Question

the function $f(x)$ is graphed below. what is true about the graph on the interval from $x = -infty$ to $x = a$?
graph of a parabola opening downward, with x-intercepts at a (left) and c (right), vertex at b above the x-axis, crossing the y-axis above the origin
answer

• it is positive and increasing
• it is positive and decreasing
• it is negative and increasing
• it is negative and decreasing

Explanation:

Step1: Analyze the sign of the function

On the interval from \( x = -\infty \) to \( x = a \), the graph of \( f(x) \) is below the x - axis (since it is below the horizontal line \( y = 0 \) in this region). So, the function values \( f(x) \) are negative in this interval.

Step2: Analyze the increasing/decreasing nature

To determine if the function is increasing or decreasing, we look at the slope of the graph. As we move from \( x = -\infty \) towards \( x = a \), the graph of the function is rising (going from a lower y - value to a higher y - value). A rising graph means that as \( x \) increases, \( f(x) \) increases. So, the function is increasing on the interval from \( x=-\infty \) to \( x = a \).

Answer:

It is negative and increasing (the option corresponding to "It is negative and increasing")