Question

quadrant graph of an even function. the function is increasing for $-9 < x < -7$. the function is increasing for $-3 < x < 0$. the point lies on the graph of the function. options: $(-5, 8)$, $(-5, -8)$, $(5, -8)$

Explanation:

Step1: Recall the property of even functions

For an even function \( f(x) \), we have \( f(x)=f(-x) \), which means the graph is symmetric about the \( y \)-axis.

Step2: Analyze the given graph (left - hand side, since it's a quadrant graph, we can infer the right - hand side from symmetry)

Looking at the left - hand side of the graph (negative \( x \)-values), we can see the behavior of the function. For an even function, if a point \( (x,y) \) is on the graph, then \( (-x,y) \) is also on the graph. But we can also analyze the sign of the \( y \)-values from the graph. The graph is in the upper half (positive \( y \)-values) for the left - hand side (negative \( x \)-values) as per the visual. Let's check the options:

• Option 1: \( (-5,8) \): Since the function is even, and for \( x = 5 \) (positive \( x \)), we can infer the \( y \)-value from the symmetry. But also, looking at the left - hand side (negative \( x \)), the \( y \)-values are positive. So \( (-5,8) \) is a candidate.
• Option 2: \( (-5,-8) \): The graph is in the upper part (positive \( y \)) for negative \( x \), so \( y \) should be positive, so this is incorrect.
• Option 3: \( (5,-8) \): For an even function, \( f(5)=f(-5) \). Since \( f(-5) \) is positive (from the graph's left - hand side), \( f(5) \) should also be positive, so this is incorrect.

Answer:

\((-5, 8)\)