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.
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
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