Final answer:
The zeros of the equation are -1 and 3. The vertex is (1, -4) and the axis of symmetry is x = 1.
The answer is option ⇒A
Step-by-step explanation:
o determine the zeros, vertex, and axis of symmetry of the given quadratic equation, we can use the properties of quadratic functions.
The zeros of a quadratic equation represent the x-values where the equation equals zero. To find the zeros, we set the equation equal to zero and solve for x. In this case, the equation is:
12x^2 - 2x - 4 = 0
To solve for x, we can factor the equation or use the quadratic formula. Factoring doesn't work easily in this case, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
From the equation 12x^2 - 2x - 4 = 0, we can identify a = 12, b = -2, and c = -4. Plugging these values into the quadratic formula:
x = (-(-2) ± √((-2)^2 - 4 * 12 * -4)) / (2 * 12)
Simplifying further:
x = (2 ± √(4 + 192)) / 24
x = (2 ± √196) / 24
x = (2 ± 14) / 24
This gives us two possible solutions for x:
x = (2 + 14) / 24 = 16 / 24 = 2/3
x = (2 - 14) / 24 = -12 / 24 = -1/2
Therefore, the zeros of the equation 12x^2 - 2x - 4 are x = 2/3 and x = -1/2.
The vertex of a quadratic equation is given by the formula (-b/2a, f(-b/2a)), where f(x) is the equation. In this case, the vertex is:
x = -(-2) / (2 * 12) = 1/12
Substituting this value of x into the equation:
y = 12(1/12)^2 - 2(1/12) - 4
y = 1/12 - 1/6 - 4
y = -23/6
So, the vertex is (1/12, -23/6).
The axis of symmetry is a vertical line that passes through the vertex of the parabola. The equation of the axis of symmetry is x = -b/2a. In this case:
x = -(-2) / (2 * 12) = 1/12
Therefore, the axis of symmetry is x = 1/12.
Based on these calculations, the correct answer is:A) Zeros: -1/2 and 2/3; Vertex: (1/12, -23/6); Axis of symmetry: x = 1/12.