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

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.

For the equation $6k{x}^2+4kx+2=0$, 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 graph of f and g intersect at points A and B.

Find the coordinates of A and B.

Find the exact length of the line AB.

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(x).

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

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.

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, $\mathrm{A}$, of each base can be modelled by the function

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

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 volume of $15400{\mathrm{cm}}^3$.

(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.

(iii) State the mathematical shape of the carboard box when its volume is a maximum.

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

Find the value of n.

Find the value of m.

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

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  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 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 $f\left(x\right)=a(x-p)(x-q)$.

(i) Find the values 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 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.

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.

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.

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 $f\left(x\right)$ .

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

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

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

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 coordinate axes.

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

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

Find the value of b.

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.

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.

(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.

Let $f\left(x\right)=2k{x}^2-(k+10)x+\frac{5}{2}+\frac{5}{8}k,$, 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 graph of has two equal roots.

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$\left(x\right)=(x-3{)}^2+k,$, 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.

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 cars less (or more) and vice versa.

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

Find the values of $x$ when $\mathrm{P}\left(x\right)=0$ and explain their significance in the context of the question.

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.

A company sells $L$ litres of water per month and their total monthly profit, $P$, can be modelled by the function

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

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

In the context of the question, explain the significance of the 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 $\mathrm{P}\left(x\right)=0$ and explain their significance in the context of the question.

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.

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

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

Find the possible values of m.