Answer
Verified
423k+ views
Hint: The general form of the parabola \[y=a{{x}^{2}}+bx+c\] is to be used while solving this question.
Complete step-by-step answer:
In the question, it is given that the axis of the parabola is parallel to the $y-$axis. So, the parabola would look like the figure below.
We know that the general formula for this form of parabola is given by,
\[y=a{{x}^{2}}+bx+c\ldots \ldots \ldots \left( i \right)\]
It is given in the question that the parabola passes through the points \[\left( 0,2 \right),\left( -1,0 \right)\] and \[\left( 1,6 \right)\]. Since the parabola passes through the three points, we know that these points must satisfy the equation of the parabola. So, we can substitute each point in the equation $\left( i \right)$ and formulate three sets of equations.
Considering the first point \[\left( 0,2 \right)\] and substituting the values of \[x=0,y=2\] in equation $\left( i \right)$, we get
\[\begin{align}
& 2=a\times 0+b\times 0+c \\
& c=2\ldots \ldots \ldots \left( ii \right) \\
\end{align}\]
Considering the second point \[\left( -1,0 \right)\] and substituting the values of \[x=-1,y=0\] in equation $\left( i \right)$, we get
\[\begin{align}
& 0=a\times {{\left( -1 \right)}^{2}}+b\times -1+c \\
& 0=a-b+c \\
\end{align}\]
Now substituting the value of \[c\] from equation $\left( ii \right)$ in the above equation, we get
\[\begin{align}
& 0=a-b+2 \\
& b-a=2\ldots \ldots \ldots \left( iii \right) \\
\end{align}\]
Considering the last point \[\left( 1,6 \right)\] and substituting the values of \[x=1,y=6\] in equation $\left( i \right)$, we get
$\begin{align}
& 6=a\times {{1}^{2}}+b\times 1+c \\
& 6=a+b+c \\
\end{align}$
Now substituting the value of \[c\] from equation $\left( ii \right)$ in the above equation, we get
\[\begin{align}
& 6=a+b+2 \\
& a+b=4\ldots \ldots \ldots \left( iv \right) \\
\end{align}\]
We have two equations \[\left( iii \right)\] and $\left( iv \right)$ to get the values of the \[a\] and $b$. So, adding the equations,
\[\dfrac{\begin{align}
& b-a=2 \\
& a+b=4 \\
\end{align}}{\begin{align}
& 2b=6 \\
& b=3 \\
\end{align}}\]
Substituting \[b=3\] in equation \[\left( iii \right)\], we get
\[\begin{align}
& 3-a=2 \\
& a=1 \\
\end{align}\]
Now we have the values as $a=1,b=3,c=2$. So, we can substitute this in equation $\left( i \right)$,
\[y={{x}^{2}}+3x+2\]
Therefore, the required equation of the parabola is obtained as \[y={{x}^{2}}+3x+2\].
Note: As three points are given in the question, we can formulate three equations and easily compute the three unknowns in the equation. If you are familiar with the cross-multiplication method, you can solve the equations and get the values of \[a,b,c\] in less time.
Complete step-by-step answer:
In the question, it is given that the axis of the parabola is parallel to the $y-$axis. So, the parabola would look like the figure below.
We know that the general formula for this form of parabola is given by,
\[y=a{{x}^{2}}+bx+c\ldots \ldots \ldots \left( i \right)\]
It is given in the question that the parabola passes through the points \[\left( 0,2 \right),\left( -1,0 \right)\] and \[\left( 1,6 \right)\]. Since the parabola passes through the three points, we know that these points must satisfy the equation of the parabola. So, we can substitute each point in the equation $\left( i \right)$ and formulate three sets of equations.
Considering the first point \[\left( 0,2 \right)\] and substituting the values of \[x=0,y=2\] in equation $\left( i \right)$, we get
\[\begin{align}
& 2=a\times 0+b\times 0+c \\
& c=2\ldots \ldots \ldots \left( ii \right) \\
\end{align}\]
Considering the second point \[\left( -1,0 \right)\] and substituting the values of \[x=-1,y=0\] in equation $\left( i \right)$, we get
\[\begin{align}
& 0=a\times {{\left( -1 \right)}^{2}}+b\times -1+c \\
& 0=a-b+c \\
\end{align}\]
Now substituting the value of \[c\] from equation $\left( ii \right)$ in the above equation, we get
\[\begin{align}
& 0=a-b+2 \\
& b-a=2\ldots \ldots \ldots \left( iii \right) \\
\end{align}\]
Considering the last point \[\left( 1,6 \right)\] and substituting the values of \[x=1,y=6\] in equation $\left( i \right)$, we get
$\begin{align}
& 6=a\times {{1}^{2}}+b\times 1+c \\
& 6=a+b+c \\
\end{align}$
Now substituting the value of \[c\] from equation $\left( ii \right)$ in the above equation, we get
\[\begin{align}
& 6=a+b+2 \\
& a+b=4\ldots \ldots \ldots \left( iv \right) \\
\end{align}\]
We have two equations \[\left( iii \right)\] and $\left( iv \right)$ to get the values of the \[a\] and $b$. So, adding the equations,
\[\dfrac{\begin{align}
& b-a=2 \\
& a+b=4 \\
\end{align}}{\begin{align}
& 2b=6 \\
& b=3 \\
\end{align}}\]
Substituting \[b=3\] in equation \[\left( iii \right)\], we get
\[\begin{align}
& 3-a=2 \\
& a=1 \\
\end{align}\]
Now we have the values as $a=1,b=3,c=2$. So, we can substitute this in equation $\left( i \right)$,
\[y={{x}^{2}}+3x+2\]
Therefore, the required equation of the parabola is obtained as \[y={{x}^{2}}+3x+2\].
Note: As three points are given in the question, we can formulate three equations and easily compute the three unknowns in the equation. If you are familiar with the cross-multiplication method, you can solve the equations and get the values of \[a,b,c\] in less time.
Recently Updated Pages
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE
Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE
What are the possible quantum number for the last outermost class 11 chemistry CBSE
Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE
Trending doubts
Assertion CNG is a better fuel than petrol Reason It class 11 chemistry CBSE
How does pressure exerted by solid and a fluid differ class 8 physics CBSE
Number of valence electrons in Chlorine ion are a 16 class 11 chemistry CBSE
What are agricultural practices? Define
What does CNG stand for and why is it considered to class 10 chemistry CBSE
The rate of evaporation depends on a Surface area b class 9 chemistry CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
State whether the following statement is true or false class 11 physics CBSE
A night bird owl can see very well in the night but class 12 physics CBSE