Reciprocal & Rational Functions (DP IB Analysis & Approaches (AA): HL): Exam Questions

Consider $f\left(x\right)=\frac{ax+b}{3x+c}$, for $x\ne -\frac{c}{3}$, where $a,b,c\in \mathrm{\mathbb{Z}}$.

The graph of $f$ has a horizontal asymptote $y=-1$ and a vertical asymptote $x=2$. The point $\mathrm{A}\left(-2,-\frac{2}{3}\right)$ lies on the graph of $f$.

Find the values of $a$, $b$ and $c$.

Consider the function $f$ defined by  $f\left(x\right)=\frac{2\left(3x-1\right)}{\left(x+3\right)\left(x-2\right)}$ , $x\in \mathrm{ℝ}$ ,  $x\ne -3,2$ .

Find the coordinates of the points where the graph of $y=f\left(x\right)$  intersects the coordinate axes.

Express $f\left(x\right)$ as partial fractions.

Hence find the equation of the horizontal asymptote to the graph of $y=f\left(x\right)$.

Consider the function  $f\left(x\right)=\frac{4x-12}{x^2-4x-5}$  ,$x\in \mathrm{ℝ}$  , $x\ne -1,5$

Find the coordinates of the points where the graph of  $y=f\left(x\right)$ intersects the

(i) $x$-axis  

(ii) $y$-axis.

Write down the equations of

(i) the vertical asymptotes

(ii) the horizontal asymptote

to the graph of $y=f\left(x\right)$.

Sketch the graph of $y=f\left(x\right)$, clearly indicating the asymptotes and any points of intersection with the axes.

Consider the function $f\left(x\right)=\frac{x^2+5x+6}{x+1}$, $x\in \mathrm{ℝ}$ , $x\ne -1$ .

Find the coordinates of the points where the graph of $y=f\left(x\right)$ intersects the

(i) $x$-axis,

(ii) $y$-axis.

Write down the equation of the vertical asymptote to the graph of $y=f\left(x\right)$.

(i) Show that  $\frac{x^2+5x+6}{x+1}=x+a+\frac{b}{x+1}$ , where $a$ and $b$ are constants to be determined.

(ii) Hence write down the equation of the oblique asymptote to the graph of    $y=f\left(x\right)$       .

Sketch the graph of $y=f\left(x\right)$. On your sketch, clearly indicate the asymptotes and the position of any points of intersection with the axes.

Let $f$ be a function defined by $f\left(x\right)=3x-4+\frac{2}{x-1}$  , $x\in \mathrm{ℝ}$ , $x\ne p$ .

Write down

(i) the value of $p$

(ii) the equation of the vertical asymptote to the graph of $y=f\left(x\right)$

(iii) the equation of the oblique asymptote to the graph of $y=f\left(x\right)$.

Show that $f\left(x\right)$ can be written in the form $\frac{a{x}^2+bx+c}{x-1}$,  where $a$, $b$ and $c$ are constants to be determined.

Use an algebraic method to show that the graph of $y=f\left(x\right)$ does not cross the $x$-axis.

Sketch the graph of $y=f\left(x\right)$, showing the coordinates of any local maximum or minimum points.

Hence write down the range of $f$.

Let $f\left(x\right)=\frac{b}{x-a}+2$, for $x\ne a.$

The line $x=-3$ is a vertical asymptote of the graph of $f$.

Write down the value of $a$.

The graph of $f$ passes through the point $\mathrm{A}\left(4,-2\right)$.

Find the value of $b$.

Find $\underset{x\to \infty }{\lim }f\left(x\right)$.

Consider $f\left(x\right)=\frac{ax+b}{cx+1}$, where $a,b,c\in \mathrm{\mathbb{Z}}$.

The graph of $f$ has a horizontal asymptote $y=4$ and a vertical asymptote $x=-1$. The point $\mathrm{P}\left(5,\frac{9}{2}\right)$ lies on the graph of $f$.

Find the values of $a$, $b$ and $c$.

Sketch the graph of $y=f\left(x\right)$. On your sketch, indicate the values of any axis intercepts and label any asymptotes with their equations.

Let $f\left(x\right)=\frac{3}{x-a}+b$, for $x\ne a.$ 

The line $x=2$ is a vertical asymptote of the graph of $f$. 

Write down the value of $a.$

The graph of $f$ passes through the point $\mathrm{A}\left(1,5\right)$.

Find the value of $b$.

Find $\underset{x\to \infty }{\lim }f\left(x\right).$

Consider the function $f$ defined by $f\left(x\right)=\frac{3x-7}{x^2-6x+5}$, for $x\ne 1$, $x\ne 5$.

Find the coordinates where the graph of $f$ intersects the coordinate axes.

(i) Express $f\left(x\right)$ as partial fractions.

(ii) Hence, write down the equation of the horizontal asymptote of the graph of $f$.

Sketch the graph of $y=f\left(x\right)$. On your sketch, indicate the values of any axis intercepts and label any asymptotes with their equations.

Consider the function $f\left(x\right)=\frac{2x^2-x-15}{x-2},x\in \mathrm{ℝ},x\ne 2.$

Find the coordinates where the graph of $f$ crosses the

(i) $x$-axis,

(ii) $y$-axis.

Find the equation of the oblique asymptote of the graph of $f$, giving the answer in the form $y=ax+b$, where $a,b\in \mathrm{\mathbb{Z}}$.

Sketch the graph of $y=f\left(x\right)$. Clearly indicate the asymptotes and give coordinates of the points where the graph intersects the axes.

