Exercise 58 Page 272

We want to draw the graph of the given quadratic function. Note that the function is already written in vertex form

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

where

a,

h,

and

k

are either positive or negative numbers.

y = (x - 2)^{2} - 1

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.
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 : y = a (x - h)^{2} + k y = 1 (x - 2)^{2} + (- 1)

We can see that

a = 1,

h = 2,

and

k = - 1 .

Since

a

is greater than

0,

the parabola will open upwards.

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 $(2, - 1) .$ 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. Let's try

x = 4 .

y = (x - 2)^{2} - 1

Substitute

$x = 4$

y = (4 - 2)^{2} - 1

▼

Simplify right-hand side

SubTerm

Subtract term

y = 2^{2} - 1

CalcPow

Calculate power

y = 4 - 1

SubTerm

Subtract term

y = 3

When

x = 4

we have

y = 3 .

Thus, the point

(4, 3)

lies on the curve. Let's plot this point and reflect it across the axis of symmetry.

Note that both points have the same $y -$ coordinate.

Step $4$

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

Graph Description

The domain of quadratic functions is all real numbers. We can see above that the minimum point of the curve is reached at the vertex. Thus, the range is all real numbers greater than or equal to

- 1 .

Domain : Range : All real numbers y \geq - 1

By looking at the graph we can state the values for the

x -

and

y -

intercepts. We see that the parabola intercepts the

y -

axis at

y = 3,

and intercepts the

x -

axis at

x = 1

and

x = 3 .

y - intercept : x - intercepts : y = 3 x = 1 and x = 3

We know that the shape of quadratic function is a parabola, and in the given function it opens upwards. Thus, the function decreases on the interval

(- \infty, 2),

it reaches the vertex at

(2, - 1),

and then it increases on the interval

(2, \infty) .

Exercise 58 Page 272

Step 1

Step 2

Step 3

Step 4

Graph Description

Step $1$

Step $2$

Step $3$

Step $4$