Grade 10 Maths · Functions

Find turning point of a parabola

Find the turning point of a parabola using x = −b ÷ 2a and then the y-value (for example y = x² − 4x + 1 turns at (2; −3)).

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Draw Parabola Grade 11 | Part 1 | Grade 11 FunctionsKevinmathscience

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Question 10.1 · November 2024 · Paper 1 · National (DBE) · 3 marks

A cyclist rode from town P and stopped at town T. The speed (in km/h) at which this cyclist rode, is represented by the equation

s^{'} (t) = - 3 t^{2} + 18 t

. NOTE: Speed is the rate of change in distance with respect to time.

10.1 Calculate the maximum speed that the cyclist reached on this ride.(3)

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

More past paper questions: find turning point of a parabola

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3. 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
4. Question 3.1.1 · March 2025 · Provincial · 2 marks
Given: $f (x) = (x - 1)^{2} - 9$
3.1 Properties of $f (x)$ .
3.1.1 Determine the turning point of $f$ .(2)
Show the memo (2 ticks)Memo
1. ✓
  Identify x-coordinate $x = 1$
  1 mark · Find turning point of a parabola
2. ✓
  Identify y-coordinate $y = - 9$
  Turning Point: $(1; - 9)$
  1 mark · Find turning point of a parabola
5. Question 5.3 · June 2024 · Paper 1 · Provincial · 2 marks
Sketched below are the graphs of $f (x) = a x^{2} + b x + c$ and $g (x) = m x + k$ . The x-intercepts of f are at points A and C. The graphs of f and g intersect at points A and B respectively. Point P is the turning point of f. Q is a point on g such that line PQ is parallel to the y-axis. Points given: $A (- 4; 0)$ , $C (2; 0)$ , $B (0; 8)$ .
5.3 Show that the coordinates of P are $(- 1; 9)$ .(2)
Show the memo (2 ticks)Memo
1. ✓
  Calculate axis of symmetry $x = \frac{- b}{2 a} = \frac{- ( - 2 )}{2 ( - 1 )} = - 1$
  1 mark · Find turning point of a parabola
2. ✓
  Calculate y-coordinate $f (- 1) = - (- 1)^{2} - 2 (- 1) + 8 = 9$
  P(-1; 9)
  1 mark · Find turning point of a parabola
6. Question 4.2 · March 2024 · Paper 1 · Provincial · 3 marks
The sketch below shows the graphs of $h (x) = - x^{2} + 2 x + p$ and $t (x) = m x + 5$ . $D$ is the $x$ -intercept, and $B$ the $y$ -intercept of $t$ . $A$ is the $y$ -intercept of $h$ . $C$ is the $x$ -intercept of $h$ . $F$ is the turning point of $h$ . $E$ is a point on $t$ , such that $EC$ is parallel to the $y$ -axis. $A B$ is 2 units and $C D$ is 7 units.
4.2 Determine the coordinates of $F$ .(3)
Show the memo (3 ticks)Memo
1. ✓
  Substitution into $x = \frac{- b}{2 a}$ $x = \frac{- b}{2 a} = \frac{- 2}{2 ( - 1 )}$
  1 mark · Find turning point of a parabola
2. ✓
  Value of x $x = 1$
  1 mark · Find turning point of a parabola
3. ✓
  Value of y $y = - (1)^{2} + 2 (1) + 3 = 4$ | $F (1; 4)$
  1 mark · Find turning point of a parabola

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