Quadratic symmetry

Quadratic symmetry

Excerpt: Are all quadratic graphs symmetrical? Students are asked to come up with a convincing argument to answer this question.

Problem

Are all quadratic graphs symmetrical?

Take a look at the graph below.

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

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

Suggestion

How would you define symmetrical in this context?

If the graph has a line of symmetry, what is it? How can we use this idea?

Does the graph look similar to any other graph you know? How can we use this idea?

Solution

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?

By completing the square, we have $y=(x−4)^2−6=(4−x)^2−6$. From this, we can see the $y$ coordinates of two points equidistant from $x=4$ must be the same and the quadratic graph is symmetrical about the line $x=4$.

Completed square form approach

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?