We want to draw the graph of a quadratic function written in standard form.

y = a x^{2} + b x + c

To do so, we will follow five steps.

Identify $a,$ $b,$ and $c .$
Calculate and sketch the axis of symmetry.
Find and plot the vertex.
Find and plot the $y -$ intercept and its symmetric point across the axis of symmetry.
Draw a smooth curve through the three plotted points.

Let's do it!

Identify $a,$ $b,$ and $c$

We will start by identifying the values of

a,

b,

and

c .

y = 3 x^{2} - 4 x + 1 ⇕ y = 3 x^{2} + (- 4) x + 1

We have identified that

a = 3,

b = - 4,

and

c = 1 .

Axis of Symmetry

The axis of symmetry is the vertical line that divides the parabola into two mirror images. Its equation follows a specific formula.

x = - \frac{b}{2 a}

Let's substitute the values

a = 3

and

b = - 4

into this equation.

x = - \frac{b}{2 a}

SubstituteII

$a = 3$ , $b = - 4$

x = - \frac{- 4}{2 ( 3 )}

▼

Evaluate right-hand side

Multiply

x = - \frac{- 4}{6}

RemoveNegFracAndNum

$- \frac{- a}{b} = \frac{a}{b}$

x = \frac{4}{6}

ReduceFrac

$\frac{a}{b} = \frac{a / 2}{b / 2}$

x = \frac{2}{3}

The axis of symmetry is the line

x = \frac{2}{3} .

Vertex

To find the vertex of the parabola, 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}))

When determining the axis of symmetry, we found that

- \frac{b}{2 a} = \frac{2}{3} .

Therefore, the

x -

coordinate of the vertex is

\frac{2}{3}

and the

y -

coordinate is

f (\frac{2}{3}) .

To find this value, we can substitute

\frac{2}{3}

for

x

in the given equation.

y = 3 x^{2} - 4 x + 1

Substitute

$x = \frac{2}{3}$

y = 3 (\frac{2}{3})^{2} - 4 (\frac{2}{3}) + 1

▼

Evaluate right-hand side

PowQuot

$(\frac{a}{b})^{m} = \frac{a ^{m}}{b ^{m}}$

y = 3 (\frac{4}{9}) - 4 (\frac{2}{3}) + 1

MoveLeftFacToNum

$a \cdot \frac{b}{c} = \frac{a \cdot b}{c}$

y = \frac{1 2}{9} - \frac{8}{3} + 1

ReduceFrac

$\frac{a}{b} = \frac{a / 3}{b / 3}$

y = \frac{4}{3} - \frac{8}{3} + 1

OneToFrac

Rewrite $1$ as $\frac{3}{3}$

y = \frac{4}{3} - \frac{8}{3} + \frac{3}{3}

AddSubFrac

Add and subtract fractions

y = - \frac{1}{3}

The vertex of the parabola is

(\frac{2}{3}, - \frac{1}{3}) .

$y -$ Intercept and Symmetric Point

Because in our equation we have that $c = 1,$ the $y -$ intercept is $1 .$ Let's plot this point and the point symmetric across the axis of symmetry.

Graph

Since $a = 3,$ which is greater than zero, we know that our parabola opens upwards. Let's draw a smooth curve connecting the three points we have. We should not use a straight edge for this!

Extra

A Common Mistake

One common mistake when identifying the key features of a parabola algebraically is forgetting to include the negative signs in the values of the constants. The standard form is addition only, so any subtraction must be treated as negative values of

a,

b,

or

c .

Let's look at an example.

y = 3 x^{2} - 4 x - 2 ⇕ y = 3 x^{2} + (- 4 x) + (- 2)

In this case, the values of

a,

b,

and

c

are

3,

- 4,

and

- 2 .

They are not

3,

4,

and

2 .

a = 3, a = 3, b = - 4, b = - 4, c = - 2 c = - 2 \times ✓

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