Understanding Graph Symmetry with Respect to the X-Axis: Explained with Examples and Equations

symmetric with the x-axis

When we say that a graph is symmetric with respect to the x-axis, it means that if we were to fold the graph in half along the x-axis, the two halves would coincide with each other

When we say that a graph is symmetric with respect to the x-axis, it means that if we were to fold the graph in half along the x-axis, the two halves would coincide with each other. In other words, if we pick a point on the graph that is above the x-axis, there would be an identical point below the x-axis at the same distance.

To determine if a graph is symmetric with the x-axis, we can look at the equation of the graph. Suppose we have a function represented by the equation y = f(x). If the graph is symmetric with the x-axis, it means that replacing y with -f(x) in the equation will result in the same equation.

Let’s take an example to understand this concept better. Consider the function y = x^2. We can plot some points on the graph of this function:

For x = -2, y = (-2)^2 = 4.
For x = -1, y = (-1)^2 = 1.
For x = 0, y = (0)^2 = 0.
For x = 1, y = (1)^2 = 1.
For x = 2, y = (2)^2 = 4.

Plotting these points, we get a symmetrical curve that is a U-shape and opens upwards. If we fold the graph along the x-axis, the left and right sides would match perfectly.

Now let’s test if y = x^2 satisfies the condition for symmetry with respect to the x-axis. Replacing y with -f(x), we get -f(x) = -(x^2) = -x^2.

Comparing this with the original equation y = x^2, we can see that they are the same. This confirms that the graph of y = x^2 is symmetric with the x-axis.

In summary, a graph is symmetric with the x-axis if replacing y with -f(x) in the equation results in the original equation. This symmetry can be observed by folding the graph along the x-axis, with the left and right sides coinciding.

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