Problem 9 - College Algebra

Question

For the following exercises, find the \(x\) - and \(y\) -intercepts of the graphs of each function. $$ f(x)=4|x-3|+4 $$

Step-by-Step Solution

Answer

Y-intercept: (0, 16); No x-intercepts.

Step 1: Understand the Function

The function given is \( f(x) = 4|x-3| + 4 \). It's an absolute value function, which typically forms a V-shape graph in the coordinate plane.

Step 2: Find the Y-Intercept

To find the y-intercept, set \( x = 0 \) in the function and solve for \( f(x) \). This gives \( f(0) = 4|0-3| + 4 = 4|3| + 4 = 12 + 4 = 16 \). Thus, the y-intercept is \( (0, 16) \).

Step 3: Find the X-Intercepts

For the x-intercepts, set \( f(x) = 0 \) and solve for \( x \). This means solving the equation \( 4|x-3| + 4 = 0 \). Simplifying gives \( 4|x-3| = -4 \). This is not possible since the absolute value term \( 4|x-3| \) is always non-negative. Therefore, the function has no x-intercepts.

Key Concepts

Absolute Value Function

An absolute value function is characterized by its distinctive V-shaped graph. These functions take the form of \( f(x) = a|x-h| + k \), where \( a \), \( h \), and \( k \) are constants. The absolute value, denoted by the vertical bars, \(|x-h|\), transforms any negative values of its argument into positive ones.

This unique feature causes the function to reflect over the line \( x = h \), which acts as the vertex of the V shape. The vertex is the point where the function changes direction. The graph is symmetric around this vertex.

In our specific case with the function \( f(x) = 4|x-3| + 4 \):

• The vertex is at the point \( (3, 4) \), shifted 3 units to the right and 4 units up.
• The slope on either side of the vertex is controlled by the coefficient 4, ensuring a steeper V shape.

Understanding the structure of absolute value functions is crucial for analyzing their intercepts and behavior.

X-Intercept

The x-intercept(s) of a function are the points where the graph crosses the x-axis. Mathematically, this occurs when the output of the function is zero. In simpler terms, to find these intercepts, we set \( f(x) = 0 \) and solve for \( x \).

For the absolute value function given, \( f(x) = 4|x-3| + 4 \), finding the x-intercept involves solving:

• \( 4|x-3| + 4 = 0 \)

This simplifies to \( 4|x-3| = -4 \). However, absolute values cannot equal negative numbers because they represent distances from zero, which are always non-negative.

Thus, no real solutions exist for this equation, meaning this function does not intersect the x-axis at any point. Hence, there are no x-intercepts for this absolute value function.

Y-Intercept

The y-intercept of a function is the point where the graph crosses the y-axis. This occurs when the input value, \( x \), is zero. Finding the y-intercept means calculating \( f(0) \).

For the function \( f(x) = 4|x-3| + 4 \), substituting \( x = 0 \) yields:

• \( f(0) = 4|0-3| + 4 \)
• This results in \( f(0) = 4 imes 3 + 4 \)
• Simplifying gives \( f(0) = 12 + 4 = 16 \)

Therefore, the y-intercept of the function is at the point \( (0, 16) \).

This intercept represents the function's vertical height when it crosses the y-axis, important for understanding the initial output of the function.