Problem 1 - Calculus with Concepts in Calculus

Question

Sketch the graph of the function. $$ f(x)=\frac{1}{2} x+1 $$

Step-by-Step Solution

Answer

The graph is a straight line with slope \( \frac{1}{2} \) and y-intercept 1.

Step 1: Identify the Type of Function

The given function is a linear function of the form \( f(x) = mx + b \), where \( m = \frac{1}{2} \) and \( b = 1 \). This means the graph of the function will be a straight line.

Step 2: Determine the Slope and Y-Intercept

The slope \( m \) is \( \frac{1}{2} \), indicating that the line rises \( 1 \) unit for every \( 2 \) units it moves to the right. The y-intercept \( b \) is 1, which means the line crosses the y-axis at the point (0, 1).

Step 3: Plot the Y-Intercept

On the graph, locate and mark the point (0, 1). This is where the line crosses the y-axis.

Step 4: Use the Slope to Find another Point

From the y-intercept (0, 1), use the slope to find another point on the line. Move 2 units to the right along the x-axis to (2, y) and then 1 unit up, giving the new point (2, 2). Plot the point (2, 2) on the graph.

Step 5: Draw the Line

Using a ruler, draw a straight line through the points (0, 1) and (2, 2). Extend the line in both directions to cover the entire coordinate plane.

Key Concepts

Graph Sketching

Graph sketching is the process of drawing the graph of a mathematical function on a coordinate plane. It involves understanding some key components, usually starting with identifying the type of function. For a linear function like the one given in the exercise, it will always produce a straight line graph.

Once you know it's linear, sketching essentially follows these steps:

• Determine important points such as the y-intercept and another point using the slope.
• Draw the line through these points.
• Extend the line fully across the graph to give a complete picture.

This procedure helps in visualizing the function's behavior across the plane, giving valuable insights into its characteristics like direction, steepness, and crossing points.

Y-Intercept

The y-intercept is one of the fundamental features in graphing a linear function. It represents the point where the graph crosses the y-axis.

In the formula of a linear function, which is written as \( f(x) = mx + b \), the y-intercept is represented by \( b \). Simply put, the y-intercept is the value of \( f(x) \) when \( x = 0 \). In our exercise, if you substitute \( x = 0 \) into the function, you'll calculate \( f(0) = \frac{1}{2} \cdot 0 + 1 \), which equals 1.

This means our line will cross the vertical axis at the point (0,1). Understanding this concept helps you quickly place a fixed crucial point on the graph, simplifying the drawing of the rest of the line.

Slope of a Line

The slope is a critical concept when dealing with linear functions as it describes the steepness or tilt of the line.

Mathematically represented by \( m \) in \( f(x) = mx + b \), the slope of a line shows how much the line rises vertically for a given horizontal movement. In simple terms, it's the 'rise over run' ratio.

• For a slope of \( \frac{1}{2} \), it indicates the line rises 1 unit vertically for every 2 units it moves horizontally to the right.

Understanding the slope helps in locating additional points on the line when graph sketching.

Once you have the y-intercept, use the slope to determine another point, as shown in the solution steps by moving from (0,1) to (2,2). Recognizing and using the slope correctly ensures you draw an accurate straight line representing the function.