Problem 26 - 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

The graph is a straight line crossing the y-axis at 5 and sloping down 3 units per 1 unit right.

Step 1: Identify the Type of Function

The function given is of the form \(f(x) = mx + c\), which is a linear function. Linear functions graph as straight lines. In this case, \(m = -3\) is the slope and \(c = 5\) is the y-intercept.

Step 2: Plot the Y-Intercept

The y-intercept of a linear function \(f(x) = mx + c\) is the point where the line crosses the y-axis. For \(f(x) = -3x + 5\), the y-intercept is \(5\). Therefore, plot the point \((0, 5)\) on the graph.

Step 3: Use the Slope to Find Another Point

The slope \(m\) indicates the rise over run. Since \(m = -3\), it implies a rise of \(-3\) for a run of \(1\). From the y-intercept \((0, 5)\), move down 3 units and right 1 unit to locate the next point \((1, 2)\). Plot this point.

Step 4: Draw the Line

Utilize the two points \((0, 5)\) and \((1, 2)\) to draw a straight line across the graph. This line represents the graph of the function \(f(x) = -3x + 5\).

Step 5: Verify with Additional Points

To confirm accuracy, choose any other value for \(x\), substitute into the function, and verify the point lies on the line. For instance, for \(x = -1\), \(f(-1) = -3(-1) + 5 = 8\), so the point \((-1, 8)\) should also lie on the line.

Key Concepts

Linear Equations

A linear equation is one of the simplest types of mathematical expressions, where the highest power of the variable is one. These equations are typically in the form of \( y = mx + c \), where:

• \( y \) represents the output value for each corresponding \( x \).
• \( m \) is the slope, which tells us how steep the line is.
• \( c \) stands for the y-intercept, the point where the line crosses the y-axis.

Linear equations produce straight-line graphs. This makes them easy to recognize and work with. They are foundational in algebra, providing a stepping stone to more complex functions. Understanding linear equations allows us to make predictions and find values quickly.

Y-Intercept

The y-intercept of a linear function is the point where the graph of the function crosses the y-axis. In terms of coordinates, this means the x-value is zero. For any linear equation in the form \( y = mx + c \), the y-intercept can be represented by the point \( (0, c) \).

For the equation \( f(x) = -3x + 5 \), the y-intercept is \(5\). This means we plot the point \( (0, 5) \) on the graph to show where the line begins its journey across the axes. Knowing the y-intercept helps plot the first crucial point, providing a foundation to graph the rest of the line.

Slope

The slope of a line indicates its steepness, showing how y-values change as x-values increase. It is a ratio representing 'rise over run.’ For the function \( f(x) = -3x + 5 \), our slope \( m \) is \(-3\).

• The "rise" is how much the line moves up or down vertically as you go from one point to another.
• The "run" is how far you move horizontally.

In \(-3\), moving from the y-intercept \( (0, 5) \) to another point involves decreasing three units vertically (downwards) across one unit horizontally to reach \( (1, 2) \). This allows us to keep plotting further points, defining the line.

Plotting Points

Plotting points is a fundamental skill necessary for graphing linear functions. It involves identifying and marking coordinates on a graph to visualize an equation.

The process begins with determining significant points:

• Start with the y-intercept: always plot this point first as \( (0, c) \).
• Use the slope to find subsequent points, following the "rise" and "run" principle.
• Verify by placing additional points using chosen x-values and calculating the corresponding y-values.

Take \( f(x) = -3x + 5 \) as an example. We start by plotting \( (0, 5) \), then find the next point \( (1, 2) \). To ensure accuracy, calculate another point like \( (-1, 8) \) and check if it lies on the drawn line. This step confirms that the graph is precise and reliable, reinforcing understanding.