Problem 5 - 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. $$f(x)=-\frac{2}{5} x+2$$

Step-by-Step Solution

Answer

The x-intercept is 5, the y-intercept is 2, the domain is all real numbers, the range is all real numbers, and the slope is -2/5.

Step 1: Identify the x-intercept

To find the x-intercept, we set \( f(x) = 0 \) and solve for \( x \). \[ 0 = -\frac{2}{5}x + 2 \]Add \( \frac{2}{5}x \) to both sides:\[ \frac{2}{5}x = 2 \]Multiply both sides by \( \frac{5}{2} \) to solve for \( x \):\[ x = 5 \times \frac{2}{2} = 5 \]Thus, the \( x \)-intercept is \( x = 5 \).

Step 2: Identify the y-intercept

To find the y-intercept, we set \( x = 0 \) in \( f(x) \):\[ f(0) = -\frac{2}{5}(0) + 2 = 2 \]This means the \( y \)-intercept is \( y = 2 \).

Step 3: Determine the Domain

The function \( f(x) = -\frac{2}{5}x + 2 \) is a linear function, which means it is defined for all real numbers. Thus, the domain is all real numbers: \((-fty, fty)\).

Step 4: Determine the Range

Since the function \( f(x) = -\frac{2}{5}x + 2 \) is linear and doesn't have restrictions, its range is also all real numbers: \((-fty, fty)\).

Step 5: Find the Slope

The slope of a linear function, given in the form \( f(x) = mx + b \), is the coefficient of \( x \). For this function, \( m = -\frac{2}{5} \). This means the slope is \(-\frac{2}{5}\).

Key Concepts

x-intercept

To find the x-intercept of a linear function like \( f(x) = -\frac{2}{5} x + 2 \), set \( f(x) \) equal to zero and solve for \( x \). The x-intercept is where the line crosses the x-axis. In this position, the value of \( y \) or \( f(x) \) is zero.

Begin by writing the equation:

• \( 0 = -\frac{2}{5} x + 2 \)
• Add \( \frac{2}{5} x \) to both sides to isolate terms involving \( x \): \( \frac{2}{5} x = 2 \)

Multiply both sides by \( \frac{5}{2} \) to solve for \( x \):

• \( x = 5 \)

Therefore, the x-intercept of this line is \( x = 5 \), which means the line crosses the x-axis at (5,0). This point is vital as it helps understand where the line begins on the x-axis.

y-intercept

Finding the y-intercept involves setting \( x = 0 \) and solving for \( f(x) \), which is essentially the \( y \)-value where the line crosses the y-axis. It is a straightforward calculation. Plug \( x = 0 \) into the function:

• \[ f(0) = -\frac{2}{5}(0) + 2 = 2 \]

Thus, the y-intercept is \( y = 2 \), indicating the line intersects the y-axis at (0,2). This point is crucial because it determines where the line hits the y-axis. Y-intercepts help in understanding the initial value of the function when no independent variable is present.

domain and range

In a linear function like \( f(x) = -\frac{2}{5}x + 2 \), the domain and range are simple to identify, since linear functions are not restricted. The domain of a function refers to the set of possible input values (x-values), and the range is the set of possible output values (y-values). For most standard straight lines, both the domain and the range are all real numbers.

• Domain: All real numbers
• Range: All real numbers

In symbols, both the domain and the range of this function are represented as

• \((-\infty, \infty)\)

These uninterrupted domains and ranges reflect the continuity of linear functions, as the line extends indefinitely in both directions on the x and y axes.

slope

The slope of a linear function is a measure of its steepness and direction, and is denoted by \( m \) in the slope-intercept form \( f(x) = mx + b \). It signifies how much \( y \) changes for a given change in \( x \).

For the function \( f(x) = -\frac{2}{5}x + 2 \),

• the slope \( m \) is \(-\frac{2}{5} \)

This negative slope indicates that as \( x \) increases, \( y \) decreases. Specifically, for every 5 units increase in \( x \), \( y \) decreases by 2 units. Understanding the slope is key in graphing the function and predicting its behavior.

• A positive slope indicates a line rising,
• while a negative slope shows a line falling.
• A larger absolute value of the slope means a steeper line.