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 = \frac{3}{2} x^{2} - 3 x + 6 \Leftrightarrow y = \frac{3}{2} x^{2} + (- 3) x + 6

We can see that

a = \frac{3}{2},

b = - 3,

and

c = 6 .

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
$a = \frac{3}{2}$ , $b = - 3$
$x = - \frac{- 3}{2 ( \frac{3}{2} )}$

▼

Simplify right-hand side

DenomMultFracToNumber
$2 \cdot \frac{a}{2} = a$
$x = - \frac{- 3}{3}$
RemoveNegFracAndNum
$- \frac{- a}{b} = \frac{a}{b}$
$x = \frac{3}{3}$
CalcQuot
Calculate quotient

$x = 1$
The axis of symmetry of the parabola is the vertical line with equation $x = 1 .$
Calculating the Vertex
To calculate the vertex, we need to think of $y$ as a function of $x,$ $y = f (x) .$ We can write the expression for the vertex by stating the $x -$ and $y -$ coordinates in terms of $a$ and $b .$
$Vertex : (- \frac{b}{2 a}, f (- \frac{b}{2 a}))$
Note that the formula for the $x -$ coordinate is the same as the formula for the axis of symmetry, which is $x = 1 .$ Thus, the $x -$ coordinate of the vertex is also $1 .$ To find the $y -$ coordinate, we need to substitute $1$ for $x$ in the given equation.
$y = \frac{3}{2} x^{2} - 3 x + 6$
Substitute
$x = 1$
$y = \frac{3}{2} (1)^{2} - 3 (1) + 6$

▼

Simplify right-hand side

CalcPow
Calculate power
$y = \frac{3}{2} (1) - 3 (1) + 6$
Multiply
Multiply
$y = \frac{3}{2} - 3 + 6$
AddSubTerms
Add and subtract terms
$y = \frac{9}{2}$
CalcQuot
Calculate quotient

$y = 4.5$
We found the $y -$ coordinate, and now we know that the vertex is $(1, 4.5) .$
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 $c .$ Therefore, the point where our graph intercepts the $y -$ axis is $(0, 6) .$ 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 $a = \frac{3}{2},$ which is positive, the parabola will open upwards. 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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