y = (x - 2)^2 - 9: which form shows where it crosses the y axis Step by Step

Step by step

The question

Question: which form of y = (x − 2)² − 9 shows where it crosses the y axis?

y = (x − 2)² − 9

1 Pick the form

A parabola can be written in three forms.

Each one shows one thing at a glance.

On the y axis, x is 0.

In the standard form, x = 0 makes every x part 0, so c is left.

So the standard form shows where it crosses the y axis.

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)

2 Open the brackets

Squared means times itself.

Open the brackets: x squared, −2x two times, and −2 times −2, that is 4.

Together: x² − 4x + 4.

(x − 2)² = (x − 2)(x − 2) = x² − 2x − 2x + 4 = x² − 4x + 4

So y is (x² − 4x + 4), minus 9.

y = (x² − 4x + 4) − 9

The plain numbers: 4 minus 9 is −5.

4 − 9 = −5

So y = x² − 4x − 5.

That is the standard form.

y = x² − 4x − 5

3 Read the y axis crossing

On the y axis, x is 0.

With x = 0, every x part is 0.

Only the plain number is left: −5.

on the y axis, x = 0: y = 0² − 4 × 0 − 5 = −5

Careful!

Not (−5, 0).

A point is written x first, then y.

Here x is 0 and y is −5.

Careful: not (−5, 0) ✗

So it crosses the y axis at (0, −5).

point = (0, −5)

4 Show it on the graph

Here is the graph of y = x² − 4x − 5.

It crosses the y axis at (0, −5).

5 Check

Let's check in the question's own form, y = (x − 2)² − 9.

x = 0 gives y = −5: the point on the y axis.

So it is right!

x = 0: y = (0 − 2)² − 9 = (−2)² − 9 = −5

Let's look back.

First: pick the form.

Then: open the brackets.

Then: read the y axis crossing.

Then: show it on the graph.

Last: check.

Answer