Question

What are the vertex, axis of symmetry, maximum or minimum value, domain and range of the function $y=-{{x}^{2}}-4x+3$?

Answer 1

Hint: To solve the given question we will use the standard formula of vertex form of parabola $y=a{{\left( x-h \right)}^{2}}+k$ , where $\left( h,k \right)$ are the coordinates of vertex and \[\left( x,y \right)\] are the coordinates of point from which parabola passes through and a is the distance of origin from the focus. Then we will find the domain and range of the given function.

Complete step-by-step solution:
We have been given a function $y=-{{x}^{2}}-4x+3$.
We have to find the vertex, axis of symmetry, maximum or minimum value, domain and range of the given function.
The given function is quadratic in nature of the form $a{{x}^{2}}+bx+c$. When we compare the given equation with the general equation we will get the values
$\Rightarrow a=-1,b=-4,c=3$
Now, we know that the general form of equation of parabola is given as $y=a{{\left( x-h \right)}^{2}}+k$ , where $\left( h,k \right)$ are the coordinates of vertex and \[\left( x,y \right)\] are the coordinates of point from which parabola passes through and a is the distance of origin from the focus.
Now, the vertex of parabola will be
$\Rightarrow h=-\dfrac{b}{2a}\text{ and }k=y\left( h \right)$
Now, substituting the values we will get
$\begin{align}
  & \Rightarrow h=-\dfrac{-4}{2\times (-1)} \\
 & \Rightarrow h=-\dfrac{-4}{-2} \\
 & \Rightarrow h=-2 \\
\end{align}$
Now,
$\begin{align}
  & \Rightarrow k=y\left( h \right) \\
 & \Rightarrow k=-{{\left( -2 \right)}^{2}}-4\left( -2 \right)+3 \\
 & \Rightarrow k=-4+8+3 \\
 & \Rightarrow k=7 \\
\end{align}$
Now, let us plot a graph for the given equation then we will get

When we observe the above graph we will find that the vertex of parabola is $\left( -2,7 \right)$.
The axis of symmetry is $x=-2$.
The maximum value is 7.
The domain of the given function is all real values. So the domain of the function is $\left( -\infty ,\infty \right)$ and the range of the function is $\left( -\infty ,7 \right]$.
Hence we get the required values.

Note: If the value of coefficient of ${{x}^{2}}$ is negative then we have to calculate the maximum value. Also if the coefficient of ${{x}^{2}}$ is negative then the parabola opens downwards. The coordinate of x is the axis of symmetry.