Question

The graph of y = StartFraction 2 Over x EndFraction – 1 is shown.

On a coordinate plane, a hyperbola has a curve in quadrant 1 and a curve in quadrant 3. The curve in quadrant 1 is asymptotic to the positive y-and x-axis, enters the grid line at (0.25, 6), and crosses the x-axis at (2, 0). The curve in quadrant 3 is asymptotic to the negative y- and x-axis, and enters the grid line at (negative 0. 4, negative 6) and (negative 3, negative 1.75).

Which statement about the graph is accurate?

The x-intercept of the graph is (2, 1).
The graph has no y-intercepts.
The y-intercept of the graph is (2, 0).
The graph has no x-intercepts.

Answer 1

Let's analyze the function \( y = \frac{2}{x} - 1 \) to determine its properties, including x-intercepts and y-intercepts.

1. **Finding the x-intercept**:
- To find the x-intercept, set \( y = 0 \):
\[
0 = \frac{2}{x} - 1
\]
\[
\frac{2}{x} = 1
\]
\[
2 = x \quad \Rightarrow \quad \text{x-intercept is } (2, 0).
\]

2. **Finding the y-intercept**:
- To find the y-intercept, set \( x = 0 \):
\[
y = \frac{2}{0} - 1 \text{ (undefined)}
\]
This means there is no y-intercept because the function does not exist at \( x = 0 \).

Now let's evaluate the statements provided:

- **The x-intercept of the graph is (2, 1)**: This is false. The correct x-intercept is (2, 0).

- **The graph has no y-intercepts**: This is true. The function is undefined at \( x = 0 \).

- **The y-intercept of the graph is (2, 0)**: This is false. There is no y-intercept.

- **The graph has no x-intercepts**: This is false. The graph does have an x-intercept at (2, 0).

The accurate statement about the graph is:
**The graph has no y-intercepts.**