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In the realm of functions and their graphs, finding the x-intercepts is a vital step in understanding the behavior of the function. The x-intercepts are the points where the graph of the function crosses the x-axis. This means that at these points, the y-coordinate is zero. To find the x-intercepts, we need to set the function equal to zero and solve for the variable, usually denoted as \(x\).

In our exercise, the function given is \(f(x) = -|x-9| + 16\). To find the x-intercepts, you substitute \(f(x)\) with 0 and solve:

• Start with the equation: \(0 = -|x-9| + 16\)
• Rearrange to isolate the absolute value: \(|x-9| = 16\)

Since the absolute value can be either positive or negative, you will have two separate equations:

• \(x - 9 = 16\)
• \(x - 9 = -16\)

Solving these gives the x-intercepts:

• For \(x - 9 = 16\), \(x = 25\)
• For \(x - 9 = -16\), \(x = -7\)

Thus, the x-intercepts are at the coordinates \((-7, 0)\) and \((25, 0)\). These points help you understand where the graph will touch or cross the x-axis.

The y-intercept of a function is another fundamental aspect of analyzing graphs. This is the point where the graph crosses the y-axis, which occurs when \(x = 0\). In simple terms, it is the value of the function when the input is zero.

To find the y-intercept for our function \(f(x) = -|x-9| + 16\), you would substitute \(x\) with 0:

• Evaluate \(f(0) = -|0-9| + 16\)
• Simplify this to: \(-|-9| + 16\)
• Become: \(-9 + 16\)

The result is \(7\), which means the y-intercept is at the point \((0, 7)\). This point indicates where the graph of the function intersects the y-axis, providing insight into the function's initial value when no other input is given.

Understanding the absolute value function is key in analyzing expressions like \(-|x-9| + 16\). The absolute value of a number is its distance from zero, regardless of direction on a number line. Therefore, it is always represented as a non-negative number.

In our function, the absolute value component \(|x-9|\) shifts the graph horizontally by 9 units. This operation means we consider both \(x-9\) and \(-x+9\), leading to a reflection effect that results in a "V-like" shape characteristic of absolute value functions. By manipulating this shape with a factor or by adding or subtracting a constant, you can move the "V" up, down, left, or right on a graph.

For \(f(x) = -|x-9| + 16\), the graph is translated upwards by 16 units because of the "+16". The negative sign before the absolute value flips the graph over the horizontal axis, turning the "V" shape upside down.

When analyzing absolute value functions, always remember:

• The point at which the variable is zero inside the absolute value determines the vertex (in this case \(x=9\));
• Positive absolute value coefficients stretch the graph, while negative create a reflection.

Absolute value functions provide valuable information about symmetry and how functions can be transformed.