Parameter values from curve properties

Edexcel P1 2019 October Q6

8 marks Standard +0.8

6. The curve $C$ has equation $y = \frac { 4 } { x } + k$, where $k$ is a positive constant.

Sketch a graph of $C$, stating the equation of the horizontal asymptote and the coordinates of the point of intersection with the $x$-axis. The line with equation $y = 10 - 2 x$ is a tangent to $C$.
Find the possible values for $k$. $\_\_\_\_$ -

CAIE FP1 2010 November Q12 EITHER

Challenging +1.8

The curve $C$ has equation $$y = \frac { x ^ { 2 } + 2 \lambda x } { x ^ { 2 } - 2 x + \lambda }$$ where $\lambda$ is a constant and $\lambda \neq - 1$.

Show that $C$ has at most two stationary points.
Show that if $C$ has exactly two stationary points then $\lambda > - \frac { 5 } { 4 }$.
Find the set of values of $\lambda$ such that $C$ has two vertical asymptotes.
Find the $x$-coordinates of the points of intersection of $C$ with
1. the $x$-axis,
2. the horizontal asymptote.
3. Sketch $C$ in each of the cases
  (a) $\lambda < - 2$,
  (b) $\lambda > 2$.

Edexcel PMT Mocks Q6

6 marks Standard +0.3

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{48f9a252-61a2-491d-94d0-8470aee96942-07_864_995_299_495} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The figure 1 shows sketch of the curve $C$ with equation $y = \mathrm { f } ( x )$. $$f ( x ) = a x ( x - b ) ^ { 2 } , x \in R$$ where $a$ and $b$ are constants.
The curve passes through the origin and touches the $x$-axis at the point $( 3,0 )$.
There is a minimum point at $( 1 , - 4 )$ and a maximum point at $( 3,0 )$.
a. Find the equation of $C$.
b. Deduce the values of $x$ for which $$\mathrm { f } ^ { \prime } ( x ) > 0$$ Given that the line with equation $y = k$, where $k$ is a constant, intersects $C$ at exactly one point,
c. State the possible values for $k$.

Edexcel Paper 1 2022 June Q6

6 marks Moderate -0.3

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{129adfbb-98fa-4e88-b636-7b4d111f3349-12_528_812_251_628} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of a curve $C$ with equation $y = \mathrm { f } ( x )$ where $\mathrm { f } ( x )$ is a cubic expression in $X$. The curve

passes through the origin
has a maximum turning point at $( 2,8 )$
has a minimum turning point at $( 6,0 )$
1. Write down the set of values of $x$ for which

$$\mathrm { f } ^ { \prime } ( x ) < 0$$ The line with equation $y = k$, where $k$ is a constant, intersects $C$ at only one point.

Find the set of values of $k$, giving your answer in set notation.

Find the equation of $C$. You may leave your answer in factorised form.

OCR MEI Further Pure with Technology 2023 June Q1

21 marks Challenging +1.2

1 A family of functions is defined as $$f ( x ) = a x + \frac { x ^ { 2 } } { 1 + x } , \quad x \neq - 1$$ where the parameter $a$ is a real number. You may find it helpful to use a slider (for $a$ ) to investigate the family of curves $y = f ( x )$. \begin{enumerate}[label=(\alph*)] \item \begin{enumerate}[label=(\roman*)] \item On the axes in the Printed Answer Booklet, sketch the curve $y = f ( x )$ in each of the following cases.

$a = - 2$
$a = - 1$
$a = 0$
State a feature which is common to the curve in all three cases, $a = - 2$, $a = - 1$ and $a = 0$.
State a feature of the curve for the cases $a = - 2 , a = - 1$ that is not a feature of the curve in the case $a = 0$.
1. Determine the equation of the oblique asymptote to the curve $\mathrm { y } = \mathrm { f } ( \mathrm { x } )$ in terms of $a$.
2. For $b \neq - 1,0,1$ let $A$ be the point with coordinates ( $- b , \mathrm { f } ( - b )$ ) and let $B$ be the point with coordinates ( $b , \mathrm { f } ( b )$ ).

Show that the $y$-coordinate of the point at which the chord to the curve $y = f ( x )$ between $A$ and $B$ meets the $y$-axis is independent of $a$.

With $\mathrm { y } = \mathrm { f } ( \mathrm { x } )$, determine the range of values of $a$ for which

Find its coordinates and fully justify that it is a cusp.

OCR MEI Further Pure with Technology 2024 June Q1

17 marks Standard +0.8

1 A family of curves is given by the equation $$y = \frac { x ^ { 2 } - x + a ^ { 2 } - a } { x - 1 }$$ where the parameter $a$ is a real number.

1. On the axes in the Printed Answer Booklet, sketch the curve in each of these cases.
  - $a = - 0.5$
  - $a = - 0.1$
  - $a = 0.5$
  - State one feature of the curve for the cases $a = - 0.5$ and $a = - 0.1$ that is not a feature of the curve in the case $a = 0.5$.
  - By using a slider for $a$, or otherwise, write down the non-zero value of $a$ for which the points on the curve (\textit{) all lie on a straight line.
  - Write down the equation of the vertical asymptote of the curve (}).
  The equation of the curve (*) can be written in the form $y = x + A + \frac { a ^ { 2 } - a } { x - 1 }$, where $A$ is a constant.
2. Show that $A = 0$.
3. Hence, or otherwise, find the value of $$\lim _ { x \rightarrow \infty } \left( \frac { x ^ { 2 } - x + a ^ { 2 } - a } { x - 1 } - x \right) .$$
4. Explain the significance of the result in part (a)(vi) in terms of a feature of the curve (*).
5. In this part of the question the value of the parameter $a$ satisfies $0 < a < 1$. For values of $a$ in this range the curve intersects the $x$-axis at points X and Y . The point Z has coordinates $( 0 , - 1 )$. These three points form a triangle XYZ.
  1. Determine, in terms of $a$, the area of the triangle XYZ.
  2. Find the maximum area of the triangle XYZ.

AQA Paper 2 2019 June Q7

10 marks Challenging +1.2

Sketch the graph of any cubic function that has both three distinct real roots and a positive coefficient of $x^3$ [2 marks]
The function f(x) is defined by $$f(x) = x^3 + 3px^2 + q$$ where $p$ and $q$ are constants and $p > 0$
1. Show that there is a turning point where the curve crosses the $y$-axis. [3 marks]
2. The equation $f(x) = 0$ has three distinct real roots. By considering the positions of the turning points find, in terms of $p$, the range of possible values of $q$. [5 marks]

SPS SPS SM 2020 June Q11

9 marks Standard +0.3

Sketch the curve with equation $$y = k - \frac{1}{2x}$$ where $k$ is a positive constant State, in terms of $k$, the coordinates of any points of intersection with the coordinate axes and the equation of the horizontal asymptote. [3]

The straight line $l$ has equation $y = 2x + 3$ Given that $l$ cuts the curve in two distinct places,

find the range of values of $k$, writing your answer in set notation. [6]