G11 AI HL UNIT TEST REVISION-FUNCTIONS
1.The following diagram shows part of the graph of f with x-intercept (5, 0) and y-intercept (0, 8).
(a.i)Find the y-intercept of the graph of f(x) + 3. [1]
(a.ii)Find the y-intercept of the graph of f(4x). [2]
(b)Find the x-intercept of the graph of f(2x). [2]
(c)Describe the transformation f(x + 1). [2]
2.Let f(x) = xz - 4x - 5. The following diagram shows part of the graph off.
(b)Find the equation of the axis of symmetry of the graph of f. [2]
The function can be written in the form f(x) = (x - h)2 + k.
(c.i)Write down the value of h. [1]
(c.ii)Find the value of k. [3]
(d)The graph of a second function, g, is obtained by a reflection of the graph of f in the y-axis, followed by a translation of
Find the coordinates of the vertex of the graph of g. [5]
3.The graph of the function f is given in the following diagram.
(a)Write down f(2) . [1]
(b)On the axes, sketch y = f-1(x). [2]
The function g is defined as g(x) = 3x - 1.
(c)Find an expression for g-1(x) [2]
(d)Find a value of x where f-1(x) = g-1(x) . [2]
4.Three towns, A, B and C are represented as coordinates on a map, where the x and y axes represent the distances east and north of an origin, respectively, measured in kilometres.
Town A is located at (−6, −1) and town B is located at (8, 6). A road runs along the perpendicular bisector of [AB]. This information is shown in the following diagram.
(a)Find the equation of the line that the road follows. [5]
(b)Town C is due north of town A and the road passes through town C.
Find the y-coordinate of town C. [2]
5.Consider the following function: for x > 1.
(a) Find h-1(1) . [2]
(b)Find the domain of h-1(x). [2]
6.The strength of earthquakes is measured on the Richter magnitude scale, with values typically between 0 and 8 where 8 is the most severe.
The Gutenberg–Richter equation gives the average number of earthquakes per year, N, which have a magnitude of at least M. For a particular region the equation is log10 N = a − M, for some a ∈ R.
This region has an average of 100 earthquakes per year with a magnitude of at least 3.
(a)Find the value of a. [2]
The equation for this region can also be written as
(b)Find the value of b. [2]
(c)Given 0 < M < 8, find the range for N. [2]
The expected length of time, in years, between earthquakes with a magnitude of at least M is N/1. Within this region the most severe earthquake recorded had a magnitude of 7.2.
(d)Find the expected length of time between this earthquake and the next earthquake of at least this magnitude. Give your answer to the nearest year. [2]
7.A scientist is conducting an experiment on the growth of a certain species of bacteria.
The population of the bacteria, P, can be modelled by the function P(t) = 1200 × kt , t ≥ 0, where t is the number of hours since the experiment began, and k is a positive constant.
(a.i)Write down the value of P(0). [1]
(a.ii)Interpret what this value means in this context. [1]
3 hours after the experiment began, the population of the bacteria is 18 750.
(b)Find the value of k. [2]
(c)Find the population of the bacteria 1 hour and 30 minutes after the experiment began. [2]
The scientist conducts a second experiment with a different species of bacteria.
The population of this bacteria, S , can be modelled by the function S(t) = 5000 × 1.65t , t ≥ 0, where t is the number of hours since both experiments began.
(d)Find the value of t when the two populations of bacteria are equal. [2]
It takes 2 hours and m minutes for the number of bacteria in the second experiment to reach 19 000.
(e)Find the value of m, giving your answer as an integer value. [4]
8.A cell phone starts charging at 07: 00. While being charged, the percentage of power, P, in the phone is modelled by the function P = 100 一 60 × a-t , where t is the number of hours after 07: 00.
(a)Find the percentage of power in the phone at 07: 00. [2]
The percentage of power in the phone reaches 75 % at 08: 00.
(b)Find the value of a. [2]
(c)Draw the graph of P = 100 一 60 × a-t on the following set of axes.
[2]
(d)State a mathematical reason why the model predicts the percentage of power in the phone will never reach 100 %. [1]
9.Consider the function f(x) = ax2 + bx + c . The graph of y = f(x) is shown in the diagram. The vertex of the graph has coordinates (0.5, −12.5). The graph intersects the x-axis at two points, (−2, 0) and (p, 0).
(a)Find the value of p. [1]
(b)Find the value of
(i) a.
(ii) b.
(iii) c . [5]
(c)
Write down the equation of the axis of symmetry of the graph. [1]
10.The pH scale is a measure of the acidity of a solution. Its value is given by the formula
pH = −log10 [H+], where [H+] is the concentration of hydrogen ions in the solution (measured in moles per litre).
(a)Calculate the pH value if the concentration of hydrogen ions is 0.0003. [2]
The pH of milk is 6.6.
(b)Calculate the concentration of hydrogen ions in milk. [2]
The strength of an acid is measured by its concentration of hydrogen ions.
A lemon has a pH value of 2 and a tomato has a pH value of 4.5.
(c)Calculate how many times stronger the acid in a lemon is when compared to the acid in a tomato. [3]