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Ray Atkinson
Ray Atkinson, Bachelor's Degree
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Sketch the graph of the function using the graph sketching

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Sketch the graph of the function using the graph sketching strategy.

f(x) = 8x + 13/
9 − 4x^2

Strategy as follows
1. Domain of f
2. Whether f is even, odd or neither
3. the x and y intercepts of f
4. the intervals on which f is positive or negative
5. the intervals of which f is increasing or decreasing and any stationary points, local maxima and local minima
6. the asymptotic behaviour of f

Need all steps of the strategy completing and a sketch of the graph if possible
1. The domain is going to be everywhere that the denominator is not zero. 9-4x²=0 --> 9=4x² --> x² = 9/4 --> x = ±3/2. There are vertical asymptotes at those points
2. It is neither even nor odd.
3. When x=0, f(x)=13/9.
Setting y=0, You need to find when 8x+13=0, so x=-13/8. Since this is not ±3/2, it is valid.
4. There is only one root as x=-13/8, so the graph is only going to cross once. Since f(0) is positive, that side of the graph is going to be positive. f(x0 and f(x>-13/8) is negative.
5. The first derivative of this is M=(32x²+104x+72)/(9-4x²)². Zero points are when 32x²+104x+72=0, which are -9/4 and -1 [I can go through the calculus if you need it]. This is where the graph flattens out. This gives 4 critical points at -9/4, -3/2, -1, and 3/2. Checking between them tells us how the graph is behaving.
Extremely negative numbers will give a positive result for M, so the graph is increasing to the left of -9/4. Looking at M(-2) we get a negative, so f(x) is decreasing from -9/4 to -3/2. Looking at M(-5/4), it is negative, so the graph is again decreasing after the asymptote. M(0)= 8/9, so the graph in increasing from -3/2 to 3/2. Looking at extreme positive, M stays positive, so the original graph increases as it heads off to the right.
This means that (-9/4, 4/9) and (-1, 1) are local extrema.
6. Since the degree of the denominator exceeds the numerator, the graph tends to 0 in both x directions.

This gives us all the information we need to make the graph.
I have uploaded it to this site. The scan is rotated 90° clockwise, so make sure you correct for it.

If you have any questions, let me know.
Ray Atkinson and other Calculus and Above Specialists are ready to help you
Customer: replied 8 years ago.

With point 2 would you be able to show me your working out. Thanks

An odd function is one that -f(x) = f(-x) and an even function is one that f(-x)=f(x) and neither of those cases is true here.
Customer: replied 8 years ago.

thanks for that :) for point 5 would you be able to go through the calculus please. Thanks

d/dx (8x+13)/(9-4x²) uses the quotient rule. (u/v)' = (vu'-uv') / v²
[(9-4x²)*(8x+13)' - (8x+13)*(9-4x²)'] / [(9-4x²)]²
[(9-4x²) * 8 - (8x+13) * -8x] / [(9-4x²)]²
72 - 32x² + 64x² + 104x / (9-4x²)²
(32x² + 104x + 72) / (9-4x²)²

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