangan.

Translation of the question:

"Given the function $f(x) = x^2 + 3x - 4$. If a translation is performed on the graph of this function with the vector $T(2,1)$, determine the equation of the transformed function."

Solution:

A translation of a function involves shifting the graph horizontally and/or vertically. A translation vector $T(2,1)$ means:

• A horizontal shift of $+2$ (rightward by 2 units),
• A vertical shift of $+1$ (upward by 1 unit).

To translate a function $f(x)$ horizontally by $h$ and vertically by $k$, the transformation is given by:

$f'(x) = f(x - h) + k$

In this case:

• $h = 2$,
• $k = 1$.

So, the transformed function is:

$f'(x) = f(x - 2) + 1$

Step 1: Calculate $f(x - 2)$

The original function is:

$f(x) = x^2 + 3x - 4$

Now substitute $x - 2$ into this function:

$f(x - 2) = (x - 2)^2 + 3(x - 2) - 4$

Step 2: Expand the expression

$(x - 2)^2 = x^2 - 4x + 4$ $3(x - 2) = 3x - 6$

Now substitute these back:

$f(x - 2) = x^2 - 4x + 4 + 3x - 6 - 4$ $f(x - 2) = x^2 - 4x + 3x + 4 - 6 - 4$ $f(x - 2) = x^2 - x - 6$

Step 3: Add the vertical shift (+1)

$f'(x) = f(x - 2) + 1 = x^2 - x - 6 + 1$ $f'(x) = x^2 - x - 5$

Final Answer:

The equation of the transformed function is:

$f'(x) = x^2 - x - 5$

Would you like a more detailed explanation on any of the steps?

Related Questions:

1. What happens if the translation vector is $T(-2,-1)$?
2. How would you translate the function vertically by 3 units?
3. How do you reflect a function across the x-axis?
4. What is the difference between horizontal and vertical translations?
5. How can we shift a function both vertically and horizontally by arbitrary values?

Tip:

For horizontal shifts, remember to modify the input of the function, $f(x - h)$, where $h$ is the shift. For vertical shifts, simply adjust the output by adding or subtracting a constant value.