Functions & Graphs

7 methods · 25 questions

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Translations, reflections and stretches act on the equation by substitution: a shift by $a$ in $x$ replaces $x$ by $x - a$ , a reflection in the $y$ -axis replaces $x$ by $- x$ , a reflection in the $x$ -axis replaces $y$ by $- y$ , a horizontal stretch by factor $s$ replaces $x$ by $x / s$ , a vertical shift adds a constant. Apply the moves in the stated order to get the new equation, then read its vertex or features off. Crucially, a horizontal shift or a reflection neither creates nor destroys $x$ -intercepts (and a pure translation cannot change the leading coefficient), whereas a vertical shift or stretch can change the number of roots, so transformation claims can be accepted or rejected almost on sight.

Trigger: the question describes shifts, reflections or stretches of $y = f (x)$ , or asks whether a given curve is such a transform.

Instances:
(i) compose the substitutions in order and expand to obtain the resulting equation;
(ii) track only the image of the vertex or stationary point rather than the whole curve;
(iii) declare a horizontal shift or reflection root-count-preserving, and match coefficients knowing a translation leaves the leading coefficient fixed.

Linked questions (3)

TMUA 2016 Paper 2 · Q11

f (x)

is a polynomial with real coefficients.

The equation

f (x) = 0

has exactly two real roots,

x = - p

and

x = p

, where

p > 0

.

Consider the following three statements:
1.

f^{'} (x) = 0

for exactly one value of

x

between

- p

and

p

2. The area between the curve

y = f (x)

, the

x

-axis and the lines

x = - p

and

x = p

is given by

2 \int_{0}^{p} f (x) d x

3. The graph of

y = - f (- x)

intersects the

x

-axis at the points

x = - p

and

x = p

only

Which of the above statements must be true?

Anone
B1 only
C2 only
D3 only
E1 and 2 only
F1 and 3 only
G2 and 3 only
H1, 2 and 3

TMUA 2018 Paper 2 · Q15

It is given that

f (x) = x^{3} + 3 q x^{2} + 2

, where

q

is a real constant.

The equation

f (x) = 0

has 3 distinct real roots.

Which of the following statements is/are necessarily true?

I. The equation

f (x) + 1 = 0

has 3 distinct real roots.

II. The equation

f (x + 1) = 0

has 3 distinct real roots.

III. The equation

f (- x) - 1 = 0

has 3 distinct real roots.

Anone of them
BI only
CII only
DIII only
EI and II only
FI and III only
GII and III only
HI, II and III

TMUA 2019 Paper 2 · Q20

When the graph of the function

y = f (x)

, defined on the real numbers, is reflected in the

y

-axis and then translated by 2 units in the negative

x

-direction, the result is the graph of the function

y = g (x)

.

When the graph of the same function

y = f (x)

is translated by 2 units in the negative

x

-direction and then reflected in the

y

-axis, the result is the graph of the function

y = h (x)

.

Which one of the following conditions on

y = f (x)

is necessary and sufficient for the functions

g (x)

and

h (x)

to be identical?

A $f (x) = f (x + 2)$ for all $x$
B $f (x) = f (x + 4)$ for all $x$
C $f (x) = f (x + 8)$ for all $x$
D $f (x) = f (- x)$ for all $x$
E $f (x) = f (2 - x)$ for all $x$
F $f (x) = f (4 - x)$ for all $x$
G $f (x) = f (8 - x)$ for all $x$