91Ó°ÊÓ

The vertical intercept of a function, often referred to as the y-intercept, is the point where the graph of the function crosses the y-axis. To find this, you simply substitute 0 for every instance of the variable \( x \) in your function's equation. The result you obtain after computing is the y-coordinate of the intercept; the x-coordinate will always be 0, as this is the nature of the y-axis.

In the exercise you provided, the function is \( f(x) = -2|x+1| + 6 \). When substituting 0 for \( x \), the calculation becomes:

• \( f(0) = -2|0+1| + 6 \)
• \( f(0) = -2(1) + 6 \)
• \( f(0) = -2 + 6 \)
• \( f(0) = 4 \)

Thus, the y-intercept, where the function crosses the y-axis, is the coordinate point \( (0,4) \). This means that at \( x = 0 \), the value of the function, \( f(x) \), is 4.

The horizontal intercept of a function, known as the x-intercept, is where the graph crosses the x-axis. To find the x-intercept, you need to set the entire function equal to zero, because at this point, the value of the function (or \( y \)-coordinate) is zero. Solve for \( x \) to determine where these intercepts occur.

In the given function, \( f(x) = -2|x+1| + 6 \), setting \( f(x) = 0 \) yields the equation:

• \( 0 = -2|x+1| + 6 \)
• \( -2|x+1| = -6 \)
• \( |x+1| = 3 \)

This opens up two possibilities due to the absolute value:

• \( x+1 = 3 \) leading to \( x = 2 \)
• \( x+1 = -3 \) leading to \( x = -4 \)

So the function has two x-intercepts: \( (2, 0) \) and \( (-4, 0) \). These occur because the absolute value function allows the quantity inside to be either a positive or a negative value.

An absolute value function is characterized by the expression \( |x| \), which represents the distance of \( x \) from zero on the number line, making it always non-negative. The graph of an absolute value function generally exhibits a V-shape, and it has unique properties and outcomes on the intercepts.

The function in this exercise, \( f(x) = -2|x+1| + 6 \), modifies an absolute value function. This specific function involves:

• A vertical stretch by a factor of 2, due to the \(-2\) multiplier, which flips it over the x-axis.
• A horizontal shift to the left by 1 unit, because of \( x+1 \) inside the absolute value.
• A vertical shift upwards by 6 units.

The combination of these transformations causes the V-shaped graph to be inverted and shifted, ensuring changes in where it crosses the axes compared to a standard \( |x| \) function. Understanding these transformations is crucial in graphing and analyzing the function's behavior.