se the image to answer the question.

Question

A downward open parabola is plotted on a coordinate plane. The x-axis ranges from negative 4 to 4 and the y-axis ranges from 0 to negative 6 in increments of 2.

How many real solutions does g(x)=−1.6−0.2x2
have, based on its graph?

(1 point)
Responses

The equation g(x)=−1.6−0.2x2
has one real solution because it crosses the y-axis once.
The equation g left parenthesis x right parenthesis equals negative 1.6 minus 0.2 x squared has one real solution because it crosses the y -axis once.

The equation g(x)=−1.6−0.2x2
has no real solutions because it has no x-intercepts.
The equation g left parenthesis x right parenthesis equals negative 1.6 minus 0.2 x squared has no real solutions because it has no x -intercepts.

The equation g(x)=−1.6−0.2x2
has no real solutions because it crosses the x-axis once.
The equation g left parenthesis x right parenthesis equals negative 1.6 minus 0.2 x squared has no real solutions because it crosses the x -axis once.

The equation g(x)=−1.6−0.2x2
has no real solutions because it has no y-intercepts.

Answer 1

To determine the number of real solutions for the equation \( g(x) = -1.6 - 0.2x^2 \), we should analyze its graph.

1. Identify the Descriptive Form of the Function: The function is a downward-opening parabola because the coefficient of \( x^2 \) is negative (specifically, \(-0.2\)).

2. Finding the Vertex: The vertex of this parabola occurs at \( x = 0 \) (since there is no linear \( x \) term). To find the y-coordinate of the vertex, substitute \( x = 0 \) into the function: \[ g(0) = -1.6 - 0.2(0^2) = -1.6 \] So the vertex is at \( (0, -1.6) \).

3. Examining the Direction of the Parabola: Since the parabola opens downwards and the vertex is at \( y = -1.6 \), which is still above the negative values on the y-axis, the graph will lie entirely below the x-axis for the given y-axis range (-2 to 0) and thus does not intersect the x-axis.

4. Conclusion on Real Solutions: Since the graph does not cross the x-axis, there are no real solutions to the equation \( g(x) = 0 \).

The appropriate response would be:

• The equation \( g(x) = -1.6 - 0.2x^2 \) has no real solutions because it has no x-intercepts.