The functions f and g have <u>same axis of symmetry</u>
The y intercept of f is <u>greater than</u> the y intercept of g.
Over the interval [-6,-3], the average rate of change of f is <u>less than</u> the average rate of change of g.
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Explanation:
Check out the diagram below.
Using the table, plot each (x,y) point onto the same xy grid the blue parabola is graphed on. This way we can compare the two graphs. I'll show these points in red. The red parabola passing through them represents all of f(x) assuming it is continuous.
When we have f(x) and g(x) plotted together, we see that they have the same axis of symmetry. Points D = (-3,-10) and K = (-3,7) have the same x coordinate. The axis of symmetry for each is x = -3.
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Furthermore, the graph shows that the y intercept of f is above the y intercept of g. Look where the graphs cross the y axis. So The y intercept of f is greater than the y intercept of g
We could note that from the table, the y intercept of f is (0,8) as the y intercept always happens when x = 0.
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Last part is the average rate of change (AROC)
AROC is the same as the slope of the line through the endpoints mentioned. The interval [-6, -3] means we go from x = -6 to x = -3. Those x values determine the endpoints.
For f(x), this means we find the slope of the line through (-6,8) and (-3,-10). Use the slope formula to get
m = (y2-y1)/(x2-x1)
m = (-10-8)/(-3-(-6))
m = (-10-8)/(-3+6)
m = -18/3
m = -6 ... negative slope goes downhill as we move from left to right
The AROC for f, on the mentioned interval, is -6
The AROC for g is done in a similar manner. We use the two points J = (-6,-2) and L = (-3,7) as the diagram shows. The slope of line JL is
m = (y2-y1)/(x2-x1)
m = (7-(-2))/(-3-(-6))
m = (7+2)/(-3+6)
m = 9/3
m = 3 ... positive slope goes uphill as we move from left to right
We get a different AROC here. More specifically, this is a larger AROC compared to what we got for f. So the AROC for f is smaller than the AROC for g.
