Question

5. Monty compared the minimum of the function f(x) = 2x2 - x + 6 to theminimum of the quadratic function that fits the values in the table below.X-3-2-101g(x)0-5-6-34What is the horizontal distance between the minimums of the twofunctions?A 0.25B. 1C. 1.5D. 12

Answer 1

The function f is given by:

`\begin{gathered} f(x)=2x^2-x+6 \\ \text{ Rewrite the quadratic function in vertex form} \\ f(x)=2(x^2-(1)/(2)x)+6 \\ =2((x-(1)/(4))^2-(-(1)/(4))^2)+6 \\ =2(x-(1)/(4))^2-2((1)/(16))+6 \\ =2(x-(1)/(4))^2+(47)/(8) \end{gathered}`

If a quadratic function is written in the form:

`\begin{gathered} a(x-h)^2+k \\ where: \\ a>0 \end{gathered}`

Then the function has a minimum point at (h,k)

And the minimum is k

In this case,

`\begin{gathered} a=2\gt0 \\ h=(1)/(4)=0.25 \\ k=(47)/(8)=5.875 \end{gathered}`

Therefore, the minimum of the function f is at (0.25, 5.875)

The minimum of the function given by the table is at (-1, -6).

Therefore, the required horizontal distance is given by:

`0.25-(-1)=1.25`

Therefore, the horizontal distance is 1.25