How To Write A Quadratic Function In Factored Form With A Vertex At X=3 And Two Distinct Roots – A Perfect Guide For Math Enthusiasts.

Explain how you could write a quadratic function in factored form that would have a vertex with an x-coordinate of 3 and two distinct roots.

sample response : The vertex is on the axis of symmetry, so the axis of symmetry is x = 3. Find any two x-intercepts that have the equivalent distance from the axis of symmetry. Use those x-intercepts to write factors of the function by subtracting their values from x. For example, 2 and 4 are each 1 unit from x = 3, so f(x) = (x – 2)(x – 4) is a possible function.

If we want to write a quadratic function in factored form that has a vertex with an x-coordinate of 3 and two distinct roots, we can use the vertex form of the quadratic equation, which is:

y = a(x – h)^2 + k

where:

– h and k are the coordinates of the vertex (h, k), and
– a is a non-zero constant that determines the shape of the parabola (if a > 0, the parabola opens upwards, and if a < 0, it opens downwards). Since we know that the vertex has an x-coordinate of 3, we can substitute h = 3 in the vertex form: y = a(x - 3)^2 + k Now, we need to determine the values of a and k in order to find the quadratic function that has two distinct roots. Since the vertex is the midpoint between the two roots, we can assume that the roots are symmetric with respect to x = 3. Let's call the distance between the vertex and one of the roots d. Then, the roots are: x1 = 3 - d x2 = 3 + d Since the quadratic function is in factored form, we know that it can be expressed as: y = a(x - x1)(x - x2) Substituting the values of x1 and x2, we get: y = a(x - (3 - d))(x - (3 + d)) Simplifying the expression, we get: y = a(x^2 - 6x + 9 - d^2) y = ax^2 - 6ax + 9a - ad^2 Now, we need to determine the values of a and d such that the quadratic function has two distinct roots. From the discriminant of the quadratic equation, we know that if the discriminant is positive, the quadratic function has two distinct roots. The discriminant is: b^2 - 4ac = (-6a)^2 - 4a(9a - ad^2) = 36a^2 - 36ad^2 For the discriminant to be positive, we need: 36a^2 - 36ad^2 > 0

Dividing both sides by 36a^2, we get:

1 – d^2/a > 0

d^2/a < 1 Since we want the roots to be distinct, we also need: d ≠ 0 Now, we can choose a value for a and solve for d. Let's choose a = 1, then: d^2 < a d^2 < 1 -1 < d < 1 Since we also know that d ≠ 0, we can choose any value of d that satisfies -1 < d < 1. For example, we can choose d = 0.5. Then, the roots are: x1 = 3 - 0.5 = 2.5 x2 = 3 + 0.5 = 3.5 Substituting a = 1 and d = 0.5 in the expression for the quadratic function, we get: y = x^2 - 6x + 9 - 0.25 y = x^2 - 6x + 8.75 Therefore, the quadratic function that has a vertex with an x-coordinate of 3 and two distinct roots is: y = (x - 2.5)(x - 3.5)

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