Package of problems

## Solution

The equation of this graph is $y = x^2 - 8x + 10$

Is this graph symmetrical? Are you convinced? How can you convince someone else?

### Approach 1

It looks like the graph has a line of symmetry with equation $x = 4$.

If we can show that each point on the curve has a mirror image about the line $x=4$, then we will be correct.

Let’s choose $x$ coordinates such that they are equidistant from the line $x=4$, e.g. $x=4+k$ and $x=4-k$.

When $x = 4 + k$, $\quad y = (4 + k)^2 - 8(4 + k) + 10 = k^2 - 6$.

When $x = 4 - k$, $\quad y = (4 - k)^2 - 8(4 - k) + 10 = k^2 - 6$.

The $y$ coordinates are equal. This shows we are correct and the graph $y = x^2 - 8x + 10$ is symmetrical about the line $x = 4$.

Alternatively, we can perform the same calculation with the completed square form. What do you notice?

What did we do in this approach? We made a conjecture (an educated guess) and then went on to prove that this was correct. This is a fundamental process in doing mathematics.

### Approach 2

It looks like the graph of $y = x^2 - 8x + 10$ has the same shape as $y = x^2$ but is in a different position.

If we can show that $y = x^2$ is symmetrical and we can translate the graph $y = x^2$ to $y = x^2 - 8x + 10$, then we have shown that $y = x^2 - 8x + 10$ is symmetrical. We do this in two steps.

Step 1: Show that the graph $y = x^2$ is symmetrical about the $y$ axis.

When $x = k$, $\quad \phantom{-}y = k^2$.

When $x = -k$, $\quad y = k^2$.

By a similar argument to Approach 1 above, the graph of $y = x^2$ is symmetrical about the $y$ axis.

Note: A function where $f(x) = f(-x)$ for all $x$ is called an even function and is symmetrical about the $y$ axis. There is more on this topic in Odd or even or ….

Step 2: Rewrite $y = x^2 - 8x + 10$ as a translation of $y = x^2$.

By completing the square, we have $y=x^2 - 8x + 10 = (x - 4)^2 - 6$ which is a translation of the graph $y = x^2$ by $\dbinom{4}{-6}$. The line of symmetry is therefore translated to $x = 4$ and the vertex is translated to ($4,-6$).

Therefore the graph $y = x^2 - 8x + 10$ is symmetrical.

How do these approaches compare?

Which is the more easily generalised?

How would these approaches help us to consider whether all quadratics are symmmetrical?