Question

What is the axis of symmetry and vertex for the graph $y = - 2{x^2} + 4x + 3$?

Answer 1

The axis of symmetry is the coordinate of the vertex. To find that, you can either use a graphical calculator or solve algebraically using the formula. In the standard form of a quadratic equation $a{x^2} + bc + c = 0.$ $a{x^2}$ is the quadratic term$bx$is the linear term. $c$ Is the constant term.

Formula used:
The standard form of the quadratic equation is $a{x^2} + bx + c = 0$
$x = - \dfrac{b}{{2a}}$
$b$-Coefficient $x$
$a$-Coefficient ${x^2}$

Complete answer:
The axis of symmetry will be parallel to the $y$ axis (normal to the $x$-axis) and pass through the vertex.
Here given quadratic equation is
$y = - 2{x^2} + 4x + 3$ …………………….(1)
Compare this equation to the standard form of a quadratic equation.
We know that the formula to find the axis of symmetry,
$x = \dfrac{{ - b}}{{2a}}$ ……………………(2)
From the given quadratic equation we have the value of $a$ and $b$
$a = - 2$
$b = 4$
Substitute the values in the equation (2) we get,
$\Rightarrow x = \dfrac{{ - \left( 4 \right)}}{{2\left( { - 2} \right)}}$
$\Rightarrow x = \dfrac{{ - 4}}{{ - 4}}$
Solve this, we get,
$x = 1$ ……………………….. (3)
This value of $+ 1$ is the value of $x$ the vertex
So $x = 1$ is the axis of symmetry
To find the value of $y$ vertex substitute the equation (3) in equation (1) we get,
$y = - 2{\left( 1 \right)^2} + 4\left( 1 \right) + 3$
$= - 2 + 4 + 3$
Solve this we get,
$= - 2 + 7$
$\Rightarrow y = 5$
Therefore the vertex
$\left( {x,y} \right) = \left( {1,5} \right)$

Note:
The axis of symmetry of a parabola is a vertical line that divides the parabola into two congruent halves. The axis of symmetry always passes through the vertex of the parabola. The x-coordinate of the vertex is the question of the axis of symmetry of the parabola.
Vertex: A vertex (in plural form: vertices or vertexes) often denoted by letters such as $P,Q,R,S$ is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices.