Question

Compare the graphs of f(x)=2x+1−−−−−√ and g(x)=2x+1−−−−−√3 . What is the y-intercept of both graphs?

Answer 1

To compare the graphs of the functions \( f(x) = \sqrt{2x + 1} \) and \( g(x) = \sqrt{2x + 1} + 3 \), we first need to find their y-intercepts.

The y-intercept of a function occurs when \( x = 0 \).

### Finding the y-intercept of \( f(x) \):
\[
f(0) = \sqrt{2(0) + 1} = \sqrt{1} = 1
\]
Thus, the y-intercept of \( f(x) \) is \( (0, 1) \).

### Finding the y-intercept of \( g(x) \):
\[
g(0) = \sqrt{2(0) + 1} + 3 = \sqrt{1} + 3 = 1 + 3 = 4
\]
Thus, the y-intercept of \( g(x) \) is \( (0, 4) \).

### Summary of the y-intercepts:
- The y-intercept of \( f(x) \) is \( 1 \) (at the point \( (0, 1) \)).
- The y-intercept of \( g(x) \) is \( 4 \) (at the point \( (0, 4) \)).

### Comparison of the Graphs:
1. **Transformations:**
- \( g(x) \) is the function \( f(x) \) vertically shifted upward by 3 units.

2. **Domain and Range:**
- The domain of both functions is \( x \geq -\frac{1}{2} \) (since \( 2x + 1 \geq 0 \)).
- The range of \( f(x) \) is \( [1, \infty) \) and the range of \( g(x) \) is \( [4, \infty) \).

In summary, both graphs are similar in shape, with \( g(x) \) being \( f(x) \) shifted up by 3 units. The y-intercepts are \( (0, 1) \) for \( f(x) \) and \( (0, 4) \) for \( g(x) \).