Modulus Functions & Further Transformations (DP IB Analysis & Approaches (AA): HL): Exam Questions

Sketch the graph of $y={\left(x-1\right)}^2-2\left|x-1\right|-1$, for $-3\le x\le 6$.

Hence, solve the equation ${\left(x-1\right)}^2-2\left|x-1\right|-1=0$.

The graph of $y=f\left(x\right)$ is given below.

On separate axes, sketch the graphs of

$y=\left|f\left(x\right)\right|$

$y={\left[f\left(x\right)\right]}^2$

Sketch the curve $y=\frac{3}{x+4}$ and line $y=4-x$ on the same axes, clearly indicating any $x$- and  $y$- intercepts and any asymptotes.

Consider the equation

$4-x=\left|\frac{3}{x+4}\right|$

(i) Explain why the cases $x<-4$, $x=-4$ and $x>-4$ must be considered separately when solving the equation.

(ii) Hence find the exact solutions of the equation.

Consider the function $f$ defined by$f\left(x\right)=3x^2\arcsin x$, $-1\le x\le 1.$

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

State the range of$f$.

Solve the inequality  $\left|3x^2\arcsin x\right|>1.$

Consider the function$f$ defined by$f\left(x\right)=\sqrt{9-x}$, where$f$ has the largest possible valid domain.

(i) Sketch the graph of $y=f\left(x\right)$, labelling the  $x$- and  $y$-intercepts. 

(ii) State the domain and range of$f$.

(i) On the same set of axes, sketch the graph of the function$f\left(\right|x\left|\right)$, labelling the $x$- and $y$-intercepts.

(ii) State the domain and range of the function$f\left(\right|x\left|\right)$.

Let $f\left(x\right)=\frac{7-9x}{cx-12}$, $x\ne \frac{12}{c}$, where $c$ is a non-zero constant.

The line $x=4$ is a vertical asymptote to the graph of $y=f\left(x\right)$.

(i) Find the value of $c$.

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

The line $y=k,$ where $k\in \mathrm{ℝ},$ intersects the graph of $y=\left|f\left(x\right)\right|$ at exactly one point. Find the possible values of $k.$

Let $f\left(x\right)=2x^3-2x$, for $x\in \mathrm{ℝ}$.

(i) Sketch the graph of $y=\left|f\left(x\right)\right|$.

(ii) Describe the transformation that maps the part of the graph of $y=f\left(x\right)$ for which $f\left(x\right)<0$ onto the graph of $y=\left|f\left(x\right)\right|$.

(i) Sketch the graph of $y=f\left(\left|x\right|\right)$.

(ii) Describe how the part of the graph of $y=f\left(\left|x\right|\right)$ for which $x<0$ is obtained from the graph of $y=f\left(x\right)$.

Let $f\left(x\right)=x\left(x-2\right)$.

Sketch the graph of $y=f\left(x\right)$, clearly showing the coordinates of the vertex and of any points where the graph meets the coordinate axes.

On the same axes, sketch the graph of $y=\frac{1}{f\left(x\right)}$, clearly showing the coordinates of any local maximum or minimum points.

Write down the equations of the asymptotes of the graph of $y=\frac{1}{f\left(x\right)}$.

Consider the function $f$ defined by $f\left(x\right)=4x^2-8x-5$.

Sketch the graph of $y=\left|f\left(x\right)\right|$. Clearly indicate any intercepts with the axes and any turning points.

Sketch the graph of $y={\left[f\left(x\right)\right]}^2$. Clearly indicate any intercepts with the axes and any turning points.

Sketch the curve $y=\frac{4}{x+3}$ and the line $y=3-x$ on the same diagram, clearly indicating any $x$- and  $y$- axes intercepts as well as any asymptotes.

Hence find the exact solutions to the equation

$3-x=\left|\frac{4}{x+3}\right|.$

By considering the inverse of an appropriate function, sketch the graph $y=\sqrt{x}$ .

The function $f$ is defined by $f\left(x\right)=\sqrt{16-2x}$ and its domain is the largest possible set of real values. 

(i) State the domain and range of $f$ . 

(ii) Sketch the graph of $y=f\left(x\right)$.
Clearly label the points where the graph intersects the axes.

On separate sets of axes, sketch the graphs of:

(i) $y=f\left(\left|x\right|\right)$

(ii) $y={\left[f\left(x\right)\right]}^2$.  

For each graph, define the domain and range and clearly label the points where the graph intersects the axes.

Consider the function $f\left(x\right)=x^2-4\left|x\right|-5,x\in \mathrm{ℝ}.$ 

Solve $f\left(x\right)=0.$

Sketch the graph of $f$.
Clearly indicate the intersections with the coordinate axes and any turning points.

Let $f\left(x\right)={\mathrm{e}}^x$, for $x\in \mathrm{ℝ}$.

On separate axes, sketch the graphs of

(i) $y=f\left(\left|x-1\right|\right)$

(ii) $y=\left|f\left(x\right)-1\right|$

(iii) $y=f\left(-\left|x\right|\right)$.

For each graph, clearly show the coordinates of any points where the graph meets the coordinate axes and of any local maximum or minimum points, and give the equations of any asymptotes. Give exact values.

The graph of $y=f\left(x\right)$ has two asymptotes, with equations $x=0.25$ and $y=0.5$, as shown in the following diagram. The graph passes through the points A(0, 6) and B(3, 0).

Sketch the graph of $y=\frac{1}{f\left(x\right)}$. Clearly indicate the points where the graph meets the axes or has a discontinuity, and give the equations of any asymptotes.

Consider the function $f$ defined by $f\left(x\right)=\ln \left(k-x\right)$, where $f$ has the largest possible domain and $k$ is a constant such that $k>1$.

Sketch the graph of $y=f\left(x\right)$, giving the equation of any asymptote and the coordinates of any points where the graph meets the axes in terms of $k$. State the domain and range of $f$.

The function $g$ is defined by $g\left(x\right)=f\left(\left|x\right|\right)$. The range of $g$ is $g\left(x\right)\le 1.$

(i) Find the exact value of $k$.

(ii) State the domain of $g$ .

(iii) Sketch the graph of $g$.

Given that $\left|g\left(x\right)\right|=p$ has exactly two distinct real solutions, find the possible values of $p$.

Find the set of values of $x$ which satisfy the inequality

$\left|2x^2-13x+15\right|>3x-15$

By considering graph transformations of an appropriate quadratic function, sketch the graph of $y={\left(x+4\right)}^2-2\left|x+4\right|-3$ . Clearly indicate any $x$-intercepts and any $y$-intercepts.

Hence, find the values of $k$ for which the equation ${\left(x+4\right)}^2-2\left|x+4\right|-3=k$ has exactly four distinct real solutions.

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

(i) Sketch the graph of $y=\left|f\left(x\right)\right|$.

(ii) Hence find the set of values of $x$ for which $f\left(x\right)<\left|f\left(x\right)\right|$

Solve the equation $f\left(x\right)=|x^2-4x+3|.$

Consider the function $f\left(x\right)=\frac{\left(x+3\right)\left(x-4\right)}{x-5},x\in \mathrm{ℝ},x\ne 5.$The graph of $f$  is shown below.

Find the equation of the oblique asymptote of the graph of $f$ .

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

Clearly indicate the points where the graph meets the coordinate axes and state the equations of the asymptotes.

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

Clearly indicate the points where the graph meets the coordinate axes and state the equations of the asymptotes.

Let $f$ be the function defined by $f\left(x\right)=(x-1)(x^2+x-26).$ 

Sketch the graph of $y=\left|f\left(x\right)\right|$.
Give the exact coordinates of the $x$-intercepts.

Find the set of values of $k$ for which the graph of $y=\left|f\left(x\right)\right|$ and the line $y=k$ have exactly four points of intersection.

The graph of the function $f\left(x\right)=a\left|x+p\right|+q$, where $a,p,q\in \mathrm{ℝ}$, is shown in the following diagram. The graph has a local maximum at the point A(3, 5) and meets the $y$-axis at the point $\left(0,-7\right)$.

(i) Find the values of $a$, $p$ and $q$.

(ii) Hence solve $f\left(x\right)=0$.

Find the solutions to the equation $f\left(x\right)=\left|7-2x\right|$.

A function $f$ is defined by $f\left(x\right)=x^2-8x+15,x\in \mathrm{ℝ}.$

Sketch the graph of $y=\frac{1}{f\left(x\right)}$.  
Clearly indicate any intercepts with the coordinate axes and state the equations of any asymptotes.
Find the coordinates of any turning points.

By sketching the graphs of  $y=f\left(x\right)$ and $y=\frac{1}{f\left(x\right)}$ on the same axes, find the values of $x$  for which  $f\left(x\right)\le \frac{1}{f\left(x\right)}$.

The graph of a function $f$ is shown in the following diagram. The equations of the asymptotes are $x=0$, $y=-1$ and $y=1$.

The point A(1, 2) lies on the graph.

On separate sets of axes, sketch the graphs defined below.

For each sketch, clearly label any asymptotes or discontinuities and give the coordinates of the image of A.

$y=\frac{1}{f\left(x\right)}+2$.

$y={\left[f\left(x\right)-2\right]}^2$.

$y=\frac{1}{{\left[f\left(x\right)\right]}^2}$

Consider the function defined by $f\left(x\right)=x\left|x\right|,x\in \mathrm{ℝ}.$

Sketch the graph of $f$.

By first sketching the graph of $f^{-1}$  on the same set of axes as the graph of $f$, solve $f\left(x\right)=f^{-1}\left(x\right).$

Given that $k$ is a real constant, find an expression for $f^{-1}\left(k\right)$ in the case when:

(i) $k\ge 0,$

(ii) $k<0$.

The graph of a function $f$ is shown below.

On separate sets of axes sketch the following graphs clearly showing any key points.

$y=\left|f\left(x\right)\right|-f\left(x\right).$

$y=f\left(x\right)\left|f\left(x\right)\right|$

$y=\frac{f\left(x\right)}{\left|f\left(x\right)\right|}$

Let $f\left(x\right)=x^4-8x^3-40x^2+224x-240$.

Find the values of $k$ for which the equation $\left|f\left(\left|x\right|\right)\right|=k$ has exactly six distinct real solutions.