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 = x^{2} + 2 x + 1 \Leftrightarrow y = 1 x^{2} + 2 x + 1

We can see that

a = 1,

b = 2,

and

c = 1 .

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
$a = 1$ , $b = 2$
$x = - \frac{2}{2 ( 1 )}$

▼

Simplify right-hand side

Multiply
Multiply
$x = - \frac{2}{2}$
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.
$y = x^{2} + 2 x + 1$
Substitute
$x = - 1$
$y = (- 1)^{2} + 2 (- 1) + 1$

▼

Simplify right-hand side

CalcPow
Calculate power
$y = 1 + 2 (- 1) + 1$
Multiply
Multiply
$y = 1 - 2 + 1$
AddSubTerms
Add and subtract terms

$y = 0$
We found the $y -$ coordinate, and now we know that the vertex is $(- 1, 0) .$
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 $(0, 1) .$ 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 $a = 1,$ 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

{{ 'ml-heading-exercise' | message }} {{ exercise.name }} {{ 'ml-label-page' | message }} {{ exercise.page }}

{{ 'ml-solutions-missing-heading' | message }}