Question

Don't understand letter Lcan u write the equation of the described function in standard form

Answer 1

We can use points (-1,12), (0,10) and (1,12) to find the quadratic equation. First, we will work with the formula:

`y=ax^2+bx+c`

With the three points mentioned above, we make the proper substitution:

`\begin{gathered} P_1=(-1,12) \\ \Rightarrow12=a(-1)^2+b(-1)+c \\ \Rightarrow12=a-b+c \\ P_2=(0,10) \\ \Rightarrow10=a(0)^2+b(0)+c \\ \Rightarrow10=c \\ P_3=(1,12) \\ \Rightarrow12=a(1)^2+b(1)+c \\ \Rightarrow12=a+b+c \end{gathered}`

So, we have the following system of equations:

`\begin{gathered} a-b+c=12 \\ a+b+c=12 \\ c=10 \end{gathered}`

Now, substitute c=10 on the other equations to get the following:

`\begin{gathered} c=10 \\ \Rightarrow a-b+10=12 \\ a+b+10=12 \\ \Rightarrow a-b=12-10=2 \\ a+b=12-10=2 \\ \Rightarrow a-b=2 \\ a+b=2 \end{gathered}`

solving by elimination we get the value for 'a':

`\begin{gathered} a-b=2 \\ a+b=2 \\ \Rightarrow2a=4_{} \\ \Rightarrow a=(4)/(2)=2 \\ a=2 \end{gathered}`

we have that a=2 and c=10. We can now find the value for b:

`\begin{gathered} a=2 \\ a+b=2 \\ \Rightarrow2+b=2 \\ \Rightarrow b=0 \end{gathered}`

Now that we have that a=2,b=0 and c=10, we can write the quadratic function:

`\begin{gathered} a=2 \\ b=0 \\ c=10 \\ y=ax^2+bx+c \\ \Rightarrow y=2x^2+0\cdot x+10 \\ y=2x^2+10 \end{gathered}`

therefore, the quadratic function in standard form is y=2x^2+10