Problem 10 - College Algebra

Question

For the following exercises, find the \(x\) - and \(y\) -intercepts of the graphs of each function. $$ f(x)=-3|x-2|-1 $$

Step-by-Step Solution

Answer

No x-intercept; y-intercept is at (0, -7).

Step 1: Identify the x-intercept

To find the x-intercept, we set the function equal to zero and solve for \(x\). This means we solve the equation \(-3|x-2|-1 = 0\).

Step 2: Solve for x

Starting from \(-3|x-2|-1 = 0\), first add 1 to both sides to get \(-3|x-2| = 1\). Then, divide each side by \(-3\) to isolate the absolute value, resulting in \(|x-2| = -\frac{1}{3}\).

Step 3: Interpret the solution for x

Since the absolute value expression \(|x-2|\) cannot equal a negative number, there is no solution for \(x\). Therefore, there is no x-intercept for the function.

Step 4: Identify the y-intercept

To find the y-intercept, we set \(x = 0\), and find \(f(0)\).

Step 5: Calculate f(0)

Substitute \(0\) into the function: \(f(0) = -3|0-2|-1 = -3|2|-1\). Simplify inside the absolute value to get \(-3(2)-1\).

Step 6: Simplify the function

Simplify \(-3(2)-1\) to get \(-6-1 = -7\). Therefore, the y-intercept is \(f(0) = -7\).

Key Concepts

Absolute Value Function

The absolute value function is a fascinating concept in mathematics. It revolves around the idea of the absolute value of a number, which is its non-negative value regardless of sign.
For instance, the absolute value of both -3 and 3 is 3, as it measures only the magnitude of the number and not its direction on the number line.
In our function, we have \( f(x) = -3|x-2|-1 \), where \(|x-2|\) represents the absolute value.
This part of the function transforms the input \(x\) into a distance from 2.

• It disregards whether \(x-2\) is positive or negative, simplifying \(x - 2\) to always output a positive number or zero.
• This property makes the absolute value graphs typically V-shaped, with a vertex where the expression inside equals zero.

When studying absolute value functions, it's important to understand that any output cannot be negative.
Thus, any equation involving an absolute value, such as \( |x-2| = -\frac{1}{3} \), has no solution because the absolute value cannot equal a negative number.

X-Intercept

Identifying an x-intercept involves finding where the graph crosses the x-axis. At this point, the y-value is zero.
For any function, to find the x-intercept, we set the function equal to zero and solve for \(x\).In our example with the function \( f(x) = -3|x-2|-1 \),
we start by setting the function to zero: \(-3|x-2|-1 = 0\).

• First, we add 1 to both sides, giving \(-3|x-2| = 1\).
• Then, divide each side by -3, yielding \(|x-2| = -\frac{1}{3}\).
• Since absolute values can't be negative, no solution exists.

This means that there is no x-intercept for \( f(x) = -3|x-2|-1 \). The absolute value condition limits the possible solutions since they cannot be negative, highlighting the unique behavior of these functions compared to linear options.

Y-Intercept

The y-intercept is the point where the graph crosses the y-axis. Here, the x-value is zero.
To find this intercept, substitute \(x = 0\) into the function and solve for \(y\).For our given function \( f(x) = -3|x-2|-1 \):

• Replace \(x\) with 0, resulting in \( f(0) = -3|0-2|-1 \).
• Simplify inside the absolute value: \( |0-2| = 2\), hence the equation becomes \(-3(2)-1\).
• Further simplify to \(-6-1\), equaling \(-7\).

Thus, the y-intercept of the graph is \( f(0) = -7 \).
This point on the graph is (0, -7), showing where the graph crosses the y-axis, which helps understand the vertical positioning within the coordinate grid.

Solving Equations

Solving equations is a fundamental skill in mathematics. It involves finding the values of variables that make the equation true.
Different types of equations, such as linear, quadratic, and absolute value, have unique methods for finding solutions.In the context of the given function \( f(x) = -3|x-2|-1 \), solving equations helps identify the intercepts. Here's a brief rundown:

• To find the x-intercept, set \( f(x) = 0\) and attempt to solve for \(x\).
• If a solution exists, it indicates where the function crosses the x-axis.
• For the absolute value function given, the solution process highlights that \(|x-2|\) being unable to equal a negative number leads to no x-intercept.
• Conversely, finding the y-intercept involves substituting \(x = 0\) into the function and simplifying to find the y-value for the point (0, y).

By practicing solving equations, students gain proficiency in handling different types of mathematical problems.
Understanding each type's nuances, like the no-solution case shown in absolute value equations, strengthens problem-solving skills across various contexts.