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Cambridge IGCSE0607

Finding a quadratic function using given information

International Mathematics 0607 Chapter Notes

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Finding a quadratic function using given information
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In this chapter

  1. 1From the vertex and another point
  2. 2From the x-intercepts and a point
  3. 3When a = 1

1. From the vertex and another point

A quadratic function with vertex (h,k) can be written in vertex form. The numbers h and k come straight from the vertex. Only a is unknown.

Substitute the coordinates of the other point for x and y. Then solve the equation for a.

Be careful with the sign of h. The vertex (2, -3) gives (x - 2), not (x + 2).

y = a(x - h)² + k

Examiner insight

“Candidates needed to use the vertex of (1, 3) to set up the equation of y = a(x – 1)2 + 3.”

Worked example 14 marks

Do not use a calculator.

A quadratic function has its vertex at (2, -3). Its graph passes through the point (4, 5). Find the function in the form y = a(x - h)² + k.

  1. 1

    vertex (2, -3) gives y = a(x - 2)² - 3 [1]

  2. 2

    substitute (4, 5): 5 = a(4 - 2)² - 3 [1]

  3. 3

    8 = 4a, so a = 2 [1]

  4. 4

    y = 2(x - 2)² - 3 [1]

  5. 5

    (expanded: y = 2x² - 8x + 5. Check: at x = 4, 32 - 32 + 5 = 5.)

2. From the x-intercepts and a point

A graph crosses the x-axis at its x-intercepts. If these are x = p and x = q, the function has factors (x -p) and (x - q). The number a multiplies them.

Substitute the coordinates of the other point for x and y. Then solve the equation for a. Use both intercepts and the point: one piece of information alone does not fix the function.

The sign again: an intercept at x = -2 gives the factor (x + 2).

y = a(x - p)(x - q)

Examiner insight

“Candidates struggled to start this question and it was rare to see the roots used correctly to form a(x + 2)(x – 3). Those who tried to substitute in points often made mistakes or only used one point.”

Worked example 14 marks

Do not use a calculator.

A quadratic function has x-intercepts at x = -1 and x = 3. Its graph passes through the point (1, -8). Find the function in the form y = ax² + bx + c.

  1. 1

    intercepts -1 and 3 give y = a(x + 1)(x - 3) [1]

  2. 2

    substitute (1, -8): -8 = a(2)(-2) [1]

  3. 3

    -8 = -4a, so a = 2 [1]

  4. 4

    y = 2(x + 1)(x - 3) = 2x² - 4x - 6 [1]

3. When a = 1

Sometimes the question says the function has a = 1. Then no other point is needed. The vertex alone, or the x-intercepts alone, fixes the function.

Write the vertex or factorised form with a = 1. Expand only if the question asks for the form x² + bx + c.

y = (x - h)² + k

y = (x - p)(x - q)

Worked example 13 marks

Do not use a calculator.

A quadratic function f(x) has a = 1 and x-intercepts at x = -2 and x = 5. Find f(x) in the form x² + bx + c.

  1. 1

    f(x) = (x + 2)(x - 5) [1]

  2. 2

    = x² - 5x + 2x - 10 [1]

  3. 3

    = x² - 3x - 10 [1]

Recap

  • Extended: vertex (h, k) and another point: y = a(x - h)² + k, then find a.
  • Extended: x-intercepts p and q and a point: y = a(x - p)(x - q), then find a.
  • Extended: when a = 1, the vertex or the x-intercepts alone give the function.
  • Take the signs from the coordinates: vertex x = h gives (x - h); intercept x = p gives (x - p).
  • Substitute the other point for x and y, then solve for a.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. State the coordinates of the vertex of y = 3(x - 2)² + 5.1 mark
  2. Do not use a calculator. A quadratic function has its vertex at (1, 4) and passes through the point (3, 0). Find the function in the form y = a(x - h)² + k.3 marks
  3. Do not use a calculator. A quadratic function has x-intercepts at x = -2 and x = 6. Its graph passes through the point (0, -24). Find the function in the form y = a(x - p)(x - q).3 marks
  4. Do not use a calculator. A quadratic function has a = 1 and its vertex is at (-4, -1). Find the function in the form x² + bx + c.3 marks
  5. Do not use a calculator. A quadratic function has a = 1 and x-intercepts at x = 3 and x = -7. Find the function in the form x² + bx + c.3 marks
  6. A quadratic function has its vertex at (3, -2) and passes through the point (5, 6). Find the function in the form y = ax² + bx + c.4 marks
  7. Do not use a calculator. A quadratic function has x-intercepts at x = 1 and x = 4. Its graph passes through the point (2, 6). Find the function in the form y = ax² + bx + c.4 marks
  8. A quadratic function has its vertex at (-1.5, 7.2) and passes through the point (2, -5.3). Use your calculator for the arithmetic. Find a in y = a(x - h)² + k, correct to 3 significant figures, and write the function.3 marks
  9. A cable hangs between two towers of a bridge. Its height above the road, h metres, is a quadratic function of the horizontal distance, x metres, from the left tower. The cable is fixed to the left tower 22 m above the road. Its lowest point is 4 m above the road, 30 m from the left tower. The right tower is 70 m from the left tower. (a) Find h in the form ax² + bx + c. (b) Find the height at which the cable is fixed to the right tower.5 marks
  10. A fountain sends a jet of water from a nozzle at ground level. The height of the water, y metres, is a quadratic function of the horizontal distance, x metres, from a marker post. The nozzle is at x = 2 and the water lands on the ground at x = 10. At x = 3 the water is 3.5 m above the ground. (a) Find y in the form ax² + bx + c. (b) A lamp hangs 5.5 m above the ground at x = 8. Find how far above or below the lamp the water passes.6 marks

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