Question

I need help with this quadratic function… I thought I knew the answer, but obviously I don’t

Answer 1

Let us start with the following quadratic function:

`f(x)=x^2-x-12`

the X-intercepts are the collection of values to X which makes f(x) = 0, and it can be calculated by the Bhaskara formula:

`x_(1,2)=(-b\pm√(b^2-4ac))/(2a)`

where the values a, b, and c are given by:

`f(x)=ax^2+bx+c`

Substituting the values from the proposed equation, we have:

`\begin{gathered} x_(1,2)=(1\pm√(1^2-4*1*(-12)))/(2*1) \\ x_(1,2)=(1\pm√(1+48))/(2)=(1\pm√(49))/(2) \\ x_(1,2)=(1\pm7)/(2) \\ \\ x_1=(1+7)/(2)=(8)/(2)=4 \\ x_2=(1-7)/(2)=-(6)/(2)=-3 \end{gathered}`

From the above-developed solution, we are able to conclude that the solution for the first box is:

(-3,0) ,(4,0)

Now, the y-intercept, is just the value of y when x = 0, which can be calculated as follows:

`\begin{gathered} f(0)=0^2-0-12=-12 \\ f(0)=-12 \end{gathered}`

From this, we are able to conclude that the solution for the second box is:

(0, -12)

Now, the vertex is the value of minimum, or maximum, in the quadratic equation, and use to be calculated as follows:

`\begin{gathered} Vertex \\ x=-(b)/(2a) \\ y=(4ac-b^2)/(2a) \end{gathered}`

substituting the values, we have:

`\begin{gathered} x=-(-1)/(2*1)=(1)/(2) \\ y=(4*1*(-12)-(-1)^2)/(4*1)=(-48-1)/(4)=(-49)/(4) \end{gathered}`

which means that the solution for the thirst box is:

(1/2, -49/4) (just as in the photo)

Now, the line of symmetry equation of a quadratic function is a vertical line that passes through the vertex, which was calculated to be in the point: (1/2, -49,4).

Because this is a vertical line, it is represented as follows:

`x=(1)/(2)`