y = (x + 1)^2 - 4: which form shows the vertex Step by Step

Step by step

The question

Question: which form of y = (x + 1)² − 4 shows the lowest point?

y = (x + 1)² − 4

1 Pick the form

A parabola can be written in three forms.

Each one shows one thing at a glance.

The vertex form has a square, and a square is never below 0.

So it shows the lowest point.

standard: ax² + bx + c → y axis at c

factored: a(x − r)(x − s) → x axis at r and s

vertex: a(x − h)² + k → turning point (h, k) ✓

It is already in vertex form.

2 Read the lowest point

A square is never below 0: a number times itself is never minus.

(x + 1) squared is smallest, 0, when x + 1 is 0: at x = −1.

(x + 1)² is never below 0; it is 0 when x = −1

There the square part is 0.

So y is just −4.

Anywhere else we add a bit more than 0, so y is bigger.

So this is the lowest point.

x = −1: y = 0 − 4 = −4

Careful!

Not 1.

x + 1 is 0 when x is −1: the sign in the bracket is the opposite.

Careful: not (x + 1)²: x = 1 ✗

So the lowest point is (−1, −4).

lowest = (−1, −4)

3 Show it on the graph

Here is the graph of y = x² + 2x − 3.

The lowest point is (−1, −4).

4 Check

Let's check in the question's own form, y = (x + 1)² − 4.

x = −1 gives y = −4, the y of the lowest point.

So it is right!

x = −1: y = ((−1) + 1)² − 4 = 0² − 4 = −4

Let's look back.

First: pick the form.

Then: read the lowest point.

Then: show it on the graph.

Last: check.

Answer