Problem 5 - A Graphical Approach to Precalculus with Limits

Question

Graph each linear function. Give the (a) \(x\) -intercept, (b) \(y\) -intercept. (c) domain, (d) range, and (e) slope of the line. $$f(x)=-\frac{2}{5} x+2$$

Step-by-Step Solution

Answer

(a) x-intercept: (5/2, 0). (b) y-intercept: (0, 2). (c) Domain: all real numbers. (d) Range: all real numbers. (e) Slope: -2/5.

Step 1: Identify the y-intercept

The equation given is in the slope-intercept form: \(f(x) = mx + b\). Here, \(b = 2\). The \(y\)-intercept is the point where the line crosses the \(y\)-axis, which occurs when \(x = 0\). Thus, the \(y\)-intercept is \((0, 2)\).

Step 2: Calculate the x-intercept

The \(x\)-intercept is the point where the line crosses the \(x\)-axis, which occurs when \(f(x) = 0\). Set the equation \(f(x) = -\frac{2}{5}x + 2 = 0\) and solve for \(x\):\[-\frac{2}{5}x + 2 = 0\]\[-\frac{2}{5}x = -2\]\[x = \frac{5}{2}\]Hence, the \(x\)-intercept is \(\left(\frac{5}{2}, 0\right)\).

Step 3: Determine the slope

The slope \(m\) of the line, given in the equation \(f(x) = -\frac{2}{5}x + 2\), is \(-\frac{2}{5}\). This means for every 5 units you move to the right along the \(x\)-axis, the line moves down 2 units.

Step 4: Define the domain

The domain of a linear function is all real numbers because the line extends infinitely in both directions along the \(x\)-axis.

Step 5: Define the range

The range of a linear function is also all real numbers because the line extends infinitely in both directions along the \(y\)-axis.

Key Concepts

x-intercept

The x-intercept of a line is the point where it intersects the x-axis. At this point, the value of the function, or \( f(x) \), is zero. To find the x-intercept of the linear function \( f(x) = -\frac{2}{5}x + 2 \), set \( f(x) \) to zero and solve for \( x \): \[ -\frac{2}{5}x + 2 = 0 \] This simplifies to: \[ -\frac{2}{5}x = -2 \] Multiply both sides by \(-\frac{5}{2}\) to isolate \(x\): \[ x = \frac{5}{2} \] Thus, the x-intercept is at the point \( \left( \frac{5}{2}, 0 \right) \). This point tells us where the line crosses the x-axis.

y-intercept

The y-intercept is the point where the line crosses the y-axis. At this point, \( x = 0 \). In the equation \( f(x) = -\frac{2}{5}x + 2 \), the y-intercept can be directly observed as the constant term, \( b = 2 \). This means the y-intercept is \((0, 2)\).

• The y-intercept provides a starting point for graphing the equation.
• It helps us understand the vertical position of the line on the coordinate plane.

The y-intercept is especially useful for visualizing how high or low the line is on the graph when \( x \) is zero.

slope

The slope of a linear function is a measure of its steepness and direction. In the equation \( f(x) = -\frac{2}{5}x + 2 \), the slope \( m \) is represented by \(-\frac{2}{5}\). The slope tells us that for every 5 units increase along the x-axis, the value along the y-axis decreases by 2 units. This gives the line a downward slant, moving from the left to the right.

• A negative slope indicates a line that falls as \( x \) increases and rises as \( x \) decreases.
• It's essential for understanding the rate of change of the function.

Knowing the slope helps predict how changes in \( x \) affect changes in \( f(x) \).

domain

The domain of a linear function refers to all possible input values \( x \) that the function can accept. For most linear functions, including \( f(x) = -\frac{2}{5}x + 2 \), the domain is all real numbers. This is because you can input any real number into a linear formula, and it will always give you a valid result.

• The domain includes every number from negative to positive infinity.
• It represents the full span of the function along the x-axis.

The domain is quite expansive for linear functions, meaning the function graph will stretch evenly along the x-axis into both directions of infinity.

range

The range of a linear function represents all possible outputs \( f(x) \) that the function can produce. For the function \( f(x) = -\frac{2}{5}x + 2 \), the range is all real numbers. This happens because a linear equation will eventually intercept any y-value at some point as it extends indefinitely in both vertical directions.

• Just like the domain, the range spans from negative to positive infinity.
• The function covers every possible y-value on the graph.

Understanding the range gives insight into what values the function can take, showing us how high and low the line will go on the graph.