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 .

f (x) = 0.5 x^{2} + x - 3 \Leftrightarrow f (x) = 0.5 x^{2} + 1 x + (- 3)

We can see that

a = 0.5,

b = 1,

and

c = - 3 .

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
$a = 0.5$ , $b = 1$
$x = - \frac{1}{2 ( 0 . 5 )}$

▼

Simplify right-hand side

Multiply
Multiply
$x = - \frac{1}{1}$
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 $x = - 1 .$ Therefore, 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.
$f (- 1) = 0.5 x^{2} + x - 3$
Substitute
$x = - 1$
$f (- 1) = 0.5 (- 1)^{2} + (- 1) - 3$

▼

Simplify right-hand side

CalcPow
Calculate power
$f (- 1) = 0.5 (1) + (- 1) - 3$
Multiply
Multiply
$f (- 1) = 0.5 - 1 - 3$
SubTerms
Subtract terms

$f (- 1) = - 3.5$
We found the $y -$ coordinate, and now we know that the vertex is $(- 1, - 3.5) .$
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 $c .$ Thus, the point where our graph intercepts the $y -$ axis is $(0, - 3) .$ 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 $a = 0.5,$ 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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