Question

Analyze the effect on the x-intercept of the graph of f(x)=x−−√3
when f(x)
is replaced by f(x+c)
and c is positive.(1 point)
Responses

The x-intercept shifts up by a distance of c.
The x -intercept shifts up by a distance of c.

The x-intercept shifts to the right by a distance of c.
The x- intercept shifts to the right by a distance of c .

The x-intercept shifts down by a distance of c.
The x -intercept shifts down by a distance of c.

The x-intercept shifts left by a distance of c.

Answer 1

The function \( f(x) = \sqrt{x} - 3 \) has an x-intercept at the point where \( f(x) = 0 \).

To find the x-intercept, we set the function equal to zero:

\[
\sqrt{x} - 3 = 0
\]
\[
\sqrt{x} = 3
\]
\[
x = 9
\]

So the original x-intercept is at \( (9, 0) \).

Now, consider the transformation of the function to \( f(x + c) \), where \( c \) is a positive constant. The new function is:

\[
f(x + c) = \sqrt{x + c} - 3
\]

To find the new x-intercept, we again set the function equal to zero:

\[
\sqrt{x + c} - 3 = 0
\]
\[
\sqrt{x + c} = 3
\]
\[
x + c = 9
\]
\[
x = 9 - c
\]

This means that when the function is transformed to \( f(x+c) \), the x-intercept shifts from \( 9 \) to \( 9 - c \). Since \( c \) is positive, \( 9 - c \) is less than \( 9 \), indicating that the x-intercept moves to the left by a distance of \( c \).

Thus, the correct response is:

**The x-intercept shifts left by a distance of c.**