IB Maths AI HL 2 Functions

Practise IB Maths AI HL functions through shared-core and HL graphs, regression, transformations, equations and technology-supported models in context.

[Maximum number: 5]

Line $L_{1}$ is tangent to the graph of a function f(x) at the point P(3,-1). Line $L_{2}$ is given by the equation $y=-\frac{1}{2} x-\frac{5}{2}$ and is perpendicular to $L_{1}$ .

(a)

Write down the gradient of $L_{1}$ .

[ 1 ]

(b)

Find the equation of $L_{1}$ in the form y=m x+c.

[ 2 ]

attempt to substitute their part (a) and point (3,-1) into the slope-intercept form or point-slope form of an equation
$-1=2 \times 3+c \quad$ OR $\quad y+1=2(x-3)$
y=2 x-7
Note: Equation must be in the form y=m x+c for A1 to be awarded.

(c)

Show that $L_{2}$ is not the line that is normal to f(x) at point P .

[ 2 ]

METHOD 1
attempt to show that P does not lie on $L_{2}$
e.g. $-\frac{1}{2}(3)-\frac{5}{2} \quad \mathbf{OR}$ graph showing $L_{2}$ and P in approximate correct locations
$-1 \neq-\frac{1}{2}(3)-\frac{5}{2}(-1 \neq-4) \mathbf{OR}(3,-1)$ does not lie on the graph of $L_{2}$
hence $L_{2}$ is not the normal line to f(x) at point P

METHOD 2
attempt to find the equation of the normal line at (3,-1)
$\left(-1=-\frac{1}{2}(3)+c \quad\right.$ OR $\left.\quad y+1=-\frac{1}{2}(x-3)\right)$
the normal line is $y=-\frac{1}{2} x+\frac{1}{2}$
hence $L_{2}$ is not the normal line to f(x) at point P

METHOD 3
attempt to find the intersection of $L_{1}$ and $L_{2}$

Intersection of y=2 x-7 and $y=-\frac{1}{2} x-\frac{5}{2}$ is (1.8,-3.4)
$x=1.8 \neq 3 \quad$ OR $\quad y=-3.4 \neq-1$
hence $L_{2}$ is not the normal line to f(x) at point P

Note: Accept equivalent written arguments provided values are seen.
Methods 1 and 2 are independent of the answers in (a) and (b) but FT marks can be given for Method 3.

[Maximum number: 7]

Walkea Tree is a furniture store. They offer a 30 % discount on all chairs priced over 250 Canadian dollars (CAD).
A different furniture store, LuxeCraft, offer customers a reduction of 150 CAD on all chairs priced over 250 CAD .

(a)

Given that x is the original price of a chair and x>250,

[ 3 ]

(i)

write down a function f(x) that models the final price of a chair at Walkea Tree, after the discount is applied.

[ 1 ]

f(x)=0.7 x OR $f(x)=x-0.3 x \quad$ A1

(ii)

write down a function g(x) that models the final price of a chair at LuxeCraft, after the reduction is applied.

A third store, Opulent Furnishings, decides to offer both a 30 % discount and a reduction of 150 CAD on all their chairs priced over 250 CAD .

[ 2 ]

$g(x)=x-150 \quad$ A1

(b)

Describe the meaning of $(f \circ g)(x)$ , in context.

[ 2 ]

attempt to discuss the order in which the reduction/discount are applied
$(f \circ g)(x)$ describes when the 150 CAD reduction is applied before the 30 % discount.

(c)

State, with justification, whether $(f \circ g)(x)$ or $(g \circ f)(x)$ is better for the customer when they buy a chair with an original price of 450 CAD .

[ 2 ]

EITHER
evidence of solving $(f \circ g)(x)$ and $(g \circ f)(x)$ for a price at 450 CAD

OR
$(f \circ g)(x)=0.7 x-105$ and $(g \circ f)(x)=0.7 x-150$ seen

OR
stating that it is better for 30 % to be deducted from a higher price

OR
the numbers 165, 210 are seen

THEN
$(g \circ f)(x)$ is better for the customer

Note: Do not award R0A1.

[Maximum number: 5]

The function f is defined by $f(x)=4 \ln (2 x-3)$ , where $x>\frac{3}{2}$ .
The graph of y=f(x) is obtained from the graph of $y=\ln x$ by a sequence of three transformations.

(a)

Describe fully these transformations, including the order in which they occur.

[ 3 ]

vertical stretch scale factor 4

Note: The vertical stretch can occur in any order.

EITHER
translation $\binom{3}{0}$ followed by horizontal stretch scale factor $\frac{1}{2}$

OR
horizontal stretch scale factor $\frac{1}{2}$ followed by translation $\binom{1.5}{0}$

Note: Award A1A0 in either approach, for correct transformations, but incorrect order.

(b)

The graph of y=g(x) is shown.
The graph passes through the points (-2,-2),(-1,0),(0,2),(1,3),(2,5) and (3,6).

Figure for Question (b) — IB Maths AI HL

Find $(f \circ g)(2)$ .

[ 2 ]

$\quad g(2)=5$ $f(5)=7.78 \quad(=7.78364 \ldots, 4 \ln 7)$

All question bank results loaded

Covered syllabus points

2 Functions question 1

Question (a)

Question (b)

Question (c)

2 Functions question 2

Question (a)

Question (i)

Question (ii)

Question (b)

Question (c)

2 Functions question 3

Question (a)

Question (b)