y = -(x - 1)(x - 5): which form shows the highest point Step by Step

Step by step

The question

Question: which form of y = −(x − 1)(x − 5) shows the highest point?

y = −(x − 1)(x − 5)

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 highest 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) ✓

2 Open the brackets

Open the brackets: each part of the first times each part of the second.

x times x is x squared.

x times −5 is −5x.

−1 times x is −x.

−1 times −5 is 5.

(x − 1)(x − 5) = x² − 5x − x + 5

The two x parts are like terms.

Together they make −6x.

−5x − x = −6x

So y is −1 times (x² − 6x + 5).

y = −(x² − 6x + 5)

Now the −1 in front multiplies each part.

−1 × each part

So y = −x² + 6x − 5.

That is the standard form.

y = −x² + 6x − 5

3 Write it in vertex form

First we take −1 out of the two x parts.

Then x squared has no number in front.

The −5 stays outside.

−x² + 6x = −(x² − 6x)

So y is minus (x² − 6x), minus 5.

y = −(x² − 6x) − 5

We want a bracket squared, like (x minus something) squared.

Its number is half of the middle number, −6: −3.

The next line shows why.

half of −6: −6 ÷ 2 = −3

Open (x − 3) squared: x squared, −3x two times, and −3 times −3, that is 9.

It starts with x² − 6x, like ours.

But it has 9 more.

(x − 3)² = x² − 3x − 3x + 9 = x² − 6x + 9

So x² − 6x is (x − 3) squared, take away the extra 9.

x² − 6x = (x − 3)² − 9

We put that in place of x² − 6x.

y = −[(x − 3)² − 9] − 5

The −1 multiplies both parts in the big bracket: −1 times minus 9 is 9.

Then 9 minus 5 is 4.

−1 × (−9) = 9

9 − 5 = 4

So y = −(x − 3)² + 4.

That is the vertex form.

y = −(x − 3)² + 4

4 Read the highest point

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

(x − 3) squared is smallest, 0, when x − 3 is 0: at x = 3.

(x − 3)² is never below 0; it is 0 when x = 3

There the square part is 0.

So y is just 4.

The minus means we take the square away from 4.

Anywhere else we take away more than 0, so y is smaller.

So this is the highest point.

x = 3: y = −1 × 0 + 4 = 4

Careful!

Not −3.

x − 3 is 0 when x is 3: the sign in the bracket is the opposite.

Careful: not (x − 3)²: x = −3 ✗

So the highest point is (3, 4).

highest = (3, 4)

5 Show it on the graph

Here is the graph of y = −x² + 6x − 5.

The highest point is (3, 4).

6 Check

Let's check in the question's own form, y = −(x − 1)(x − 5).

x = 3 gives y = 4, the y of the highest point.

So it is right!

x = 3: y = −(3 − 1)(3 − 5) = −2 × (−2) = 4

Let's look back.

First: pick the form.

Then: open the brackets.

Then: write it in vertex form.

Then: read the highest point.

Then: show it on the graph.

Last: check.

Answer