IB Math Studies Review Ch. 17: Quadratics
1. The graph of the quadratic function f (x) = ax2 + bx + c intersects the y-axis at the point A(0, 5) and has its vertex at the point B(4, 13).
(a) Write down the value of c. [1]
(b) By using the coordinates of the vertex, B, or otherwise, write down two equations in a and b.
[3]
(c) Find the values of a and b. [2]
2. The surface of a red carpet is shown below. The dimensions of the carpet are in meters.
(a) Writer down an expression for the area, A, in m2, of the carpet. [1]
The area of the carpet is 10 m2.
(b) Calculate the value of x. [3]
(c) Hence, write down the value of the length and of the width of the carpet in meters. [2]
3. The front view of the edge of a water tank is drawn on a set of axes shown below. The edge is modelled by y = ax2 + c.
Point P has coordinates (-3, 1.8), point O has coordinates (0, 0) and point Q has coordinates (3, 1.8).
(a) Write down the value of c. [1]
(b) Find the value of a. [2]
(c) Hence, write down the equation of the quadratic function which models the edge of the water tank. [1]
4. A building company has many rectangular construction sites, of varying widths, along a road. The area, A, of each site is given by the function Ax=x(200-x) where x is the width of the side in meters and 20<x<80.
(a) Site S has a width of 20m. Write down the area of S. [1]
(b) Site T has the same area as site S, but a different width. Find the width of T. [2]
When the width of the construction site is b meters, the site has a maximum area.
(c) (i) Write down the value of b. [2]
(ii) Write down the maximum area.
The range of A(x) is m≤A(x)≤n.
(d) Hence write down the value of m and of n. [1]
5. The profit (P) in Swiss Francs made by three students selling homemade lemonade is modeled by the function
P = – + 5x – 30
where x is the number of glasses of lemonade sold.
(a) Copy and complete the table below
x / 0 / 10 / 20 / 30 / 40 / 50 / 60 / 70 / 80 / 90P / 15 / 90 / 75 / 50
(3)
(b) On graph paper draw axes for x and P, placing x on the horizontal axis and P on the vertical axis. Use suitable scales. Draw the graph of P against x by plotting the points. Label your graph. (attach the graph)
(5)
(c) Use your graph to find
(i) the maximum possible profit;
(1)
(ii) the number of glasses that need to be sold to make the maximum profit;
(1)
(iii) the number of glasses that need to be sold to make a profit of 80 Swiss Francs;
(2)
(iv) the amount of money initially invested by the three students.
(1)
(d) The three students Baljeet, Jane and Fiona share the profits in the ratio of 1:2:3 respectively. If they sold 40 glasses of lemonade, calculate Fiona’s share of the profits.
(2)
(Total 15 marks)