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A parabola has zeros of 7 and 1 and passes through the point (3,4). Determine the equation of the parabola.

Sagot :

semsee45

Answer:

The zeros are the points where the parabola intercepts the x-axis.

[tex]\sf x = 7 \implies x - 7 = 0[/tex]

[tex]\sf x = 1 \implies x - 1 = 0[/tex]

[tex]\sf \implies y = a(x - 7)(x - 1)[/tex]  where a is some constant

If the parabola passes through point (3, 4) then:

[tex]\sf \implies a(3 - 7)(3 - 1) = 4[/tex]

[tex]\sf \implies a(-4)(2) = 4[/tex]

[tex]\sf \implies -8a = 4[/tex]

[tex]\sf \implies a = -\dfrac12[/tex]

So the equation of the parabola is:

[tex]\sf y = -\dfrac12(x - 7)(x - 1)[/tex]

Or in standard form:

[tex]\sf y = -\dfrac12x^2+4x-\dfrac72[/tex]

Spot the quadratic equation:-

  • (x-1)(x-7)
  • x(x-7)-1(x-7)
  • x²-7x-x+7
  • x²-8x+7

Find a through (3,4)

  • a(x²-8x+7)=4
  • a(9-24+7)=4
  • a(-8)=4
  • a=-1/2

Equation:-

  • y=-1/2(x-7)(x-1)
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