Graph y = -1/2 x^2 from y = x^2 Step by Step

Step by step

The question

Question: graph y = minus one half times x², from y = x².

y = −12x²

We know the graph of y = x².

It is a U.

we know: y = x²

1 Read the changes

The one half in front makes every y one half as big.

Smaller y's: the graph gets wider and flatter.

12 in front → every y × 12

Careful!

A number bigger than 1 makes it narrower.

A number between 0 and 1 makes it wider.

Careful: a number between 0 and 1 in front: wider, not narrower

The minus in front changes the sign of every y.

So the graph flips upside down.

− in front → flip over the x-axis

2 Start from y = x²

Draw the grid.

Draw y = x² first.

It goes through (−2, 4), (0, 0) and (2, 4).

3 Shrink the y's

(−2, 4): minus one half of 4 is −2.

It goes to (−2, −2).

(0, 0) stays: one half of 0 is 0.

(2, 4): minus one half of 4 is −2.

It goes to (2, −2).

4 Draw the new graph

Join them with the same shape, wider and upside down.

This is the new graph.

5 Check

Let's check: put x = 2 in the new rule.

One half of 4 is −2.

We get the point (2, −2).

It works!

x = 2: y = −12(2)² = −12 × 4 = −2

Let's look back.

First: read the changes.

Then: start from y = x².

Then: shrink the y's.

Then: draw the new graph.

Last: check.

Answer