Question

A quadratic function is given in vertex form: 𝑦 = 2(𝑥 − 1)^2 + 3.
a) Use the values of a and q to find the number of x-intercepts.

Answer 1

clearly, there are no intercepts.
The vertex is at (1,3) and the parabola opens upward.

Answer 2

The quadratic function given is in vertex form 𝑦 = a(𝑥 - q)^2 + p, where (q, p) is the vertex of the parabola.

In this case, the vertex form is 𝑦 = 2(𝑥 - 1)^2 + 3, so a = 2 and q = 1.

To find the number of x-intercepts, we can look at the discriminant, which is the value under the square root in the quadratic formula.

The formula for the discriminant is Δ = b^2 - 4ac, where a, b, and c are the coefficients of the quadratic function in standard form (𝑦 = ax^2 + bx + c).

In this case, the quadratic function is already in vertex form, so we'll need to convert it to standard form to find the values of a, b, and c.

Expanding 𝑦 = 2(𝑥 - 1)^2 + 3, we get 𝑦 = 2(𝑥^2 -2𝑥 + 1) + 3.
Simplifying further, we have 𝑦 = 2𝑥^2 - 4𝑥 + 2 + 3.
Combining like terms, we get 𝑦 = 2𝑥^2 - 4𝑥 + 5.

Now we can see that a = 2, b = -4, and c = 5. Calculating the discriminant, we have Δ = (-4)^2 - 4(2)(5) = 16 - 40 = -24.

Since the discriminant is negative (Δ < 0), this means that the quadratic function has no x-intercepts.

Answer 3

Apologies for the confusion in my previous response. You are correct.

The given quadratic function is in vertex form 𝑦 = 2(𝑥 - 1)^2 + 3, where a = 2 and q = 1.

The vertex of the parabola is (1, 3). Since the coefficient of the squared term (a = 2) is positive, the parabola opens upward.

Since the parabola does not intersect or touch the x-axis, there are no x-intercepts.