Problem 14 - College Algebra with Corequisite Support

Question

For the following exercises, find the \(x\) - and \(y\) -intercepts of the graphs of each function. \(f(x)=|-2 x+1|-13\)

Step-by-Step Solution

Answer

The x-intercepts are \(x = -6\) and \(x = 7\). The y-intercept is \(y = -12\).

Step 1: Find the x-intercept(s)

To find the x-intercept(s) of the function, set \( f(x) = 0 \). This occurs when \( |-2x + 1| - 13 = 0 \).First, solve the equation: \[ |-2x + 1| - 13 = 0 \rightarrow |-2x + 1| = 13 \]Thus, we have two equations to solve due to the absolute value:1. \(-2x + 1 = 13\)2. \(-2x + 1 = -13\)Solving each separately:1. \(-2x + 1 = 13 \rightarrow -2x = 12 \rightarrow x = -6\)2. \(-2x + 1 = -13 \rightarrow -2x = -14 \rightarrow x = 7\)So, the x-intercepts are \( x = -6 \) and \( x = 7 \).

Step 2: Find the y-intercept

To find the y-intercept of the function, set \( x = 0 \) in the equation and solve for \( f(x) \).Plug \( x = 0 \) into the function:\[f(0) = |-2(0) + 1| - 13 = |1| - 13 = 1 - 13 = -12\]Therefore, the y-intercept is \( f(0) = -12 \).

Key Concepts

x-intercepts

When working with algebraic functions, finding the x-intercepts is essential. The x-intercepts of a function represent the points on the graph where the function crosses the x-axis. In other words, these are the values of \( x \) that make the function value (or \( f(x) \)) equal to zero.

In this exercise, the function given is \( f(x) = |-2x + 1| - 13 \). To find the x-intercepts, set the function equal to zero and solve for \( x \):

• Start with \( |-2x + 1| - 13 = 0 \).
• Simplify to \( |-2x + 1| = 13 \).
• Since you have an absolute value equation, split it into two cases: \( -2x + 1 = 13 \) and \( -2x + 1 = -13 \).
• For the first case, solve \( -2x + 1 = 13 \), which simplifies to \( x = -6 \).
• For the second case, solve \( -2x + 1 = -13 \), leading to \( x = 7 \).

Therefore, the x-intercepts of the function are at \( x = -6 \) and \( x = 7 \). These points show where the function touches the x-axis on a graph.

y-intercepts

Finding the y-intercept of a function is quite straightforward. The y-intercept is where the graph intersects the y-axis, corresponding to \( x = 0 \). In essence, it is the value of the function when \( x \) is zero.

Let's apply this to the given function: \( f(x) = |-2x + 1| - 13 \). To find the y-intercept:

• Substitute \( x = 0 \) into the function.
• Calculate \( f(0) = |-2(0) + 1| - 13 = |1| - 13 \).
• This simplifies to \( 1 - 13 = -12 \).

Thus, the y-intercept is \( f(0) = -12 \). This value points out where the function graph crosses the y-axis, specifically at the point \( (0, -12) \).

absolute value equations

Absolute value equations require a special approach when solving, due to the nature of the absolute value operation itself, which always results in a non-negative number.

In the function \( f(x) = |-2x + 1| - 13 \), the absolute value is indicated by the vertical bars. To solve, we remove the absolute value by considering both possibilities:
- The expression inside the absolute value is equal to the positive of the number outside.- The expression inside is equal to the negative of that number.

• First equation: \( -2x + 1 = 13 \)
• Second equation: \( -2x + 1 = -13 \)

This technique ensures that both scenarios are taken into account, capturing all possible x-values that satisfy the equation.

Understanding this two-case method is crucial. It's handy when dealing with any problem involving absolute value equations, as it covers all potential solutions the variable could satisfy within the given context.