Quadratic Functions & Graphs (DP IB Analysis & Approaches (AA): HL): Exam Questions

Write the quadratic function $y=x^2+8x-9$ in the form $y=a{\left(x+b\right)}^2+c$, where $a$, $b$ and $c$ are integers to be found.

Write down the minimum point on the graph of $y=x^2+8x-9$.

Sketch the graph of $y=x^2+8x-9$, clearly labelling the minimum point and any point where the graph intersects the coordinate axes.

Solve the equation $2x^2+x-6=0$.

Find the coordinates of the turning point on the graph of $y=2x^2+x-6$.

Sketch the graph of $y=2x^2+x-6$, labelling the turning point and any points where the graph crosses the coordinate axes.

Find the minimum value of the function $\mathrm{f}\left(x\right)=x^2+4x+5$.

Hence, or otherwise, show that the function $f\left(x\right)=x^2+4x+5$ has no real roots.

The function $f\left(x\right)=k{x}^2+2kx-3$ has two real, distinct roots.

Show that $4k^2+12k>0$.

Hence show that $k<-3$ or $k>0$.

The equation $2x^2-4x+3-2k=0$ has real roots.

Show that $16k-8\ge 0$.

Hence find the possible values of $k$.

The equation $x^2+px+q=0$ has no real roots.

Show that $p^2<4q$.

Given that $q=4$, find the set of possible values of $p$.

The graph below shows the curve $f\left(x\right)=4-\frac{x^2}{8}$.

The curve is to be used as the model for the arch on a bridge, where the water level under the bridge is represented by the $x$-axis. All measurements are in metres.

Write down the maximum height of the bridge above the water.

Determine whether the bridge is wide enough to span a river of width 11 m. Justify your answer.

The diagram below shows the graph of $y=f\left(x\right)$, where $f\left(x\right)$ is a quadratic function. The intercepts with the $x$-axis and the vertex have been labelled.

Write down the equation of the axis of symmetry for the graph of $y=f\left(x\right)$.

The function $f$ can be written in the form $f\left(x\right)=a{\left(x-h\right)}^2+k$.

Find the values of $a$, $h$ and $k$.

Solve the equation $m^2-13m+36=0$.

Hence, or otherwise, solve the equation $x^4-13x^2+36=0$.

Let $f\left(x\right)=2p{x}^2+\left(2p-5\right)x+p-\frac{5}{2}$, for $x\in \mathrm{ℝ}$, where $p\in \mathrm{\mathbb{Q}}$ and $p\ne 0$.

Show that the discriminant of $f$ is $-4p^2+25$.

Find the values of $p$ for which $f\left(x\right)$ has two real, distinct roots.

A factory produces cardboard boxes in the shape of a cuboid, with a fixed height of 25 cm and a base of varying area. The area, $A$, of each base can be modelled by the function

$A\left(x\right)=x\left(50-x\right),10\le x\le 40$,

where $x$ is the width of the base of the cardboard box in centimetres.

Cardboard box M has a width of 12 cm.

Find the volume of cardboard box M.

Find the possible dimensions of a cardboard box with a volume of $15 400{\text{ cm}}^3$.

(i) Find the value of $x$ that makes the volume of the cardboard box a maximum.

(ii) Write down the maximum volume of the cardboard box.

Consider $f\left(x\right)=m\left(x-n\right)\left(x-2\right)$. The graph of $y=f\left(x\right)$ has axis of symmetry $x=4$ and $y$-intercept at $\left(0,-6\right)$.

Find the value of $n$.

Find the value of m.

Write $f\left(x\right)$ in the form $f\left(x\right)=a{x}^2+bx+c$.

For the equation $6k{x}^2+4kx+2=0$, where $k\ne 0$, find the possible values of $k$ which will give

(i) two distinct real roots

(ii) two equal real roots

(iii) no real roots.

Let $f\left(x\right)=3x^2-3x+2$.

Find the coordinates of the vertex of $f$.

Let $g\left(x\right)=3x+2$. The graphs of $f$ and $g$ intersect at the points A and B.

Find the coordinates of A and B.

Find the exact value of AB.

Let $f\left(x\right)=2x^2+16x+29.$

Write down the coordinates of the $y$-intercept.

The function $f$ can be written in the form $f\left(x\right)=a{\left(x-h\right)}^2+k$.

(i) Find the values of $a$, $h$ and $k$.

(ii) Hence write down the coordinates of the vertex and state whether it is a maximum or minimum point.

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

Let $f\left(x\right)=-x^2+7x-10$.

Write down the coordinates of the y-intercept.

The function $f$ can be written in the form $f\left(x\right)=a\left(x-p\right)\left(x-q\right)$.

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

(ii) Hence write down the coordinates of the $x$-intercepts.

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

Let $f\left(x\right)=2x^2-12x+c$, for $x\in \mathrm{ℝ}$, where $c\in \mathrm{\mathbb{Z}}$. The graph of $f$ intersects the $x$-axis at $x=6$.

Find the equation of the axis of symmetry of the graph of $f$.

Find the coordinates of the other point where the graph of f intersects the x-axis.

Find the value of c.

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 graphs of $f$ and $g$ intersect at the points A and B.

Find the coordinates of A and B.

Find AB.

Let $f\left(x\right)=2k{x}^2-\left(k+10\right)x+\frac{5}{2}+\frac{5}{8}k$, for $x\in \mathrm{ℝ}$, where $k\in \mathrm{\mathbb{Z}}$.

Show that the discriminant of $f$ is $100-4k^2$.

Find the values of $k$ such that the equation $f\left(x\right)=0$ has two equal roots.

