To draw the graph of the given quadratic function, written in standard form, we must start by identifying the values of

a,

b,

and

c .

y = - 4 x^{2} + 8 x + 2 \Leftrightarrow y = - 4 x^{2} + 8 x + 2

We can see that

a = - 4,

b = 8,

and

c = 2 .

Now, we will follow four steps to graph the function.

Find the axis of symmetry.
Calculate the vertex.
Identify the $y -$ intercept and its reflection across the axis of symmetry.
Connect the points with a parabola.
Finding the Axis of Symmetry
The axis of symmetry is a vertical line with equation $x = - \frac{b}{2 a} .$ Since we already know the values of $a$ and $b,$ we can substitute them into the formula.
$x = - \frac{b}{2 a}$
SubstituteII
$a = - 4$ , $b = 8$
$x = - \frac{8}{2 ( - 4 )}$

▼

Simplify right-hand side

Multiply
Multiply
$x = - \frac{8}{- 8}$
RemoveNegFracAndDenom
$- \frac{a}{- b} = \frac{a}{b}$
$x = \frac{8}{8}$
CalcQuot
Calculate quotient

$x = 1$
The axis of symmetry of the parabola is the vertical line with equation $x = 1 .$
Calculating the Vertex
Note that the formula for the $x -$ coordinate of the vertex is the same as the formula for the axis of symmetry, which is $x = 1 .$ To find the $y -$ coordinate of the vertex, we need to substitute $1$ for $x$ in the given equation.
$y = - 4 x^{2} + 8 x + 2$
Substitute
$x = 1$
$y = - 4 (1)^{2} + 8 (1) + 2$

▼

Simplify right-hand side

CalcPow
Calculate power
$y = - 4 (1) + 8 (1) + 2$
Multiply
Multiply
$y = - 4 + 8 + 2$
AddTerms
Add terms

$y = 6$
We found the $y -$ coordinate, and now we know that the vertex is $(1, 6) .$
Identifying the $y -$ intercept and its Reflection

The $y -$ intercept of the graph of a quadratic function written in standard form is given by the value of $c .$ Thus, the point where our graph intercepts the $y -$ axis is $(0, 2) .$ Let's plot this point and its reflection across the axis of symmetry.

Connecting the Points

We can now draw the graph of the function. Since $a = - 4,$ which is negative, the parabola will open downwards. Let's connect the three points with a smooth curve.

Finding the Axis of Symmetry

Calculating the Vertex

Identifying the y-intercept and its Reflection

Connecting the Points

Identifying the $y -$ intercept and its Reflection

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