y = (x - 2)(x - 6): which form shows where it crosses the x axis Step by Step

Step by step

The question

Question: which form of y = (x − 2)(x − 6) shows where it crosses the x axis?

y = (x − 2)(x − 6)

1 Pick the form

A parabola can be written in three forms.

Each one shows one thing at a glance.

On the x axis, y is 0.

The factored form is a product, and a product is 0 when one bracket is 0.

So the factored form shows where it crosses the x 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)

It is already in factored form.

2 Read the x axis crossings

On the x axis, y is 0.

So the product must be 0.

on the x axis, y = 0: (x − 2)(x − 6) = 0

A product is 0 only when one of its parts is 0.

x − 2 is 0 when x = 2.

x − 6 is 0 when x = 6.

A × B = 0 → A = 0 or B = 0

x − 2 = 0 → x = 2

x − 6 = 0 → x = 6

Careful!

Not −2.

x − 2 is 0 when x is 2: the sign in the bracket is the opposite.

Careful: not (x − 2): x = −2 ✗

So it crosses the x axis at x = 2 and x = 6: the points (2, 0) and (6, 0).

points = (2, 0) and (6, 0)

3 Show it on the graph

Here is the graph of y = x² − 8x + 12.

Each square up is 2.

It crosses the x axis at 2 and 6.

4 Check

Let's check in the question's own form, y = (x − 2)(x − 6).

x = 2 gives y = 0: on the x axis.

x = 2: y = (2 − 2)(2 − 6) = 0 × (−4) = 0

And x = 6 gives y = 0: on the x axis.

So it is right!

x = 6: y = (6 − 2)(6 − 6) = 4 × 0 = 0

Let's look back.

First: pick the form.

Then: read the x axis crossings.

Then: show it on the graph.

Last: check.

Answer