Grade 10 Maths · Functions

Shift or reflect a graph

Describe in words how a graph has moved, flipped or stretched to make a new graph, or write the new graph's equation.

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Grade 12 Functions Student Question Find equation of p(x) if f(x) is shifted 2 units left | NTENTE \ New Technological Education

Functions Grade 11 & 12 | Part 5: Reflections & TranslationsMaths With Mudinda

Transforming Algebraic Functions: Shifting, Stretching, and ReflectingProfessor Dave Explains

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Question 5.5 · November 2024 · Paper 1 · National (DBE) · 2 marks

Sketched below is the graph of

f (x) = \frac{a}{x + p} + q

having the domain

(- \infty; 1) \cup (1; \infty)

. The graph of

f

cuts the y-axis at

(0; 1)

. A line of symmetry of

f

is given by

g (x) = x - 3

5.5 Graph

f

undergoes a transformation to

h

where: The domain and range of

h

are the same as that of

f

h^{'} (x)

, the derivative of

h

, is negative on its domain. Describe a possible transformation that

f

could have undergone to result in

h

.(2)

Work it out on paper first. Then mark yourself with the official memo, one tick at a time.

More past paper questions: shift or reflect a graph

Work each one out first, then open the memo and check every tick.

3. Question 8.5 · March 2024 · Paper 1 · Provincial · 2 marks
In the diagram below, the graphs of $f (x) = cos x + m$ and $g (x) = sin (x + n)$ are drawn on the same set of axes for $x \in [- 24 0^{\circ}; 24 0^{\circ}]$ . $A$ is an $x$ -intercept of $f$ and has coordinates $(6 0^{\circ}; 0)$ . $B$ is a turning point of $g$ and has coordinates $(6 0^{\circ}; 1)$ .
8.5 Describe the transformations that the graph of $g$ has to undergo to form the graph of $p$ , where $p (x) = - cos x$ .(2)
Show the memo (1 tick)Memo
1. ✓
  Identify shift and reflectionShift $6 0^{\circ}$ to the left.
  Reflection in the $x$ -axis.
  2 marks · Shift or reflect a graph
4. Question 6.4 · November 2024 · Paper 1 · National (DBE) · 5 marks
In the diagram below, the graphs of $f (x) = - x^{2} + 4 x + 5$ and $g$ , a straight line, are drawn. $C (3; 8)$ is a point of intersection of $f$ and $g$ . $E H$ is drawn parallel to the y-axis, with $E$ a point on $f$ and $H$ a point on $g$ .
6.4 Given: $k (x) = f (x + m) = - x^{2} - 2 m x - m^{2} + 4 x + 4 m + 5$ . Determine the value of $m$ such that $g$ is a tangent to $k$ .(5)
Show the memo (3 ticks)Memo
1. ✓
  METHOD 1: Using Derivative $k^{'} (x) = - 2 x - 2 m + 4$
  1 mark · Differentiate using the power rule
2. ✓
  Equate derivative to gradient of g $k^{'} (x) = 2$
  $- 2 x - 2 m + 4 = 2$
  $- 2 x = 2 m - 2$
  $x = 1 - m$
  1 mark · Find equation of a tangent
3. ✓
  Substitute and solve $k (1 - m) = g (1 - m)$
  $- (1 - m)^{2} - 2 m (1 - m) - m^{2} + 4 (1 - m) + 4 m + 5 = 2 (1 - m) + 2$
  Simplify: $- 2 m - 4 = 0$
  $m = - 2$
  3 marks · Find equation of a tangent
5. Question 5.4 · November 2025 · Paper 1 · Provincial · 3 marks
The graphs of $f (x) = \frac{a}{x + p} + q$ and $g (x) = \frac{x}{3} + c$ are drawn below. C $(3; - 1)$ is the point of intersection between the asymptotes of $f$ . A and B are $x$ - and $y$ -intercepts of $f$ and $g$ respectively. E $(4; 2)$ is a point on $f$ in the first quadrant. $Diagram for this question$
5.4 Write down the domain of $h$ , if $h (x) = f [- (x - 4)]$ (3)
Show the memo (3 ticks)Memo
1. ✓
  Replacing $x$ by $- (x - 4)$ Vertical asymptote of $f$ : $x - 3 = 0$ , so for $h$ :
  $- (x - 4) - 3 = 0$
  1 mark · Shift or reflect a graph
2. ✓
  $x - 1 = 0$ / $x = 1$ $- x + 4 - 3 = 0$
  $- x + 1 = 0$
  $x - 1 = 0$
  V.A.: $x = 1$
  1 mark · State the asymptotes
3. ✓
  AnswerDomain: $x \in R$ ; $x \neq = 1$
  1 mark · State the domain and range
6. Question 5.3 · March 2023 · Provincial · 3 marks
The graph of $f (x) = \frac{2}{x + p} + q$ is sketched.
5.3 If $h (x) = - 2 [g (x)]$ , determine the equation of the inverse of $h$ , $h^{- 1}$ , in the form $h^{- 1} (x) = m x + c$ .(3)
Show the memo (3 ticks)Memo
1. ✓
  Determine equation for h(x) $g (x) = - x - 4$
  $h (x) = - 2 (- x - 4) = 2 x + 8$
  1 mark · Shift or reflect a graph
2. ✓
  Swap x and y for inverse $x = 2 y + 8$
  1 mark · Find the equation of an inverse
3. ✓
  Solve for y $2 y = x - 8$
  $y = \frac{x}{2} - 4$
  $h^{- 1} (x) = \frac{1}{2} x - 4$
  1 mark · Find the equation of an inverse

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