Quadratic Functions & Graphs (DP IB Maths: AA HL)

Consider , for , where . 

The graph of f  has a local maximum at . 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 , find the possible values of k, which will give

Let  .

Find the coordinates of the vertex of

Let . 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 intersects the y-axis at -8 and has an x-intercept at . The function can be obtained by an appropriate shift of the graph .

Find the values of and .

Find the other x-intercept of f(x).

Determine the coordinates of the maximum value of .

A fence of length 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 per metre, while on the other three sides the cost is  per metre. The total cost of the fence is .

Calculate the maximum area of the paddock.

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

the total length 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, , of each base can be modelled by the function

where  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 .

Consider . The graph of has axis of symmetry and  y-intercept at .

Find the value of n.

Find the value of m.

Write n the form .

Let

Write down the coordinates of the y-intercept.

The function f can be written in the form.

Sketch the graph of , clearly labelling the vertex and any points where the graph intersects coordinate axes.

Let .

Write down the coordinates of the y-intercept.

The function f can be written in the form .

Sketch the graph of , clearly labelling the vertex and any points where the graph intersects coordinate axes.

Let, for , where . The graph of f intersects the x-axis at

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 .

Solve the equation .

The function f is a quadratic in the form

The graph of f  has x-intercepts and .

Find the values of a and b.

Sketch the graph of , clearly labelling the vertex and any points where the graph intersects the coordinate axes.

Let . The diagram below shows part of the graph of .

Another function is defined by 

Sketch the graph of  on the axes above.

The graph of and  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  intercepts the -axis at  and has an -intercept at  The function can be obtained by an appropriate shift of the graph .

Find the values of  and .

Find the other -intercept of .

Determine the coordinates of the maximum value of .

Consider  The graph of  has no real roots.

Show that and explain why  must be a positive value.

The minimum point on the graph of  is

Find the value of b and the value of c.

Sketch the graph of clearly labelling the minimum point and any points where the graph intersects coordinate axes.

Consider the function , for , where . 

The equation  has exactly one solution. 

Find the value of b.

Let  , for , where k is a constant. The line  does not intersect the graph of .

Find the possible values of k.

The graph of a quadratic function has equation , where  and the axis of symmetry is .

Draw the axis of symmetry on the grid above.

The graph of the quadratic function intersects the -axis at points  and B.

Find the values of  and .

Let , for  where .

Show that the discriminant of f is .

Find the values of k such that the graph of has two equal roots.

Let for  . The following diagram shows part of the graph of f. Let .

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

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

Find the area of ABCD.

Let g, for , 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

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

It is given that  and .

Find the values of  and .

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, .

Find the values of  when  and explain their significance in the context of the question.

Calculate

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

where  is the sale price of each litre sold, in dollars, at and  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  and .

Write down the function of , in the form , where  and  are constants.

Find the values of  when  and explain their significance in the context of the question.

Calculate

Let  and , where . The vertex of the graph of  is at and the vertex of the graph of is at , where . The graphs of and intersect at exactly one point. Find the value of .

Consider The graph of f has an axis of symmetry at and  y-intercept at  The line is a tangent to the graph of f.

Find the possible values of m.