Problem 2 - Calculus with Concepts in Calculus

Question

Sketch the graph of the function. $$ f(x)=1-3 x \text { for }-1 \leq x \leq 2 $$

Step-by-Step Solution

Answer

The graph is a line from \((-1, 4)\) to \((2, -5)\).

Step 1: Identify the Type of Function

The given function is a linear function of the form \( f(x) = ax + b \). Here, \( a = -3 \) and \( b = 1 \). Linear functions graph as straight lines.

Step 2: Determine Key Points

We need at least two points to graph a line. Calculate \( f(x) \) at \( x = -1 \) and \( x = 2 \).For \( x = -1 \): \[ f(-1) = 1 - 3(-1) = 1 + 3 = 4 \]For \( x = 2 \):\[ f(2) = 1 - 3(2) = 1 - 6 = -5 \]

Step 3: Plot the Points

Plot the points \((-1, 4)\) and \((2, -5)\) on a graph. These are the endpoints of the segment of the line we'll draw, since the function is only defined for \(-1 \leq x \leq 2\).

Step 4: Draw the Line Segment

Draw a straight line through the points \((-1, 4)\) and \((2, -5)\). This line segment represents the graph of the function for the interval \(-1 \leq x \leq 2 \).

Step 5: Label the Graph

Label the graph with key points and mark the segment endpoints to indicate the domain, \(-1 \leq x \leq 2\), over which the function is defined. Optionally, label the line with its equation \( f(x) = 1 - 3x \).

Key Concepts

Linear Equation

A linear equation is one of the most fundamental concepts in algebra and is easy to recognize by its general form, \( f(x) = ax + b \). Here, \( a \) and \( b \) are constants that determine the slope and y-intercept of the line, respectively. In the given function \( f(x) = 1 - 3x \), the slope is \(-3\) and the y-intercept is \(1\). These lines have a constant rate of change, meaning for any change in \( x \), the change in \( f(x) \) stays consistent, represented by the slope.

• Slope (\( a \)): Indicates the steepness and direction of the line. A positive slope means the line rises as it moves from left to right, whereas a negative slope means it falls.
• Y-Intercept (\( b \)): This is the value of \( f(x) \) when \( x \) is zero, helping in determining where the line crosses the y-axis.

Linear equations simplify the process of graphing because once the slope and y-intercept are known, the entire graph can be drawn with ease.

Domain and Range

The domain and range of a function are essential for understanding its behavior on a graph. In our exercise, the domain is given as \(-1 \leq x \leq 2\). This means we're only considering input values (\( x \)) between -1 and 2 inclusive.

• Domain: This is all the possible input values (\( x \)) for which the function is defined. In our function, the domain is limited to the segment \([-1, 2]\).
• Range: This is determined by the output values (\( f(x) \)) as \( x \) varies over the domain. Calculating \( f(x) \) at the endpoints of the domain helps specify the range.

For \( f(x) = 1 - 3x \), the endpoints yield outputs of 4 and -5, hence the range is \([-5, 4]\). Understanding the range helps us know the vertical extent of the graph of the function.

Plotting Points

Plotting points is a critical step in graphing linear functions. It involves identifying specific coordinates that lie on the graph of the equation. For our linear function \( f(x) = 1 - 3x \), we found the points \((-1, 4)\) and \((2, -5)\).

• Choosing Points: To accurately graph the line, it's crucial to select points that are evenly distributed across the domain, often the endpoints.
• Calculate Outputs: For each chosen \( x \)-value, substitute it into the equation to determine the corresponding \( f(x) \)-value.
• Recording Coordinates: Each calculated \( x \) and \( f(x) \) pair represents a point, such as \((-1, 4)\) or \((2, -5)\).

Once these points are plotted on the graph, simply connect them with a straight line. This shows the function's entire graph over the specified domain. Plotting points and drawing lines make abstract functions clear and visually interpretable.