We want to draw the graph of the given quadratic function. To do so, we will rewrite it in vertex form,

g (x) = a (x - h)^{2} + k,

where

a,

h,

and

k

are either positive or negative numbers.

g (x) = (x - 2)^{2} - 4 ⇕ g (x) = 1 (x - 2)^{2} - 4

To draw the graph, we will follow four steps.

Identify the constants $a,$ $h,$ and $k .$
Plot the vertex $(h, k)$ and draw the axis of symmetry $x = h .$
Plot any point on the curve and its reflection across the axis of symmetry. We can also find the $x -$ intercepts.
Sketch the curve.

Let's get started.

Step $1$

We will first identify the constants

a,

h,

and

k .

Recall that if

a < 0,

the parabola will open downwards. Conversely, if

a > 0,

the parabola will open upwards.

Vertex Form : Function : g (x) = - a (x - h)^{2} + k g (x) = 1 (x - 2)^{2} - 10

We can see that

a = 1,

h = 2,

and

k = - 4 .

Since

a

is greater than

0,

the parabola will open downwards.

Step $2$

Let's now plot the vertex $(h, k)$ and draw the axis of symmetry $x = h .$ Since we already know the values of $h$ and $k,$ we know that the vertex is $(2, - 4) .$ Therefore, the axis of symmetry is the vertical line $x = 2 .$

Step $3$

We will now plot a point on the curve by choosing an

x -

value and calculating its corresponding

y -

value. We can also find the

x -

intercepts and plot them. Since we want to find the

x -

intercepts, we will find them now and plot them. To do this, we will solve the equation

g (x) = 0 .

g (x) = 0

Substitute

$g (x) = (x - 2)^{2} - 4$

(x - 2)^{2} - 4 = 0

▼

Solve for

x

AddEqn

$LHS + 4 = RHS + 4$

(x - 2)^{2} = 4

SqrtEqn

$LHS = RHS$

x - 2 = \pm 4

CalcRoot

Calculate root

x - 2 = \pm 2

AddEqn

$LHS + 2 = RHS + 2$

x = 2 \pm 2

Now we can calculate the first

x -

intercept using the positive sign and the second one using the negative sign.

$x = 2 \pm 2$
$x = 2 + 2$	$x = 2 - 2$
$x = 4$	$x = 0$

Therefore, the $x -$ intercepts are $x = 4$ and $x = 0 .$

Note that the point $(0, 0),$ which is an $x -$ intercept, is also the $y -$ intercept.

Step $4$

Finally, we will sketch the parabola which passes through the three points. Remember not to use a straightedge for this!

Step 1

Step 2

Step 3

Step 4

Step $1$

Step $2$

Step $3$

Step $4$

Exercise