Problem 8 - A Graphical Approach to College Algebra

Question

Give the (a) \(x\) -intercept, (b) \(y\) -intercept, (c) domain, (d) range, and (e) slope of the line. Do not use a calculator. $$f(x)=-0.5 x$$

Step-by-Step Solution

Answer

(a) x-intercept: (0, 0), (b) y-intercept: (0, 0), (c) domain: all real numbers, (d) range: all real numbers, (e) slope: -0.5.

Step 1: Find the x-intercept

To find the x-intercept, set \( f(x) \) to zero and solve for \( x \).\[0 = -0.5x \]Divide both sides by \(-0.5\):\[x = 0\]. Therefore, the x-intercept is \((0, 0)\).

Step 2: Find the y-intercept

To find the y-intercept, evaluate \( f(x) \) at \( x = 0 \).\[f(0) = -0.5 \times 0 = 0\] Thus, the y-intercept is \((0, 0)\).

Step 3: Determine the Domain

The domain of a linear function is all real numbers because there are no restrictions on \( x \). Hence, the domain is \((-\infty, \infty)\).

Step 4: Determine the Range

The range of a linear function is all real numbers as well because the output \( f(x) \) can take any real value. Therefore, the range is \((-\infty, \infty)\).

Step 5: Determine the Slope

The slope of the line can be identified directly from the function \( f(x) = -0.5x \). The coefficient of \( x \) is the slope, which is \(-0.5\).

Key Concepts

x-intercept

The **x-intercept** of a linear function is the point where the graph of the function crosses the x-axis. This means that at the x-intercept, the value of the function, or \(f(x)\), is zero. To find it, you set \(f(x)\) to zero and solve for \(x\). For the function \(f(x) = -0.5x\), setting \(0 = -0.5x\) and solving gives \(x = 0\). This tells us that the x-intercept is at the point \((0, 0)\). In simple terms, it's where the function touches the x-axis. Knowing this concept helps you better understand how graphs behave on a coordinate grid.

y-intercept

The **y-intercept** is where a graph crosses the y-axis. At this point, the value of \(x\) is always zero because the line touches the y-axis. To find the y-intercept of our function \(f(x) = -0.5x\), we evaluate the function at \(x = 0\). This gives us \(f(0) = -0.5 \times 0 = 0\). Hence, the y-intercept is at the point \((0, 0)\).

• The y-intercept is crucial in determining the starting point of a line on a coordinate plane.
• It helps us quickly understand how the line behaves as it intercepts the y-axis.

domain and range

When discussing linear functions, the **domain** and **range** are crucial aspects. The domain is the set of all possible \(x\)-values that you can input into the function. For linear functions like \(f(x) = -0.5x\), the domain is all real numbers because there is no restriction on which \(x\) values to use. Mathematically, this is written as \((-\infty, \infty)\). The **range** reflects all possible outputs of the function, given the domain. For our linear function, the range is also all real numbers, \((-\infty, \infty)\), since any real number can be a resultant value of \(f(x)\). Understanding domain and range helps you recognize how far and in what ways a line stretches across both the x-axis and y-axis.

slope of a line

The **slope of a line** in the linear equation \(f(x) = -0.5x\) is the value that determines how steep the line is. You can identify the slope as the coefficient of \(x\) in the function. In this case, it's \(-0.5\). A negative slope indicates that as you move from left to right on the graph, the line falls downward. Some important aspects of slopes:

• Positive slopes mean the line ascends as it moves from left to right.
• Zero slope results in a flat horizontal line.
• A steeper slope, either positive or negative, means a more dramatic rise or fall.

Understanding the slope helps in predicting how the function behaves and what kind of changes occur in its graphical representation.