A quadratic function is given in vertex form: 𝑦 = 2(𝑥 − 1)^2 + 3.
b) Change the function to standard form and use the discriminant to check your
conclusion from part (a) of this question
To change the function to standard form, we need to expand and simplify the expression:
y = 2(x - 1)^2 + 3
y = 2(x^2 - 2x + 1) + 3
y = 2x^2 - 4x + 2 + 3
y = 2x^2 - 4x + 5
Now we can use the discriminant to check our conclusion. The discriminant is given by the formula:
Δ = b^2 - 4ac
In our case, a = 2, b = -4, and c = 5. Substituting these values into the formula:
Δ = (-4)^2 - 4(2)(5)
Δ = 16 - 40
Δ = -24
Since the discriminant is negative (-24 < 0), this means the quadratic function does not have any real solutions. This confirms our conclusion from part (a) that the function does not intersect the x-axis.