How do you sketch the curve of $f(x) = (2x-3) / (2 x -9)^2$ ?

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How do you sketch the curve of $f(x) = (2x-3) / (2 x -9)^2$ ?

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Answer 1 · 2015-09-11T20:07:59

See explanation section.

Explanation:

$f(x) = (2x-3) / (2 x -9)^2$

Domain: $RR " - " {9/2}$

$x$ -intercept: $3/2$
$y$ intercept $-1/27$

Symmetry: None

Vertical Asymptote: $x = 9/2$
Horizontal Asymptote: $y = 0$ (a.k.a. the $x$ -axis)

$f'(x) = (-2(2x+3))/(2x-9)^3$

Sign analysis of $f'$ shows that:
$f$ is decreasing on $(-oo, -3/2)$ and on $(9/2, oo)$ and
$f$ is increasing on $(-3/2, 9/2)$

Local Minimum $-1/24$ (at $x=-3/2$ )

$f''(x) = (8(2x+9))/(2x-9)^4$
Sign analysis of $f''$ shows that:
$f$ is concave down on $(-oo, -9/2)$ and
$f$ is concave up on $(-9/2, 9/2)$ and on $(9/2, oo)$

$(-9/2, -1/27)$ is the only inflection point.

The graph looks like this:

graph{y=(2x-3) / (2 x -9)^2 [-6.16, 11.62, -2.91, 5.985]}

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How do you sketch the curve of $f(x) = (2x-3) / (2 x -9)^2$ ?

1 Answer

Explanation:

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