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find the vertex and zeroes of x^2-12y+5=0

Sagot :

y = ax² + bx + c    ⇒    y = f(x)
f(x) = ax² + bx + c

Given:  x² - 12y + 5 = 0
Convert to y = ax² + bx + c
        x² - 12y + 5 = 0
        x² + 5 = 12y     
       12y = x² + 5
       12y/12 = x²/12 + 5/12
        y = [tex] \frac{ x^{2} }{12} + \frac{5}{12} [/tex]

A.)  Set y to = 0
      [tex] \frac{ x^{2} }{12} + \frac{5}{12} =0[/tex]

      Solve for roots (zeroes) using the method extracting the square roots.
      Use this method when b = 0 in equation ax² + bx + c = 0.

      [tex]12( \frac{ x^{2} }{12} + \frac{5}{12} =0)[/tex]

      x² + 5 = 0

      x² = -5

      [tex] \sqrt{ x^{2} } = \frac{+}{-} \sqrt{-5} [/tex]

      x₁ = [tex]i \sqrt{5} [/tex]

      x₂ = [tex]-i \sqrt{5} [/tex]

THE ZEROES (ROOTS) are [tex]i \sqrt{5} [/tex]  and [tex]-i \sqrt{5} [/tex].

It means that the equation has no real roots, and the graph (parabola) that opens upward is above the x-axis.

B.)  Find the vertex of the parabola. 
       Since the equation has a positive leading leading term ([tex] \frac{ x^{2} }{12} [/tex]), the parabola opens upward (u-shaped), and the vertex is the minimum.

Vertex = (h, k)

h = [tex] \frac{-b}{2a} [/tex] 

h = [tex] \frac{0}{2( \frac{1}{12}) } [/tex]

h = 0

k = f(h)
Plug -in  the value of h (0) to x in equation [tex] \frac{x ^{2} }{12} + \frac{5}{12} [/tex]

k = [tex] \frac{0 ^{2} }{12} + \frac{5}{12} [/tex]

k = 0 + ⁵/₁₂

k = ⁵/₁₂

Vertex = (h, k)
Vertex = (0, ⁵/₁₂)
   
FINAL ANSWER:  The vertex is (0, ⁵/₁₂) and the zeroes (roots) are [tex]i \sqrt{5} [/tex] and [tex]-i \sqrt{5} [/tex].

Please click image to see the graph of the given equation.
View image Аноним