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
To find the number of x-intercepts, we can look at the discriminant of the quadratic function, which is given by the formula 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 the vertex form, 𝑦 = 2(𝑥 − 1)^2 + 3, we can see that a = 2, b = 0, and c = 3. Plugging these values into the discriminant formula:
Δ = 0^2 - 4(2)(3)
Δ = 0 - 24
Δ = -24
The discriminant is negative (-24), which means that there are no real solutions for x. In other words, the quadratic function does not intersect the x-axis and does not have any x-intercepts.