Problem 19 - A Graphical Approach to College Algebra

Question

Sketch the graph of \(f\) by hand. $$f(x)=3$$

Step-by-Step Solution

Answer

The graph of \(f(x) = 3\) is a horizontal line at \(y = 3\).

Step 1: Identify the Function Type

The given function is a constant function because it does not have any variable terms, and is expressed in the form \(f(x) = c\) where \(c\) is a constant. In this case, \(f(x) = 3\).

Step 2: Understand the Graph of a Constant Function

A constant function like \(f(x) = 3\) represents a horizontal line on the graph. The function value is the same for all \(x\)-values. This means that regardless of the \(x\) value, the output \(f(x)\) will always be 3.

Step 3: Determine the Y-intercept

For the function \(f(x) = 3\), the graph touches the y-axis at \((0, 3)\). This point is called the y-intercept and represents where the graph will cross the y-axis.

Step 4: Draw the Horizontal Line

On a graph, mark the point \((0, 3)\) on the y-axis, since this is the y-intercept. Then, draw a horizontal line across the plane that passes through the y-value of 3. This line should extend infinitely to the left and right, parallel to the x-axis.

Key Concepts

graphing functions

Graphing functions is a foundational concept in mathematics that helps us visualize how functions behave. When graphing functions, we represent them as lines or curves on a coordinate plane. For each input value, or 'x' value, we have a corresponding output value or 'y' value, which we plot as a point.
Graphing makes it easy to see patterns or trends. For example, with a linear function, you'll see a straight line. When considering a function like the constant function \(f(x) = 3\), its graph focuses on giving a clearer visual of constant values and behaviors.

• Start by plotting points on a coordinate plane.
• Ensure there's a consistent relationship between x and y values.
• Connect the points smoothly if applicable (though, for constant functions, you'll see a straight line).

Visualizing the function helps to identify key characteristics such as intercepts and the overall shape.

y-intercept

A y-intercept is a crucial point where the graph of a function crosses the y-axis. In mathematics, this is the \((0, y)\) coordinate. The y-intercept occurs when the input value \(x\) is zero. This means you look at where the function intersects the vertical y-axis.
For constant functions, like \(f(x) = 3\), identifying the y-intercept is straightforward. Here, the y-intercept is the point \((0, 3)\), representing the function's output when the input is zero.

• Examine the function equation and set \(x = 0\) to find the y-intercept.
• Plot this point on the y-axis.
• The y-coordinate at this point gives the function's constant value.

Being able to find and plot the y-intercept aids in understanding the position and height of the function on the graph.

horizontal line

A horizontal line is a straight line that runs parallel to the x-axis. This kind of line shows that a function maintains a constant value, regardless of the x-values. It's a key feature of constant functions. For \(f(x) = 3\), the graph is a horizontal line positioned at y = 3.
Horizontal lines reflect great stability in function values, which means the output does not change across different x-values.

• Locate the function's y-value.
• Draw a line parallel to the x-axis at this y-value.
• This line extends infinitely in both horizontal directions.

Understanding horizontal lines clarifies how constant functions behave and appear graphically. These lines emphasize that while x changes, y remains steady.