Question

How do you write the equation of the parabola in vertex form given vertex $\left( 4,-1 \right)$ and y-intercept $\left( 0,15 \right)$ ?

Answer 1

Here in this question we have been asked to write the equation of the parabola having vertex $\left( 4,-1 \right)$ and y-intercept $\left( 0,15 \right)$ in the vertex form. From the basic concepts of parabola we know the general vertex form equation of a parabola is given as $y=a{{\left( x-h \right)}^{2}}+k$ where $\left( h,k \right)$ is center of the given parabola.

Complete step by step answer:
Now considering from the question we have been asked to write the equation of the parabola having vertex $\left( 4,-1 \right)$ and y-intercept $\left( 0,15 \right)$ in the vertex form.
From the basic concepts of parabola we know the general vertex form equation of a parabola is given as $y=a{{\left( x-h \right)}^{2}}+k$ where $\left( h,k \right)$ is center of the given parabola.
Hence we can say that the parabola equation is given as $y=a{{\left( x-4 \right)}^{2}}-1$ using the center substitution.
We can say that the given parabola passes through $\left( 0,15 \right)$ by substituting this point we will have
$\begin{aligned} & \Rightarrow 15=a{{\left( 0-4 \right)}^{2}}-1 \\\ & \Rightarrow 15=16a-1 \\\ & \Rightarrow 16=16a \\\ & \Rightarrow a=1 \\\ \end{aligned}$ .

Therefore we can conclude that the equation of the parabola having vertex $\left( 4,-1 \right)$ and y-intercept $\left( 0,15 \right)$ in the vertex form is given as $y={{\left( x-4 \right)}^{2}}-1$

Note: During the process of answering questions of this type we should be sure with the parabola concepts and the calculations that we are going to apply and perform in between the steps. This is a very simple and easy question and can be answered accurately in a short span of time. Very few mistakes are possible in questions of this type. If someone had confused and taken $\left( k,h \right)$ as the center then they will have
$\begin{aligned} & \Rightarrow 15=a{{\left( 0+1 \right)}^{2}}+4 \\\ & \Rightarrow 15=a+4 \\\ & \Rightarrow 11=a \\\ \end{aligned}$
Then the answer will be $y=11{{\left( x+1 \right)}^{2}}+4$ which is a wrong conclusion.