Question

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

The y-intercept shifts to the left by a distance of d.
The y -intercept shifts to the left by a distance of d .

The y-intercept shifts up by a distance of d.
The y -intercept shifts up by a distance of d .

The y-intercept shifts to the right by a distance of d.
The y -intercept shifts to the right by a distance of d .

The y-intercept shifts down by a distance of d.

Answer 1

To analyze the effect on the y-intercept when \( f(x) \) is replaced by \( f(x) + d \) where \( d \) is positive, we start with the original function:

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

The y-intercept occurs when \( x = 0 \). To find it, we substitute \( x = 0 \):

\[ f(0) = \sqrt{0} - 3 = -3 \]

So the original y-intercept is at the point \( (0, -3) \).

Now, we consider the new function \( f(x) + d \):

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

To find the new y-intercept, we again substitute \( x = 0 \):

\[ f(0) + d = \sqrt{0} - 3 + d = -3 + d \]

Since \( d \) is positive, this means:

\[ -3 + d > -3 \]

Thus, the new y-intercept is at the point \( (0, -3 + d) \).

This shows that the y-intercept has shifted up by a distance of \( d \).

Therefore, the correct response is:

**The y-intercept shifts up by a distance of d.**