Question

1. find the x-intercepts of the function graphed below. then average rate of change over the interval (-2, 3)

Explanation:

Step1: Find x-intercepts

x-intercepts are where \( y = 0 \). From the graph, the function crosses the x - axis at \( x=-1 \) (wait, no, looking at the graph: let's check the points. Wait, the left part: when does y=0? Wait, the graph: the left segment goes from, maybe, (-2,8) to (0, -8)? No, wait the grid: the x - axis crossings. Wait, the graph: let's see, the function has x - intercepts at \( x = - 1\)? Wait no, looking at the graph, the left part: when x = -1? Wait no, the graph: the first x - intercept (where y=0) is at x = -1? Wait no, let's re - examine. Wait, the graph: the left line: let's see the points. The left line: from, maybe, (-2,8) to (0, -8)? No, the open circle at (0, -8). Then the middle line: from (0, -8) open circle to (4,6)? Wait no, the middle line goes up to (4,6)? Wait no, the graph has a peak at x=4? Wait no, the x - axis: the function crosses the x - axis at x = -1? Wait no, wait the graph: the left line: when does y=0? Let's calculate the equation of the left line. The left line passes through (-2,8) and (0, -8) (but (0, -8) is an open circle). The slope of the left line is \( m=\frac{-8 - 8}{0-(-2)}=\frac{-16}{2}=-8 \). The equation is \( y - 8=-8(x + 2) \), \( y=-8x-16 + 8=-8x - 8 \). Set \( y = 0 \): \( 0=-8x - 8\), \( 8x=-8\), \( x=-1 \). Then the right line: let's find its equation. The right line has a peak, and crosses the x - axis at x = 4? Wait no, the right line: let's take two points. The peak is at (4,6)? Wait no, the grid: the y - axis has 8,4,0,-4,-8. The right line: from (4,0) to (6, -8) (open circle at (6, -8))? Wait no, the x - intercepts: when y = 0, the function crosses the x - axis at x=-1 and x = 4? Wait, let's check the graph again. The left part: when x=-1, y=0 (from the equation \( y=-8x - 8 \), when x=-1, y=0). The middle part: the line from (0, -8) open circle to (4,6) (peak) and then to (6, -8) open circle. Wait, the middle line: let's find its equation. Let's take two points: (0, -8) open circle and (4,6). The slope is \( m=\frac{6-(-8)}{4 - 0}=\frac{14}{4}=\frac{7}{2} \). The equation is \( y+8=\frac{7}{2}(x - 0) \), \( y=\frac{7}{2}x-8 \). Set y = 0: \( 0=\frac{7}{2}x-8 \), \( \frac{7}{2}x=8 \), \( x=\frac{16}{7}\approx2.2857 \)? No, that can't be. Wait, maybe I misread the graph. Wait, the graph: the x - intercepts are at x=-1 and x = 4? Wait, looking at the graph, the left line crosses the x - axis at x=-1, and the right line crosses the x - axis at x = 4. So x - intercepts are \( x=-1 \) and \( x = 4 \).

Step2: Find average rate of change over \((-2,3)\)

The average rate of change of a function \( f(x) \) over the interval \([a,b]\) is \( \frac{f(b)-f(a)}{b - a} \). First, find \( f(-2) \) and \( f(3) \).

For \( x=-2 \): From the left line (since x=-2 is in the domain of the left line, as the left line is from x=-2 (closed circle?) Wait, the left line: the point at x=-2: is it a closed circle? The graph shows a closed arrow at x=-2? Wait, the left end: the first segment: from x=-2 (closed) to x=0 (open). So \( f(-2) \): on the left line, when x=-2, \( y = 8 \) (from the graph, at x=-2, y=8).

For \( x = 3 \): x=3 is on the middle line (the line from x=0 (open) to x=4 (closed? Wait, the middle line: from x=0 (open) to x=4 (closed? The peak at x=4? Wait, the middle line: let's find f(3). The equation of the middle line: we know that at x=0, the open circle is at y=-8, and at x=4, what's the y - value? The peak at x=4: from the graph, the peak is at (4,6)? Wait no, the grid: the y - axis at x=4, the y - value is 6? Wait, no, let's recast. Wait, the middle line: let's take two points. The middle line goes from (0, -8) open circle to (4,6) (peak). The slope \( m=\frac{6-(-8)}{4-0}=\frac{14}{4}=\frac{7}{2} \). So the equation is \( y=\frac{7}{2}x-8 \) (for \( 0\lt x\lt4 \)). So when x = 3, \( f(3)=\frac{7}{2}(3)-8=\frac{21}{2}-8=\frac{21 - 16}{2}=\frac{5}{2}=2.5 \).

Now, average rate of change \(=\frac{f(3)-f(-2)}{3-(-2)}=\frac{\frac{5}{2}-8}{3 + 2}=\frac{\frac{5 - 16}{2}}{5}=\frac{\frac{-11}{2}}{5}=-\frac{11}{10}=-1.1 \)? Wait, no, maybe I made a mistake in f(3). Wait, maybe the middle line's equation is wrong. Wait, let's check the graph again. Wait, the left line: from x=-2 (closed) to x=0 (open), equation \( y=-8x - 8 \) (as before). At x=-2, y=8 (correct, since \( y=-8(-2)-8 = 16 - 8 = 8 \)). At x=0, y=-8 (open circle).

The middle line: from x=0 (open) to x=4 (closed). Let's find two points on the middle line. Let's say at x=1, what's y? Wait, maybe the middle line passes through (0, -8) open and (4,6) is wrong. Wait, maybe the middle line is from (0, -8) open to (4,6) is incorrect. Wait, the graph: the middle line goes up to a peak, then down to x=4 (where y=0? No, x=4 is an x - intercept? Wait, no, earlier we thought x=4 is an x - intercept, but when x=4, y=0? Wait, maybe my x - intercepts were wrong. Wait, let's re - examine the graph. The right line: from x=4 (open) to x=6 (open). Wait, the x - intercepts: the function crosses the x - axis at x=-1 and x=4. So at x=4, y=0. So the middle line: from x=0 (open, y=-8) to x=4 (closed, y=0). Then the slope of the middle line is \( m=\frac{0-(-8)}{4 - 0}=\frac{8}{4}=2 \). Ah, that's a mistake earlier. So the equation of the middle line (from x=0 open to x=4 closed) is \( y=2x-8 \) (because when x=0, y=-8 (open), when x=4, y=0 (closed)). Let's check: when x=4, \( y=2(4)-8=0 \), correct. When x=0, \( y=-8 \), correct. So that's the correct equation.

So now, find \( f(-2) \): x=-2 is on the left line. The left line: from x=-2 (closed) to x=0 (open). The equation of the left line: passes through (-2,8) and (0, -8) (open). The slope \( m=\frac{-8 - 8}{0-(-2)}=\frac{-16}{2}=-8 \). Equation: \( y-8=-8(x + 2) \), \( y=-8x-16 + 8=-8x - 8 \). So when x=-2, \( f(-2)=-8(-2)-8=16 - 8 = 8 \).

For x=3: x=3 is in the interval (0,4), so use the middle line equation \( y=2x-8 \). So \( f(3)=2(3)-8=6 - 8=-2 \).

Now, average rate of change over (-2,3) is \( \frac{f(3)-f(-2)}{3-(-2)}=\frac{-2 - 8}{3 + 2}=\frac{-10}{5}=-2 \).

Answer:

x - intercepts: \( x=-1 \) and \( x = 4 \); Average rate of change: \(-2\)