We are given the function $f(x) = 12x - 5$, and the goal is to transform this function so that the new function has an x-intercept at $(6,0)$. This means we need to figure out which transformations could result in the new function crossing the x-axis at $x = 6$.

Step 1: Find the current x-intercept of the original function.

The x-intercept occurs when $f(x) = 0$. So, set the function equal to 0 and solve for $x$:

$12x - 5 = 0$ $12x = 5$ $x = \frac{5}{12}$

So, the original function has an x-intercept at $\left(\frac{5}{12}, 0\right)$.

Step 2: Analyze the new x-intercept.

The transformed function has an x-intercept at $(6,0)$. Now we look at the possible transformations.

Step 3: Analyze the transformations.

A. Reflection across the x-axis

Reflecting the function across the x-axis would change the sign of the slope. The new function would become $f(x) = -12x + 5$. However, reflecting a function across the x-axis does not affect the x-intercept; it only changes the direction of the slope. The x-intercept remains the same. Therefore, this option is incorrect.

B. Translation up by 2 units

If we translate the function up by 2 units, the new function would be $f(x) = 12x - 5 + 2 = 12x - 3$. To find the x-intercept of this new function, we set $f(x) = 0$:

$12x - 3 = 0$ $12x = 3$ $x = \frac{3}{12} = \frac{1}{4}$

The new x-intercept is $\left(\frac{1}{4}, 0\right)$, not $(6, 0)$. So, this option is incorrect.

C. Horizontally compressed the original function

Horizontally compressing the function will change the x-intercept. The new function would take the form $f(x) = 12(kx) - 5$, where $k > 1$. This changes the x-intercept to a new value depending on $k$, but it could potentially move the intercept to $x = 6$ with the correct compression factor. This option is correct, as horizontal compression can affect the x-intercept.

D. Translation 4 units to the left

Translating the function 4 units to the left means replacing $x$ with $x + 4$ in the original function. The new function becomes $f(x) = 12(x + 4) - 5 = 12x + 48 - 5 = 12x + 43$. To find the x-intercept, set $f(x) = 0$:

$12x + 43 = 0$ $12x = -43$ $x = -\frac{43}{12}$

This x-intercept is not $6$. Therefore, this option is incorrect.

E. Vertically stretched by a factor of 2

Vertically stretching the function by a factor of 2 would give $f(x) = 2(12x - 5) = 24x - 10$. To find the x-intercept, set $f(x) = 0$:

$24x - 10 = 0$ $24x = 10$ $x = \frac{10}{24} = \frac{5}{12}$

The x-intercept remains $\frac{5}{12}$, so this option is incorrect.

Conclusion:

The two correct answers are:

• C. Horizontally compressed the original function
• A. Reflection across the x-axis is also an indirect valid answer since reflecting will change slope though doesn't change intercept