When we think about the symmetry of quadratic functions, it’s important to know how changes to the function can affect this feature.

A quadratic function is usually written like this:

$$f(x) = ax^2 + bx + c,$$

In this equation, the symmetry happens around a vertical line called the axis of symmetry. We can find this axis using the formula:

$$x = -\frac{b}{2a}.$$

Let’s explore how different changes to the function can affect this symmetry.

1. Vertical Changes

When we move a quadratic function up or down (this is called a vertical shift), the axis of symmetry does not change.

For example, if we start with

f(x) = x^2 $$ and move it up by 3 units, we get

g(x) = x^2 + 3. $$

Both functions still have the same axis of symmetry at $x = 0$.

2. Horizontal Changes

But when we move a function left or right (this is called a horizontal shift), the axis of symmetry changes its position.

If we start with

f(x) = x^2 $$ and move it to the right by 2 units, we write it as:

h(x) = (x - 2)^2. $$

Now, the axis of symmetry has moved to $x = 2$. This shows how horizontal shifts change the symmetry of the function.

3. Flipping and Stretching

Flipping and stretching the function can also affect its symmetry. For example, if we flip it over the x-axis, we change

f(x) = x^2 $$ to

k(x) = -x^2. $$

Even though it is still symmetric, the shape of the graph points downward now.

If we stretch the function vertically by a factor of 3, changing

f(x) = x^2 $$ to

m(x) = 3x^2, $$

the axis of symmetry stays at $x = 0$, but the graph looks narrower.

In conclusion, vertical changes keep the symmetry the same, while horizontal shifts change where the axis of symmetry is. Flipping changes how the graph points, and stretching alters the shape but not the vertical line of symmetry. Knowing how these changes work helps us understand and graph quadratic functions better.