Consider the function $f$ defined by $f\left(x\right)=\frac{ax+b}{2x^2+cx+d},$ where $a,b,c,d\in \mathrm{\mathbb{Z}}$. The lines $x=-\frac{1}{2}$ and $x=3$ are vertical asymptotes of the graph of $f$ as shown below. The graph crosses through the points $\mathrm{A}\left(0,4\right)$ and $\mathrm{B}\left(2,0\right)$.

Find the values of $a,b,c$ and $d$.

Sketch the graph of the function $f$ defined by $f\left(x\right)=\frac{4x-1}{2x^2-x-6}.$ Clearly indicate the coordinates where the graph intersects the axes and state the equation of any asymptotes.

Sketch the graph of the function $g$ defined by $g\left(x\right)=\frac{2x^2-x-6}{4x-1}$. Clearly indicate the coordinates where the graph intersects the axes and state the equation of any asymptotes.

The graph of a function $f$ is shown below. The equations of the asymptotes are $x=0.25$ and $y=0.5.$ The graph crosses the coordinate axes at the points $\mathrm{A}(0,m)$ and $\mathrm{B}(5,0)$.

Find an equation for $f\left(x\right)$ in the form  $\frac{ax+b}{cx+d}$, where $a,b,c,d\in \mathrm{\mathbb{Z}}$.

Write down the value of $m$.

The graph of a function $f$ is shown below. The equations of the asymptotes are $x=3$ and $y=3x-2$. The graph crosses the $y$-axis at $\left(0,-2.5\right)$.

The function can be written as $f\left(x\right)=ax+b+\frac{3}{cx+d}$, where $a,b,c,d\in \mathrm{\mathbb{Z}}.$ 

Find the values of $a,b,c$ and $d$.

Find the exact coordinates of the points where $y=f\left(x\right)$ crosses the $x$-axis.

Given that $f\left(x\right)=k$ has no real solutions where $k\in \mathrm{ℝ}$, find the set of possible values of $k$. Give the bounds correct to 2 decimal places.

The graph of a function $f$ is shown below. The graph crosses the $x$-axis at the point $\mathrm{A}$ and the $y$-axis at the point $\mathrm{B}$. The asymptotes intersect at the point $\mathrm{C}\left(4,-\frac{8}{3}\right)$. The area of triangle $\mathrm{AOB}$ is $3$.

Find an equation for $f\left(x\right)$ in the form $\frac{ax+b}{cx+d}$, where $a,b,c,d\in \mathrm{\mathbb{Z}}$.

Let the function $f$ be defined by $f\left(x\right)=3-\frac{a}{2x-5},x\in \mathrm{ℝ},x\ne p$, where $a$ is a positive constant. 

Write down the value of $p$.

Sketch the graph of $f$ . State the equations of the asymptotes and give the coordinates of the intersections with the coordinate axes in terms of $a$.

Given that the graph of $f$ intersects the line $y=x$, find the set of possible values of $a$.

Consider the function $f$ defined by $f\left(x\right)=\frac{4-2x}{x^2-3x+3}$. The graph of $f$ is shown below. The graph has a maximum point at $\mathrm{A}$ and a minimum point at $\mathrm{B}$.

Show that $f$ is defined for all $x\in \mathrm{ℝ}$.

(i) Given that the graph of $f$ intersects the line $y=k$, show that $3k^2-4k-4\le 0$.

(ii) Hence, find the range of $f$.

Hence, find the coordinates of A and B.

The graph of a function $f$  is shown below. The graph crosses the $x$-axis at the points $\mathrm{A}\left(-0.5,0\right)$ and $\mathrm{B}\left(3,0\right)$. The graph crosses the $y$-axis at $\mathrm{C}\left(0,-3\right)$. The equation of the vertical asymptote is $x=2$.

The function $f$ can be written in the form $f\left(x\right)=\frac{a{x}^2+bx+c}{dx+e}$, where $a,b,c,d,e\in \mathrm{\mathbb{Z}}.$ 

Given that $ce+2=b$ , find the values of $a,b,c,d$and $e$.

Hence find the equation of the oblique asymptote of the graph of $f$.

Consider the function $f$ defined by $f\left(x\right)=\frac{\left(x-p\right)\left(x-q\right)}{x-n}$, where $n,p$ and $q$are positive constants and $p<q$.

In the case that $p,q$and $n$ are distinct:

(i) write down the coordinates of the points where the graph of $f$ intersects the axes,

(ii) write down the equation of the vertical asymptote,

(iii) show that the equation of the oblique asymptote is  $y=x+(n-p-q)$.

Sketch the graph of $f$ in the case that:

(i) $n<p<q$,

(ii) $p<n<q$,

(iii) $n=p$.

Clearly indicate where the graph crosses the coordinate axes and any asymptotes or discontinuities.

Consider the function $f$ defined by $f\left(x\right)=\frac{ax+b}{x^2+cx+d}$,  where $a,b,c,d\in \mathrm{\mathbb{Z}}$. The line $x=2$ is the only vertical asymptote of the graph of $f$ as shown below. The graph crosses through the points $\mathrm{A}\left(0,k\right)$ and $\mathrm{B}\left(k,0\right)$ where $k$ is a positive constant. The line $y=-1$ is a tangent to the graph of $f$ at the point $\mathrm{C}$.

Find the values of $a,b,c,d$ and $k$.

Find the coordinates of $\mathrm{C}$.

The diagram below shows the graphs of two linear functions $f$ and $g$ and a quadratic function $h$.

Sketch the graph of $y=\frac{f\left(x\right)}{g\left(x\right)}$ . Clearly indicate where the graph intersects the $x$-axis and the location of any vertical asymptotes.

Sketch the graph of $y=\frac{g\left(x\right)}{h\left(x\right)}$. Clearly indicate where the graph intersects the $x$-axis and the location of any vertical asymptotes.