Grade 10 Maths · Functions

Find a horizontal or vertical length

Find a vertical or horizontal distance as top minus bottom or right minus left (for example the vertical gap between f and g is f(x) − g(x)).

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Vertical Distance Grade 11 FunctionsKevinmathscience

Distance vertical between functions with top minus bottom rule (mathdou) Gr 11+Dou Kleefstra

Maximum Distance Grade 11 FunctionsKevinmathscience

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Question 4.6 · May/June 2025 · Paper 1 · National (DBE) · 2 marks

The graph of

f (x) = \frac{4}{x - 3} + 4

is drawn below. M is the point where the asymptotes of

f

intersect. C and D are the x- and y-intercepts respectively of

f

. A is the point on

f

that is closest to M. Diagram for this question

4.6 A single transformation is applied to

f

to obtain a new graph defined as

h (x) = \frac{- 4}{x + 3} + 4

A^{'}

is the image of A under this transformation. Calculate the length of

A A^{'}

.(2)

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

More past paper questions: find a horizontal or vertical length

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

3. Question 4.4 · June 2024 · Paper 1 · Provincial · 2 marks
Consider the functions $f (x) = a x^{2} + c$ and $g (x) = b x + q$ . $f$ and $g$ intersect at points $A (- 1; - 3)$ and $B (2; 0)$ .
4.4 Determine the value of $g (x) - f (x)$ (2)
Show the memo (2 ticks)Memo
1. ✓
  Calculate difference at y-intercepts (implied by memo) $q - c = - 2 - (- 4)$
  1 mark · Find a horizontal or vertical length
2. ✓
  Final answer $= 2$
  1 mark · Find a horizontal or vertical length
4. Question 6.3 · November 2024 · Paper 1 · National (DBE) · 4 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 $g$ .
6.3 Calculate the maximum length of $E H$ for $f > g$ .(4)
Show the memo (4 ticks)Memo
1. ✓
  Formulate expression for EH $E H = f (x) - g (x)$
  $E H = (- x^{2} + 4 x + 5) - (2 x + 2)$
  1 mark · Find a horizontal or vertical length
2. ✓
  Simplify expression $E H = - x^{2} + 2 x + 3$
  1 mark · Collect like terms
3. ✓
  Find x-value for maximum $x = \frac{- b}{2 a} = \frac{- 2}{2 ( - 1 )}$
  $x = 1$
  1 mark · Find turning point of a parabola
4. ✓
  Calculate maximum lengthMax length $= - (1)^{2} + 2 (1) + 3$
  $= 4$
  1 mark · Find turning point of a parabola
5. Question 9.5 · May/June 2025 · Paper 1 · National (DBE) · 6 marks
Given: $f (x) = x^{3} - 6 x^{2} + 9 x - 4 = (x - 4) (x - k)^{2}$
9.5 Calculate the maximum vertical distance between $f$ and $h$ for $1 < x < 3$ , if $h (x) = - 2 f^{'} (x)$ .(6)
Show the memo (6 ticks)Memo
1. ✓
  Establish the expression for the distance $h (x) = - 2 (3 x^{2} - 12 x + 9) = - 6 x^{2} + 24 x - 18$
  $Distance = h (x) - f (x)$
  $= (- 6 x^{2} + 24 x - 18) - (x^{3} - 6 x^{2} + 9 x - 4)$
  1 mark · Find a horizontal or vertical length
2. ✓
  Simplify the distance expression $= - 6 x^{2} + 24 x - 18 - x^{3} + 6 x^{2} - 9 x + 4$
  $= - x^{3} + 15 x - 14$
  1 mark · Collect like terms
3. ✓
  Differentiate the distance expression and equate to 0To find max distance, let $D^{'} (x) = 0$ .
  $D^{'} (x) = - 3 x^{2} + 15 = 0$
  1 mark · Find stationary points
4. ✓
  Solve the derivative equation for x squared $3 x^{2} = 15$
  $x^{2} = 5$
  1 mark · Solve a linear equation
5. ✓
  Find the correct x-value within the given domain $x = \pm 5$
  But the domain is restricted to $1 < x < 3$ .
  $∴ x = 5$ (since $- 5$ is not in the interval)
  1 mark · Solve a quadratic by factorising
6. ✓
  Calculate the maximum vertical distanceSubstitute $x = 5$ into the distance equation:
  $Max distance = - (5)^{3} + 15 (5) - 14$
  $= - 55 + 155 - 14$
  $= 105 - 14 \approx 8, 36$
  1 mark · Find stationary points
6. Question 6.5 · May/June 2024 · Paper 1 · National (DBE) · 5 marks
The graphs of $f (x) = x^{2} - 2 x - 3$ and $g (x) = m x + c$ are drawn below. D and E are the $x$ -intercepts and P is the $y$ -intercept of $f$ . The turning point of $f$ is T $(1; - 4)$ . The graphs of $f$ and $g$ intersect at P and E.
6.5 Determine the maximum vertical distance between $h$ and $g$ if $h (x) = - f (x)$ for $x \in [- 2; 3]$ .(5)
Show the memo (5 ticks)Memo
1. ✓
  Equation of h $h (x) = - x^{2} + 2 x + 3$
  1 mark · Reflect a graph
2. ✓
  Distance between the graphs $D = - x^{2} + 2 x + 3 - (x - 3) = - x^{2} + x + 6$
  1 mark · Find a horizontal or vertical length
3. ✓
  Method for the maximum $D^{'} (x) = - 2 x + 1 = 0$ or $x = - \frac{b}{2 a} = - \frac{1}{2 ( - 1 )}$
  1 mark · Find turning point of a parabola
4. ✓
  x-value $x = \frac{1}{2}$
  1 mark · Find turning point of a parabola
5. ✓
  Maximum distance $D (\frac{1}{2}) = - (\frac{1}{2})^{2} + \frac{1}{2} + 6 = 6, 25$ units
  1 mark · Find turning point of a parabola

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