2.2 Quadratic Functions & Graphs

Write the quadratic function   in the form  where a, b and c are integers to be found.

Write down the minimum point on the graph of .

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

Solve the equation .

Find the coordinates of the turning point on the graph of .

Sketch the graph of , labelling the turning point and any points where the graph crosses the coordinate axes.

Find the minimum value of the function .

Hence, or otherwise, prove that the function  has no real roots.

The function has two distinct real roots.

Show that .

The equation has real roots.

Find the possible values of k.

The equation has no real roots. Show that .

The graph below shows the curve .

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 meters.

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

Is the bridge wide enough to span a river of width 11 m?

The diagram below shows the graph of , where  is a quadratic function.
The intercepts with the x-axis and the turning point have been labelled.

Write down the equation of the axis of symmetry for the graph of 

The function can be written in the form of 
Find the values of  and .

Solve the equation .

Solve .

Let , for  where .

Show that the discriminant of f is .

Find the values of  so that the function f(x) has two distinct roots.

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

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.