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 the points $\mathrm{A}\left(-6,0\right)$ and B.

(i) Write down the coordinates of B.

(ii) 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.

A company sells water, and its total monthly profit, $P$, can be modelled by the function

$P\left(x\right)=\left(x-0.45\right)\times N\left(x\right)$,

where $x$ is the sale price of each litre sold, in dollars, and $N$ is the linear function for the number of litres the company can sell per month at each given sale price.

Interpret, in context, the value 0.45.

It is given that $N(0.5)=400$ and $N(1.25)=250$.

Write down the function of $N$, in the form $N\left(x\right)=mx+c$, where $m$ and $c$ are constants.

Find the values of $x$ when $P\left(x\right)=0$ and interpret, in context, these values.

Calculate

(i) the maximum monthly profit.

(ii) the sale price needed to generate the maximum monthly profit.

(iii) the number of litres sold to generate the maximum monthly profit.

Consider $f\left(x\right)=-2x^2+bx+c$, for $x\in \mathrm{ℝ}$, where $b,c\in \mathrm{\mathbb{Z}}$. 

The graph of f  has a local maximum at $x=6$. The distance between the two x-intercepts of the graph of  f   is 10 units.

Find the coordinates of the two x-intercepts.

Find the value of b and the value of c.

Find the coordinates of the local maximum.

The function $f\left(x\right)=a{x}^2+bx+c$ intersects the $y$-axis at $-8$ and has an $x$-intercept at $x=-4$. The function can be obtained by an appropriate shift of the graph $y=-2x^2$.

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

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

Find the coordinates of the maximum point on the graph of $f$.

A fence of length $L$ is made to go around the perimeter of a rectangular paddock that borders a straight river. The cost of the fence along the river is $15 per metre, while on the other three sides the cost is $10 per metre. The total cost of the fence is $2000.

Calculate the maximum area of the paddock.

Using the value for the area from part (a), calculate

(i) the side lengths

(ii) the total length $L$ of the fence.

Solve the equation $4\sqrt{x}=21-x$.

Solve the equation $13x^2=x^4+36$.

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

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.

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

The function $f\left(x\right)=a{x}^2+bx+c$ intersects 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 $f\left(x\right)$.

Find the coordinates of the maximum point on the graph of $f$.

Solve $x^{\frac{2}{5}}+x^{\frac{1}{5}}=6$.

Consider $f\left(x\right)=x^2+bx+c$. The equation $f\left(x\right)=0$ has no real roots.

Show that $b^2<4c$ and explain why $c$ must be positive.

The minimum point on the graph of $y=f\left(x\right)$ is $\left(3,2\right)$.

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

Sketch the graph of $y=f\left(x\right)$, clearly labelling the minimum point and any points where the graph intersects the coordinate axes.

Let $f\left(x\right)=2k+\frac{9}{x}$, for $x\ne 0$, where $k$ is a constant. The line $y=k-x$ does not intersect the graph of $y=f\left(x\right)$.

Find the possible values of $k$.

Let $f\left(x\right)=9-x^2$, for $x\in \mathrm{ℝ}$. The following diagram shows part of the graph of $f$. Let $\mathrm{OD}=d$.

Find an expression in terms of $d$ for the area of the rectangle ABCD.

The coordinates of A are (-2, 0).

Find the area of ABCD.

Let $g\left(x\right)={\left(x-3\right)}^2+k$, for $x\in \mathrm{ℝ}$, where $k$ is a constant.

Given that the graphs of $f$ and $g$ intersect exactly once, find the value of $k$.

Consider the function $f\left(x\right)={\log }_b\left(6x-x^2\right)$, for $0<x<6$, where $b>0$.

The equation $f\left(x\right)=2$ has only one solution.

Find the value of $b$.

A tunnel is being constructed and its opening can be modelled by the quadratic function

$h\left(x\right)=ax(b-x),x\ge 0,$

where $h$ is the height of the tunnel, in metres, and $x$ is the width of the tunnel, in metres.

It is given that $h\left(10\right)=10$ and $h\left(20\right)=15$.

Find the values of $a$ and $b$.

The height required for a lane of traffic is 5 m and each lane requires a width of 2.8 m.

Find the number of lanes of traffic the tunnel can fit.

A company sells 55 cars per month for a sale price of $2000, whilst incurring costs for supplies, production and delivery of $890 per car. Reliable market research shows that for each increase (or decrease) of the sale price by $50 the company will sell 5 fewer (or 5 more) cars.

Find an expression for the total monthly profit, $P$, in terms of the sale price, $x$.

Find the values of $x$ when $P\left(x\right)=0$ and interpret, in context, these values.

Calculate

(i) the maximum monthly profit, giving your answer to the nearest dollar.

(ii) the sale price needed to generate the maximum monthly profit.

(iii) the number of cars sold to generate the maximum monthly profit.

Let $f\left(x\right)=-{\left(x-a\right)}^2+b$ and $g\left(x\right)=4{\left(x-c\right)}^2+d$, where $a,b,c,d\in \mathrm{ℝ}$. The vertex of the graph of $f$ is at $\left(2k,8k^2\right)$ and the vertex of the graph of $g$ is at $\left(k,2k\right)$, where $0<k<1$. The graphs of $f$ and $g$ intersect at exactly one point.

Find the value of $k$.

Consider $f\left(x\right)=r\left(x-s\right)\left(x+3\right)$. The graph of $f$ has an axis of symmetry at $x=-1$ and $y$-intercept at $\left(0,-6\right)$. The line $y=mx-14$ is a tangent to the graph of $f$.

Find the possible values of $m$.