Problem 12 - College Algebra with Corequisite Support

Question

For the following exercises, find the \(x\) - and \(y\) -intercepts of the graphs of each function. \(f(x)=-5|x+2|+15\)

Step-by-Step Solution

Answer

The x-intercepts are (1, 0) and (-5, 0); the y-intercept is (0, 5).

Step 1: Find the x-intercepts

To find the x-intercepts, set the function equal to 0 and solve for x: \[ -5|x+2| + 15 = 0 \]This gives us: \[ |x+2| = 3 \] Which results in two scenarios: 1. \(x+2 = 3\) leading to \(x = 1\)2. \(x+2 = -3\) leading to \(x = -5\) Thus, the x-intercepts are \((1, 0)\) and \((-5, 0)\).

Step 2: Find the y-intercept

To find the y-intercept, evaluate the function at \(x = 0\): \[ f(0) = -5|0+2| + 15 \]Calculate the expression:\[ f(0) = -5(2) + 15 = -10 + 15 = 5 \]Therefore, the y-intercept is \((0, 5)\).

Key Concepts

Understanding the Absolute Value Function

The absolute value function is a vital concept in mathematics, often represented as \(|x|\). This function measures the distance a number is from zero on the number line, without considering which side of zero it is. For example, both \(|3|\) and \(|-3|\) equal 3 because each number is three units away from zero.

The absolute value function affects equations by changing how we consider them. Solving an equation involving absolute values typically results in two different scenarios:

• If \(|x+2| = 3\), this means the expression inside the absolute value sign, here \(x+2\), can either be 3 or -3. This is because the absolute value of both 3 and -3 is 3.

Understanding this concept influences our ability to solve problems that involve absolute values effectively.

Solving Equations Involving Absolute Values

When solving equations that involve absolute values, it's important to first isolate the absolute value expression and then consider each possible scenario separately. Take the equation \(-5|x+2| + 15 = 0\). Here, the goal is to solve for \(x\). By isolating the absolute value, we rearrange the equation to form \(|x+2| = 3\).

This equation

• requires solving two separate linear equations: \(x+2=3\) and \(x+2=-3\).
• For \(x+2=3\), subtract two from both sides to find \(x = 1\).
• For \(x+2=-3\), again subtract two from both sides to discover \(x = -5\).

These steps lead to the solutions for the x-intercepts \((1, 0)\) and \((-5, 0)\).
In this context, understanding how to handle equations with absolute values is crucial for finding accurate x-intercepts.

Graphical Interpretation of Functions

Interpreting functions graphically involves analyzing the shape and position of the graph for a given function. For functions involving absolute values, such as \(f(x) = -5|x+2| + 15\), the graph often forms a "V" shape. The vertex of this "V" indicates the lowest or highest point, depending on the function's orientation.Finding intercepts is part of understanding these graphs:

• X-intercepts occur where the graph crosses the x-axis, usually when the output of the function is zero. For our given function, these were found at points \((1,0)\) and \((-5,0)\).
• Y-intercepts occur where the graph intersects the y-axis. This is determined by evaluating the function at \(x = 0\). In our example, the y-intercept is \((0,5)\).

The graphical interpretation aids in visualizing how a function behaves and its key points, which helps deepen understanding of its algebraic properties.