Grade 10 Maths · Functions

Find the equation of a parabola

Find a parabola's equation from its x-intercepts or turning point and one more point (for example use y = a(x − 1)(x − 3), then find a).

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Parabola Equation Grade 11 | Grade 11 FunctionsKevinmathscience

Parabola Equation In Turning Point Form | Part 2 | Grade 11 FunctionsKevinmathscience

Parabola How to Find the Equation Grade 10Miss Martins Maths and Science

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Question 5.1 · May/June 2025 · Paper 1 · National (DBE) · 3 marks

The graph of

f (x) = a (x + p)^{2} + q

is drawn below.

C (- 1; 4)

is the turning point of

f

B (- 3; - 4)

is a point on

f

5.1 Show that

f (x) = - 2 x^{2} - 4 x + 2

.(3)

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

More past paper questions: find the equation of a parabola

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

3. Question 4.1 · September 2024 · Paper 1 · Provincial · 2 marks
The graph of $f (x) = a x^{2}$ , $x \leq 0$ , is sketched below. The point $P (- 6; - 8)$ lies on the graph of f.
4.1 Calculate the value of a.(2)
Show the memo (2 ticks)Memo
1. ✓
  Substitute point P $- 8 = a (- 6)^{2}$
  1 mark · Find the equation of a parabola
2. ✓
  Answer $a = - \frac{2}{9}$
  1 mark · Find the equation of a parabola
4. Question 8.3 · May/June 2025 · Paper 1 · National (DBE) · 6 marks
8.3 Given that $y = 4 x - 14$ is a common tangent to $f (x) = 2 x^{2} - 4 x - 6$ and $g (x) = a x^{2} + b x - 18$ , calculate the values of $a$ and $b$ .(6)
Show the memo (6 ticks)Memo
1. ✓
  Equate the derivative of f(x) to the gradient of the tangentThe gradient of the tangent $y = 4 x - 14$ is $m = 4$ .
  $f^{'} (x) = 4 x - 4$
  $∴ 4 x - 4 = 4$
  1 mark · Find equation of a tangent
2. ✓
  Solve for the x-coordinate of the point of tangency $4 x = 8$
  $x = 2$
  1 mark · Find equation of a tangent
3. ✓
  Calculate the y-coordinate at the point of tangencySubstitute $x = 2$ into the tangent equation:
  $y = 4 (2) - 14 = - 6$
  Point of tangency is $(2; - 6)$
  1 mark · Find equation of a tangent
4. ✓
  Use the point of tangency to form an equation with g(x)Since it is a common tangent, $g (x)$ also passes through $(2; - 6)$ .
  $g (2) = a (2)^{2} + b (2) - 18 = - 6$
  $4 a + 2 b - 18 = - 6$
  $4 a + 2 b = 12$ ... (Equation 1)
  1 mark · Find the equation of a parabola
5. ✓
  Use the gradient to form a second equation with g'(x)The gradient of $g (x)$ at $x = 2$ must also be $4$ .
  $g^{'} (x) = 2 a x + b$
  $g^{'} (2) = 2 a (2) + b = 4$
  $4 a + b = 4$ ... (Equation 2)
  1 mark · Find equation of a tangent
6. ✓
  Solve the equations simultaneously for a and bSubtract (2) from (1):
  $(4 a + 2 b) - (4 a + b) = 12 - 4$
  $b = 8$
  Substitute $b = 8$ into (2):
  $4 a + 8 = 4$
  $4 a = - 4$
  $a = - 1$
  1 mark · Solve simultaneous equations by elimination
5. Question 3.2.2 · March 2025 · Provincial · 4 marks
3.2 Sketched below is the parabola $f$ and the straight line $g (x) = 3 x + 6$ . R and S are the x-intercepts of $f$ , and $P (- \frac{1}{2}; 9)$ is its turning point. $g$ has its x-intercept at R.
3.2.2 Determine the equation of the parabola.(4)
Show the memo (4 ticks)Memo
1. ✓
  Use Turning Point form $y = a (x + \frac{1}{2})^{2} + 9$
  1 mark · Find the equation of a parabola
2. ✓
  Substitute point R(-2; 0) $0 = a (- 2 + \frac{1}{2})^{2} + 9$
  $0 = a (- \frac{3}{2})^{2} + 9$
  $0 = \frac{9}{4} a + 9$
  1 mark · Find the equation of a parabola
3. ✓
  Solve for a $\frac{9}{4} a = - 9 \to a = - 4$
  1 mark · Solve a linear equation
4. ✓
  Write final equation $y = - 4 (x + \frac{1}{2})^{2} + 9$ or $y = - 4 x^{2} - 4 x + 8$
  1 mark · Find the equation of a parabola
6. Question 9.1.2 · September 2024 · Paper 1 · Provincial · 4 marks
9.1 The graphs of $g^{'} (x) = a x^{2} + b x + c$ and $h (x) = 2 x - 4$ are sketched below. The graph of $g^{'} (x) = a x^{2} + b x + c$ is the derivative graph of a cubic function g. The graphs of h and $g^{'}$ have a common y-intercept at point E. C(-2;0) and D(6;0) are the x-intercepts of the graph of $g^{'}$ . Point A is the x-intercept of h and point B is the turning point of $g^{'}$ . Line AB is parallel to the y-axis.
9.1.2 Determine the equation of the graph of $g^{'}$ in the form $y = a x^{2} + b x + c .$ (4)
Show the memo (4 ticks)Memo
1. ✓
  Substitute roots and point $y = a (x + 2) (x - 6)$ $- 4 = a (2) (- 6)$
  1 mark · Find the equation of a parabola
2. ✓
  Value of a $12 a = 4$ $a = \frac{1}{3}$
  1 mark · Solve a linear equation
3. ✓
  Substitute a and simplify $y = \frac{1}{3} (x + 2) (x - 6) = \frac{1}{3} (x^{2} - 4 x - 12)$
  1 mark · Find the equation of a parabola
4. ✓
  Equation $y = \frac{1}{3} x^{2} - \frac{4}{3} x - 4$
  1 mark · Find the equation of a parabola

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