Question
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
b) Change the function to standard form and use the discriminant to check your
conclusion from part (a) of this question
Answers
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Answered by
GPT 3.5
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.
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.
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