Further Functions & Graphs (DP IB Applications & Interpretation (AI): HL): Exam Questions

Let $f\left(x\right)=x^2-3x+2$. The diagram below shows part of the graph of $f$.

Another function is defined by $g\left(x\right)=2-x.$

Sketch the graph of $g$ on the axes above.

The graph of $f$ and $g$ intersect at points A and B.

Find the coordinates of A and B and label them on the diagram above.

Find the length of the line AB.

Let $f\left(x\right)=\frac{7}{2(x-7)}-5$, for $x\ne 7$.

For the graph of $f$, find the:

(i) $x$-intercept

(ii) $y$-intercept.

For the graph of $f$, write down the equation of any asymptotes.

Let $g\left(x\right)=2(1-2x),$for $x\in \mathrm{ℝ}.$ The graphs of $f$ and $g$ intersect at points $\mathrm{P}$ and $\mathrm{Q}$.

Write down the coordinates of P and Q.

Find the distance of $\mathrm{PQ}$.

The perimeter, P, and area, A, of a given square can be expressed by $\mathrm{P}=4x$ and $\mathrm{A}=x^2$ respectively, where  is the length of the side of the square.

Write down an expression for:

(i) P in terms of A, $\mathrm{P}\left(\mathrm{A}\right)$

(ii) A in terms of P, $\mathrm{A}\left(\mathrm{P}\right).$

${\mathrm{P}}^{-1}\left(40\right)=\mathrm{A}\left(k\right).$

Find the value of $k$ and $\mathrm{A}\left(k\right)$.

Consider the function $f\left(x\right)=a(0.75{)}^x+b$ where $a$ and $b$ are constants. The graph of $f$ passes through the points $(0,18)$ and   $(2,11)$  and is shown below.

Write down two equations relating $a$and $b$.

Find the value of $a$and $b$.

Write down the equation of the horizontal asymptote of the graph of $f$.

The function $f\left(x\right)=a{x}^2+bx+c$ intercepts the $y$-axis at $-12$ and has an $x$-intercept at $x=3.$ The function can be obtained by an appropriate shift of the graph $y=-4x^2$.

Find the values of $a,b$ and $c$.

Find the other $x$-intercept of $f\left(x\right)$.

Determine the coordinates of the maximum value of $f\left(x\right)$ .

A function is defined by $f\left(x\right)=\frac{1}{(x-3{)}^2}+2,x\ne p.$

Find the value of $p.$

For the graph of $f$ write down the equation of any asymptotes.

Find the range of $f$.

The line  intersects the graph of $f$ when $x=1$ and when $x=4.$

Find the equation of $l$. Give your answer in the form $ax+by+d=0$, where $a,b$ and $d$ are integers.

A function is defined by $f\left(x\right)=4-\frac{12}{(5x+9)},x\ne a.$

Find the value of $a$. Give your answer as a fraction.

Find the range of $f$.

Find the value of $f^{-1}\left(2\right)$. Give your answer as a fraction.

The diagram below shows the graph of the function $f\left(x\right)=8\cos \left(48x\right)-3$, for $-5\le x\le 5$.

For the graph of $f$, find the:

(i) $x$-intercepts

(ii) $y$-intercept.

Write down the range of $f$.

Three lines are drawn connecting the two local minimum points and the local maximum point forming a triangle. Calculate the area of the triangle.

Let  $f\left(x\right)=2-5\cos \left(30\left(3x-1\right)\right),$ for $x>0.$

The $n$th maximum point on the graph of $f$ has $x$ coordinate $x_n$, where $n\in {\mathrm{\mathbb{Z}}}^{+}$.

Given that $x_n=x_1+\left(n-1\right)d$, find $x_1$ and $d$.

Using sigma notation, write down an expression for $x_1+x_2+x_3+\cdots x_8$.

Find the value of the sum from part (b) (i). Give your answer to 2 decimal places.

The average fat-free mass, M, in kg, of footballers as a function of their age, $a$, in years, can be given by the logarithmic function:

$\mathrm{M}\left(a\right)=10\log \left(a-15\right)+50,16\le a\le 25.$

Calculate the average fat free mass of players aged:

(i) 16 years

(ii) 25 years.

Find an expression for a linear model using your answers to part (a) (i) and (ii).

Calculate the percentage error from using the linear model found in part (b) to approximate the average fat free mass of a player aged 20 years old.

The axes below shows the graph of the piecewise function, $f$

The gradient of the graph of $f$ is 0 for $a\le x<b$.

Find the values of $a$ and $b$.

Complete the following piecewise function for $f$

$$\begin{gathered} f\left(x\right)=\{0\le x<2 \\ 2\le x<3 \\ 3\le x\le 4 \end{gathered}$$

A function is defined by $f\left(x\right)=e^{x^2+bx+4}$. The graph of  has an axis of symmetry of $x=2.$

Find the value of $b$.

