The curve $C$ has equation $y = \frac{ax^2 + bx - 5}{x + c}$ where $a$, $b$, $c$ are

Question

The curve $C$ has equation $y = \frac{ax^2 + bx - 5}{x + c}$ where $a$, $b$, $c$ are constants and $x \neq -c$. Given that $x = 1$ is an asymptote of $C$ and $C$ has a turning point on the $y$-axis, determine the values of $b$ and $c$. Given also that $C$ has no $x$-intercept, show that $a < -\frac{5}{4}$. Sketch the curve $C$ for $-5 < a < -\frac{5}{4}$, stating clearly the coordinates of any stationary point, point of intersection with the axes, and the equations of any asymptotes.

Accepted Answer

(i) $b=5,\ c=-1$. (ii) $a<-\frac54$. (iii) Minimum $(0,5)$, maximum $(2,4a+5)$; asymptotes $x=1$ and $y=ax+a+5$; no $x$-intercepts. See the worked sketch.

Graphing Techniques - Sketching Rational Functions

Graphing Techniques - Sketching Rational Functions

Solution:

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