91Ó°ÊÓ

Y-axis symmetry in calculus refers to a situation where the graph of a function remains unchanged when reflected across the y-axis. This means that if you fold the graph along the y-axis, both halves will match perfectly. To check for y-axis symmetry, replace every instance of \(x\) with \(-x\) in the equation and simplify. If the new equation simplifies back to the original one, then the function is symmetric with respect to the y-axis.

For example, consider the equation \(y = 4 - \sqrt{x+3}\). By substituting \(x\) with \(-x\), we get \(y_1 = 4 - \sqrt{-x+3}\). In this case, \(y\) is not equal to \(y_1\), indicating that the function does not have y-axis symmetry.

This type of symmetry typically appears in even functions. Common examples are parabolas that open upwards or downwards, like \(y = x^2\). However, our specific example doesn't fit this pattern, which is why it's not symmetric about the y-axis.

X-axis symmetry is when a function's graph remains unchanged if it were flipped vertically around the x-axis. This would mean that for every point \((x, y)\) on the graph, the point \((x, -y)\) also exists. To determine if a graph is symmetric about the x-axis, you replace \(y\) with \(-y\) in the function and solve for symmetry.

In our given example \(y = 4 - \sqrt{x+3}\), modifying the equation to check for x-axis symmetry involves substituting \(y\) with \(-y\). The resulting function \(-y = 4 - \sqrt{x+3}\) does not simplify back to the original equation \(y = 4 - \sqrt{x+3}\). Therefore, the function does not have x-axis symmetry either.

Often, functions that are symmetric with respect to the x-axis are not functions in the traditional sense, because they can fail the vertical line test. An example of a curve with x-axis symmetry would be a circle, such as \(x^2 + y^2 = r^2\). Our function does not share this property.

Origin symmetry occurs if a graph remains unchanged when rotated 180 degrees around the origin, essentially meaning every point \((x, y)\) has a counterpart \((-x, -y)\) on the graph. To test this, substitute both \(x\) and \(y\) with their negatives in the equation and see if it still holds true.

For our function \(y = 4 - \sqrt{x+3}\), replacing \(x\) with \(-x\) and \(y\) with \(-y\) gives us \(-y = 4 - \sqrt{-x+3}\). As \(-y\) does not equal the original \(y\) expression, the function is not symmetric about the origin.

Origin symmetry is common in odd functions, such as \(y = x^3\) or linear equations like \(y = x\). These situations exhibit the unique property whereby rotating the entire graph 180 degrees maps it onto itself. Unfortunately, the structure of our function does not support this type of symmetry either.