Problem 26 - Brief Applied Calculus

Question

Graph each function "by hand." [Note: Even if you have a graphing calculator, it is important to be able to sketch simple curves by finding a few important points.] $$ f(x)=-3 x+5 $$

Step-by-Step Solution

Answer

Plot the y-intercept at (0, 5), use slope -3, and draw the line through points (0, 5) and (1, 2).

Step 1: Identify the Type of Function

The given function is a linear function because it can be written in the form of \(f(x) = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. Here, \(m = -3\) and \(c = 5\).

Step 2: Determine the Y-Intercept

To determine the y-intercept, set \(x = 0\) in the function. Calculate \(f(0) = -3(0) + 5 = 5\). Thus, the y-intercept is \(5\), meaning the graph will cross the y-axis at the point \((0, 5)\).

Step 3: Determine Another Point Using the Slope

The slope \(m = -3\) indicates that for every unit increase in \(x\), \(f(x)\) decreases by 3 units. Starting from point \((0, 5)\), choose \(x = 1\). Calculate \(f(1) = -3(1) + 5 = 2\). So, another point on the graph is \((1, 2)\).

Step 4: Plot the Points on a Graph

On a Cartesian plane, plot the points \((0, 5)\) and \((1, 2)\). These points represent the intercept and an additional point along the line of the function.

Step 5: Draw the Line

Use a ruler to draw a straight line through the points \((0, 5)\) and \((1, 2)\). This line represents the graph of the function \(f(x) = -3x + 5\). Extend the line in both directions to cover the graph.

Key Concepts

Graphing

Graphing a linear function involves plotting points on a Cartesian coordinate plane and connecting them with a straight line.
To graph the function, start by identifying key points such as the y-intercept and additional points derived from the slope.
This process visually represents the equation on a two-dimensional plane, providing a clear picture of the function's behavior.

• Begin at the y-intercept, the point where the curve crosses the y-axis.
• Locate additional points using the slope to ensure accuracy.
• Draw a straight line through these points, extending the line across the plane.

By sketching the line manually, you gain a better understanding of its direction and intersecting points, crucial for solving and predicting outcomes in algebraic contexts.

Slope

The slope of a linear function measures how steep the line is. It tells you how much the function value changes for one unit increase in the x-variable.
The slope, represented by the variable \(m\), is a key component in the slope-intercept form of a linear equation, \(f(x) = mx + c\).
In the given function, \(f(x) = -3x + 5\), the slope is \(-3\).

• A negative slope indicates that the line slopes downward as you move from left to right.
• The slope of \(-3\) means for every increase of 1 in \(x\), \(f(x)\) decreases by 3 units.

Understanding slope is essential for predicting the movement of the line and evaluating the rate of change. In real-world contexts, slope can represent mortality trends, economic inflation, or any scenario where changes occur at a consistent rate.

Y-Intercept

The y-intercept is the point where a line crosses the y-axis. It is a vital component of a linear function, as it indicates where the line will intersect the vertical axis when \(x = 0\).
In the function \(f(x) = -3x + 5\), the y-intercept is the constant \(c\), which is \(5\).

• Locate the y-intercept by setting \(x = 0\) in the equation.
• Calculate the resulting \(f(x)\) to find the intercept point on the y-axis.
• This point helps define the initial position of the line before slope adjustments are applied.

Recognizing the y-intercept allows you to establish the starting point for graphing the line and provides a foundational point from which to plot additional points using the slope.