Problem 9 - College Algebra with Corequisite Support

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

The function has no x-intercepts and y-intercept at (0, 16).

Step 1: Find the x-intercept

To find the x-intercept, we need to set the function equal to zero and solve for x, because the x-intercept occurs when y = 0. Therefore, set \(f(x) = 0\):\[ 4|x - 3| + 4 = 0 \]Subtract 4 from both sides:\[ 4|x - 3| = -4 \]Divide by 4:\[ |x - 3| = -1 \]Since an absolute value cannot be negative, there is no solution. This means the function has no x-intercepts.

Step 2: Find the y-intercept

To find the y-intercept, we set \(x = 0\) and solve for \(f(x)\), since the y-intercept occurs when x = 0. Substitute \(x = 0\) into the function:\[ f(0) = 4|0 - 3| + 4 \]Simplify the expression:\[ = 4 imes 3 + 4 \]\[ = 12 + 4 \]\[ = 16 \]So, the y-intercept is \((0, 16)\).

Key Concepts

Absolute Value Function

An absolute value function is a special kind of function in mathematics that's represented as \( f(x) = a|x - h| + k \). Here, the symbols \(a\), \(h\), and \(k\) are constants which affect the shape and position of the graph.
The main feature of an absolute value function is its 'V' shape. This shape occurs because the absolute value operator \(|x|\) changes any negative values to positive. Thus, it makes all outputs at least zero or more.
Understanding absolute values involves two key ideas:

• The absolute value of a number is always nonnegative.
• The expression inside the absolute value signs, for example, \(x - 3\) in our function, affects the position of the 'V' vertex on the graph.

In the given function \(f(x) = 4|x-3| + 4\), the graph is shifted 3 units to the right due to \(x - 3\). The entire graph is also shifted up by 4 units because of the \(+ 4\) at the end. The coefficient 4 in front of the absolute value stretches the 'V' shape vertically, making it steeper compared to a basic \( |x|\) graph.

Graphing Functions

Graphing functions is all about visualizing mathematical expressions on a coordinate plane. By plotting points and drawing curves or lines, one can understand how the function behaves.
The shape of the graph can tell us a lot about the function, as observed with the absolute value function \(f(x) = 4|x-3| + 4\). Here, due to the absolute value, the graph takes on a 'V' shape with a vertex at the point \((3, 4)\). This vertex comes from setting the expression inside the absolute value \(x-3 = 0\), solving to get \(x = 3\), then substituting back to find the corresponding \(y\)-value.
Understanding the graph of an absolute value function involves:

• Identifying the vertex, which is the turning point of the graph.
• Determining the direction and steepness of the 'V' shape, influenced by the coefficient of the absolute value term.
• Recognizing any shifts left/right or up/down, as determined by the terms \(h\) and \(k\).

This visualization helps in making sense of intercepts and the overall geometry of functions.

Intercepts of Functions

Intercepts are fundamentally important for understanding graphs of functions as they show where the graph crosses the axes.
There are two main types of intercepts:

• x-intercepts: The points where the graph crosses the x-axis. These occur where \(y = 0\).
• y-intercepts: The points where the graph crosses the y-axis. These occur where \(x = 0\).

For the function \(f(x) = 4|x-3| + 4\), we discovered:

• No x-intercept: By substituting \(f(x) = 0\), it was shown that the equation \(4|x-3| = -4\) has no solution because absolute values are nonnegative and cannot equal a negative number.
• Y-intercept at (0, 16): Substituting \(x = 0\) into the function finds \(f(0) = 16\). Thus, the graph crosses the y-axis at \((0, 16)\).

Finding intercepts can provide key points for sketching graphs and understanding where they interact with the coordinate axes.