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

Feb 25, 2016

Axis of symmetry$\text{ } \to x - 1$
$\textcolor{w h i t e}{.}$
Vertex$\text{ } \to \left(x , y\right) \to \left(1 , 5\right)$

#### Explanation:

First consider the $- 2 x$. As this is negative the general shape of the graph is $\cap$

The axis of symmetry will be parallel to the y-axis (normal to the x-axis) and pass through the vertex
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This next bit is a variant on the vertex form equation

Given:$\text{ "y=-2x^2+4x+3" }$........................................(1)

Write as:$\text{ } y = - 2 \left({x}^{2} - \frac{4}{2} x\right) + 3$

Consider the $- \frac{4}{2} \text{ from } - \frac{4}{2} x$

Apply this process:$\text{ } \left(- \frac{1}{2}\right) \times \left(- \frac{4}{2}\right) = + 1$

This value of $+ 1$ is the value of ${x}_{\text{vertex}}$

$\textcolor{b r o w n}{\text{So "x=1 " is the axis if symmetry.}}$
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Substitute $x = 1$ into equation (1) to find" "y_("vertex")

$\implies y = - 2 {\left(1\right)}^{2} + 4 \left(1\right) + 3 = 5$

$\textcolor{b r o w n}{\text{Vertex} \to \left(x , y\right) \to \left(1 , 5\right)}$