Find the range of $f$ .

Another function is defined by $g\left(x\right)=-\frac{\left(x^2-25\right)}{5}$. The graph of $f$ and $g$ intersect at points A and B.

Find the equation of the line passing through points A and B. Give your answer in the form $y=mx+c.$

Find the distance of the line AB.

Consider the function $f\left(x\right)=5-\log (6-4x).$ The line $l_1$ intersects the graph of $f$ at point A$(-1,y)$ and B$(x,5)$.

Find the value of $x$ and $y$.

Find the equation of $l_1$. Give your answer in the form $y=mx+c$, where $m$ and $c$ are fractions.

The line $l_2$ is the perpendicular bisector to [AB].

Find the equation of $l_2$. Give your answer in the form $ax+by+d=0$, where $a,b$ and $d$ are integers.

A rectangle has length $l=4x$ and width $w=x$.

Find an expression for

(i) the perimeter of the rectangle, P, in terms of $x$.

(ii) the area of the rectangle, A, in terms of $x$.

Show that $P\left(\mathrm{A}\right)=5\sqrt{\mathrm{A}}$.

The graph of the function P, for $0\le A\le 4$, is shown below.

On the grid above, draw the graph of the inverse function $P^{-1}$.

The function $f$ is a quadratic in the form $f\left(x\right)=a{x}^2+bx-2$, for $-10\le x\le 10.$

The graph of $f$ has $x$-intercepts $\left(\frac{1+\sqrt{5}}{2},0\right)$ and $\left(\frac{1-\sqrt{5}}{2},0\right).$

Find the values of $a$ and $b.$

Another function can be defined by $g\left(x\right)=6{\left(0.8\right)}^{-x}-1$, for $-10\le x\le 10$.

The graph of $f$ and $g$ intersect at points $\mathrm{A}$ and $\mathrm{B}$.

Find the coordinates of $\mathrm{A}$ and $\mathrm{B}$.

Solve the inequality $f\left(x\right)<g\left(x\right)$.

Let $f\left(x\right)=e^{2cos\left(30x\right)}+1,$ for $x>0.$

The $n$th minimum point on the graph of $f$ has $x$ coordinate $x_n$, where $n\in {\mathrm{\mathbb{Z}}}^{+}$.

Given that $x_n=x_1+(n-1)d$, find $x_1\mathrm{and}d$.

(i) Using sigma notation, write down an expression for $x_1+x_2+x_3+\cdots +x_{16}.$

(ii) Find the value of the sum from part (b) (i). Give your answer to the nearest integer.

Let $f\left(x\right)=2x^4-2x^3-4x^2+x+1$ where $x\in \mathrm{ℝ}$.

Solve the inequality $f\left(x\right)<0$.

For the graph of $f$, find the coordinates of the

(i) local maximum point.

(ii) local minimum points.

Write down the possible domains of $f$ for which $f$ has an inverse and explain why the domain must be restricted.

The graph of a quadratic function has equation $y=\frac{1}{4}{x}^2+bx+c$, where $b,c\in \mathrm{\mathbb{Z}},$ and the axis of symmetry is $x=-4$.

Draw the axis of symmetry on the grid above.

The graph of the quadratic function intersects the $x$-axis at points $\mathrm{A}(-6,0)$ and B.

Write down the coordinates of B.

Find the values of $b$ and $c$.

(i) Mark and label A and B on the grid above.

(ii) Write down the coordinates of the Vertex, V, and label it on the grid above.

(iii) Write down the coordinates of the $y$-intercept, C, and label it on the grid above.

(iv) Draw the graph of the quadratic function on the grid above.

The diagram below shows part of the graph of the function $f\left(x\right)=5\cos (12x-30)+6$, where points A, B, C and D represent stationary points. The points A, B, C and D are joined together forming a quadrilateral ABDC.

Write down the coordinates for

(i) $\mathrm{A}$.

(ii) $\mathrm{B}$.

(iii) $\mathrm{C}.$

(iv) $\mathrm{D}.$

Find the area of the quadrilateral ABDC.

The intensity of light, I, is assumed to be 100% at the surface of the ocean and decreases with depth, $d$, and can be estimated by the function

$I\left(d\right)=k{(1.08)}^{-d}$

where $I$ is expressed as a percentage, $d$ is the depth below the surface, in metres, and $k$ is a constant.

Calculate the value of $k$.

State the domain and range of $I$.

On a particularly bright day the intensity of light 6.2 m below the surface is 75%.

Calculate the percentage error between the estimate made by the function and the actual intensity of light.

Graph the following piecewise function on the grid below

$$\begin{gathered} f\left(x\right)=\{-x+5,-2\le x<2 \\ -4x+11,2\le x<3 \\ 2x-7,3\le x\le 6 \end{gathered}$$

Find the values of $x$ when $f\left(x\right)=0.$