Question

for items 1–4, find each key feature of the function shown in the graph, 1. range 2. domain 3. intervals where the graph is positive 4. what is the average rate of change for the function over the interval -6, -4? a 2 b -10 c -1 d -2

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( f(x) \) over the interval \([a, b]\) is given by \(\frac{f(b) - f(a)}{b - a}\).

Step2: Identify the points at \( x = -6 \) and \( x = -4 \)

From the graph, at \( x = -6 \), the \( y \)-value (let's call it \( f(-6) \)) is \( 4 \) (since there's an open circle? Wait, no, looking at the graph: the left part has a point at \( x = -8 \) with \( y = 4 \), then goes down to \( x = -4 \) with \( y = -4 \)? Wait, no, maybe I misread. Wait, the graph: from \( x = -8 \) (open circle? No, the left point is a closed circle? Wait, the first segment: from \( x = -8 \) (closed circle, \( y = 4 \)) to \( x = -4 \) (where \( y = -4 \))? Wait, no, the grid: let's check the coordinates. Let's assume the grid has each square as 1 unit. So at \( x = -6 \), what's \( f(-6) \)? Let's see, the first line: from \( (-8, 4) \) to \( (-4, -4) \)? Wait, no, maybe the first segment is from \( x = -8 \) (closed circle, \( y = 4 \)) to \( x = -4 \) (where \( y = -4 \))? Wait, no, the problem is about the interval \([-6, -4]\). So \( a = -6 \), \( b = -4 \). Let's find \( f(-6) \) and \( f(-4) \).

Looking at the graph: the first part (left) is a line from \( x = -8 \) (with \( y = 4 \)) to \( x = -4 \) (with \( y = -4 \)). So the slope of that line is \( \frac{-4 - 4}{-4 - (-8)} = \frac{-8}{4} = -2 \). But for the interval \([-6, -4]\), let's find \( f(-6) \) and \( f(-4) \).

At \( x = -6 \): since it's on the line from \( (-8, 4) \) to \( (-4, -4) \), we can calculate \( f(-6) \). The change in \( x \) from -8 to -6 is \( 2 \) units. The slope is \( -2 \) (as above), so the change in \( y \) is \( -2 \times 2 = -4 \). So \( f(-6) = 4 + (-4) = 0 \)? Wait, no, that can't be. Wait, maybe the first point is \( (-8, 4) \) (closed circle) and then goes down to \( (-4, -4) \) (closed circle? No, the graph shows a closed circle at \( x = -4 \) with \( y = -4 \)? Wait, the original graph: let's re-express.

Wait, the graph:

• Left segment: from \( x = -8 \) (closed circle, \( y = 4 \)) to \( x = -4 \) (closed circle, \( y = -4 \))?

• Then from \( x = -4 \) to \( x = 0 \), maybe? Wait, no, the problem is about the interval \([-6, -4]\). So \( x = -6 \) is between -8 and -4, and \( x = -4 \) is the endpoint.

So \( f(-4) \): looking at the graph, at \( x = -4 \), the \( y \)-value is \( -4 \) (since the segment ends there).

\( f(-6) \): let's calculate. The line from \( (-8, 4) \) to \( (-4, -4) \). The equation of this line: using point-slope form. \( y - 4 = -2(x + 8) \) (since slope is -2). So \( y = -2x - 16 + 4 = -2x - 12 \).

So at \( x = -6 \): \( y = -2(-6) - 12 = 12 - 12 = 0 \). Wait, that's \( f(-6) = 0 \)?

At \( x = -4 \): \( y = -2(-4) - 12 = 8 - 12 = -4 \), so \( f(-4) = -4 \).

Now, the average rate of change is \( \frac{f(-4) - f(-6)}{-4 - (-6)} = \frac{-4 - 0}{2} = \frac{-4}{2} = -2 \). Wait, but let's check again.

Wait, maybe I made a mistake in the equation. Let's re-calculate the slope. From \( (-8, 4) \) to \( (-4, -4) \): change in \( y \) is \( -4 - 4 = -8 \), change in \( x \) is \( -4 - (-8) = 4 \), so slope is \( -8/4 = -2 \). So the equation is \( y = -2(x + 8) + 4 = -2x - 16 + 4 = -2x - 12 \). So at \( x = -6 \), \( y = -2(-6) -12 = 12 -12 = 0 \). At \( x = -4 \), \( y = -2(-4) -12 = 8 -12 = -4 \). So the average rate of change is \( (f(-4) - f(-6))/(-4 - (-6)) = (-4 - 0)/(2) = -4/2 = -2 \). So the answer is D.

Answer:

D. -2