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g(r)=p2−6p−55g(r) =p^2-6p -55 \newline11) What are the zeros of the function? \newline22) What is the vertex of the parabola?

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Q. g(r)=p2−6p−55g(r) =p^2-6p -55 \newline11) What are the zeros of the function? \newline22) What is the vertex of the parabola?
  1. Find Zeros of Function: Find the zeros of the function g(r)=p2−6p−55g(r) = p^2 - 6p - 55.\newlineTo find the zeros, we need to solve the quadratic equation p2−6p−55=0p^2 - 6p - 55 = 0.
  2. Factor Quadratic Equation: Factor the quadratic equation p2−6p−55=0p^2 - 6p - 55 = 0. We look for two numbers that multiply to −55-55 and add up to −6-6. These numbers are −11-11 and 55. So, we can write the equation as (p−11)(p+5)=0(p - 11)(p + 5) = 0.
  3. Solve for Zeros: Solve for pp to find the zeros.\newlineSetting each factor equal to zero gives us the solutions p=11p = 11 and p=−5p = -5.\newlineSo, the zeros of the function are p=11p = 11 and p=−5p = -5.
  4. Identify Smaller and Larger Zeros: Identify the smaller and larger zeros.\newlineThe smaller zero is p=−5p = -5 and the larger zero is p=11p = 11.
  5. Find Vertex of Parabola: Find the vertex of the parabola g(r)=p2−6p−55g(r) = p^2 - 6p - 55. The vertex form of a parabola is y=a(x−h)2+ky = a(x - h)^2 + k, where (h,k)(h, k) is the vertex. To find the vertex of the parabola, we can use the formula h=−b2ah = -\frac{b}{2a} for the xx-coordinate of the vertex.
  6. Calculate X-coordinate of Vertex: Calculate the x-coordinate hh of the vertex.\newlineFor the quadratic equation p2−6p−55p^2 - 6p - 55, a=1a = 1 and b=−6b = -6.\newlineSo, h=−(−6)/(2⋅1)=6/2=3h = -(-6)/(2\cdot1) = 6/2 = 3.
  7. Calculate Y-coordinate of Vertex: Calculate the y-coordinate kk of the vertex.\newlineSubstitute p=3p = 3 into the equation g(r)=p2−6p−55g(r) = p^2 - 6p - 55 to find kk.\newlinek=(3)2−6(3)−55=9−18−55=−64k = (3)^2 - 6(3) - 55 = 9 - 18 - 55 = -64.
  8. Write Vertex of Parabola: Write the vertex of the parabola.\newlineThe vertex of the parabola is (h,k)=(3,−64)(h, k) = (3, -64).

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