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The x-intercepts of a graph are the points where the graph crosses the x-axis. To find these points, you need to set the y-value in the equation equal to zero, since every point along the x-axis has a y-coordinate of zero.

In our given equation, which is \( y = 16 - x^4 \), we find the x-intercepts by setting \( y = 0 \). This gives us the equation:

• \( 16 - x^4 = 0 \)

Solve for \( x \) by rearranging the equation to get \( x^4 = 16 \). To isolate \( x \), take the fourth root of both sides:

• \( x = \pm 2 \)

This means the x-intercepts occur where the graph crosses the x-axis at the points (2, 0) and (-2, 0).

Remember, x-intercepts serve as an important feature when graphing polynomial functions, as they signal where the graph changes from positive to negative values or vice versa.

The y-intercept of a graph is the point where the graph intersects the y-axis. At this intersection, the x-coordinate is always zero. To find the y-intercept, substitute \(x = 0\) into the equation of the function.

For the function \(y = 16 - x^4\), set \(x = 0\), resulting in:

• \(y = 16 - (0)^4 = 16\)

This calculation indicates that the y-intercept of the graph is at the point (0, 16).

Finding the y-intercept helps with quickly sketching the graph, providing a reliable starting point and anchor for the curve of the function.

Symmetry in graphs signifies that the graph is a mirror image over a particular line. In polynomial functions, we frequently check for symmetry about the y-axis and the origin.

In this case, to test for symmetry about the y-axis, we replace \(x\) with \(-x\) in the function and evaluate if \(f(-x) = f(x)\). For the equation \(y = 16 - x^4\), substituting \(-x\) yields:

• \(f(-x) = 16 - (-x)^4 = 16 - x^4\), which simplifies to \(f(x)\)

As a result, since \(f(-x) = f(x)\), the function is symmetrical with respect to the y-axis.

This symmetry is visually reflected as you graph it, ensuring that if a point \((a, b)\) lies on the graph, its symmetric counterpart \((-a, b)\) will also be present. It simplifies understanding and sketching as it allows predicting the shape without plotting every point explicitly.