# IBDP Maths analysis and approaches Topic: SL 2.4 :Investigation of key features of graphs, such as maximum HL Paper 1

### Question:

Consider the function f defined by f (x) = ln (x2 – 16) for x > 4 .
The following diagram shows part of the graph of f which crosses the x-axis at point A, with
coordinates ( a , 0 ). The line L is the tangent to the graph of f at the point B.

(a) Find the exact value of a . [3]

Ans: When f(x)=0, we have
ln(x2−16)=0

x2−16=1

x=±√17
However, since x>4, we have x=√17

(b) Given that the gradient of L is $$\frac{1}{3}$$ , find the x-coordinate of B. [6]

Ans: Differentiating f(x) with respect to x, we have f′(x)= $$\frac{2x}{x^2-16}$$
At B, f′(x)=$$\frac{1}{3}$$ , i.e., we have
x2−16=6x

x2−6x−16=0

(x+2)(x−8)=0.
Thus, x=−2 (reject) or x=8.

### Question

The graph of $$y = \frac{{a + x}}{{b + cx}}$$ is drawn below.

(a)     Find the value of a, the value of b and the value of c.

Ans: an attempt to use either asymptotes or intercepts     (M1)

$$a = – 2,{\text{ }}b = 1,{\text{ }}c = \frac{1}{2}$$     A1A1A1

(b)     Using the values of a, b and c found in part (a), sketch the graph of $$y = \left| {\frac{{b + cx}}{{a + x}}} \right|$$ on the axes below, showing clearly all intercepts and asymptotes.

Ans:

A4

Note: Award A1 for both asymptotes,

A1 for both intercepts,

A1, A1 for the shape of each branch, ignoring shape at $$(x = – 2)$$.

[8 marks]

### Question

Let $$f(x) = {x^3} + a{x^2} + bx + c$$ , where a , b , $$c \in \mathbb{Z}$$ . The diagram shows the graph of y = f(x) .

a.Using the information shown in the diagram, find the values of a , b and c .[4]

Ans: METHOD 1

f(x) = (+ 1)(− 1)(− 2)     M1

$$= {x^3} – 2{x^2} – x + 2$$     A1A1A1

a = −2 , b = −1 and c = 2

METHOD 2

from the graph or using f(0) = 2

c = 2     A1

setting up linear equations using f(1) = 0 and f(–1) = 0 (or f(2) = 0)     M1

obtain a = −2 , b = −1     A1A1

[4 marks]

b. If g(x) = 3f(x − 2) ,

(i)     state the coordinates of the points where the graph of g intercepts the x-axis.

(ii)     Find the y-intercept of the graph of .[3]

Ans:

(i)     (1, 0) , (3, 0) and (4, 0)     A1

(ii)     g(0) occurs at 3f(−2)     (M1)

= −36     A1

[3 marks]

### Question

The diagram below shows a sketch of the graph of $$y = f(x)$$.

a. Sketch the graph of $$y = {f^{ – 1}}(x)$$ on the same axes.[2]

Ans:

shape with y-axis intercept (0, 4)     A1

Note:     Accept curve with an asymptote at $$x = 1$$ suggested.

correct asymptote $$y = 1$$     A1

[2 marks]

b. State the range of $${f^{ – 1}}$$.[1]

Ans: range is $${f^{ – 1}}(x) > 1{\text{ (or }}\left] {1,{\text{ }}\infty } \right[)$$     A1

Note:     Also accept $$\left] {1,{\text{ 10}}} \right]$$ or $$\left] {1,{\text{ 10}}} \right[$$.

Note:     Do not allow follow through from incorrect asymptote in (a).

[1 mark]

c. Given that $$f(x) = \ln (ax + b),{\text{ }}x > 1$$, find the value of $$a$$ and the value of $$b$$.[4]
Ans: $$(4,{\text{ }}0) \Rightarrow \ln (4a + b) = 0$$     M1
$$\Rightarrow 4a + b = 1$$     A1
asymptote at $$x = 1 \Rightarrow a + b = 0$$     M1
$$\Rightarrow a = \frac{1}{3},{\text{ }}b = – \frac{1}{3}$$     A1