Use the image to answer the question.

Question

A downward open parabola, made of a solid line is plotted on a coordinate plane. The x-axis ranges from negative 5 to 5 and the y-axis ranges from 0 to negative 10, both in increments of 5.

How many real solutions does f(x)=−2x2−1 have, based on the graph?

(1 point)
Responses

no real solutions because the graphs neither touches nor crosses the x-axis
no real solutions because the graphs neither touches nor crosses the x -axis

two real solutions because the graph is symmetric about the y-axis
two real solutions because the graph is symmetric about the y -axis

one real solution because the graph intersects the y-axis once
one real solution because the graph intersects the y -axis once

one real solution because the graph touches y-axis at y=−1

Answer 1

To determine how many real solutions the equation \( f(x) = -2x^2 - 1 \) has based on the provided description of the graph, we need to evaluate how the parabola behaves.

• The equation \( f(x) = -2x^2 - 1 \) describes a downward-opening parabola.
• The vertex of this parabola occurs at \( y = -1 \) when \( x = 0 \) (the maximum point).
• Since the parabolic curve opens downwards and has its maximum at \( y = -1 \), and only goes further down, it will never touch or cross the x-axis.

Thus, the line \( y = 0 \) (which represents the x-axis) will not intersect the parabola at any point.

Based on this information, the correct response is:

no real solutions because the graphs neither touches nor crosses the x-axis.