Problem 18 - A Graphical Approach to College Algebra

Question

Sketch the graph of \(f\) by hand. $$f(x)=1-2 x$$

Step-by-Step Solution

Answer

The graph is a straight line with a slope of -2 that crosses the y-axis at 1.

Step 1: Identify the Function Type

The given function is \(f(x) = 1 - 2x\), which is a linear function. The general form of a linear equation is \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.

Step 2: Determine the Slope

In the function \(f(x) = 1 - 2x\), we can rewrite it as \(f(x) = -2x + 1\). Thus, the slope \(m = -2\). This means for every increase of 1 unit in \(x\), \(f(x)\) decreases by 2 units.

Step 3: Identify the Y-intercept

The y-intercept \(c\) is the constant term in the equation \(f(x) = -2x + 1\); hence, \(c = 1\). This means the graph will intersect the y-axis at the point \((0, 1)\).

Step 4: Find a Second Point

Choose an arbitrary value for \(x\) to find a second point. Let's use \(x = 1\):\[f(1) = 1 - 2 \times 1 = -1\]. The second point is \((1, -1)\).

Step 5: Draw the Coordinate Axes

On a graph, draw the x-axis and y-axis perpendicular to each other, ensuring equal-scale measurements on both axes if possible.

Step 6: Plot the Points

Plot the y-intercept, \((0, 1)\), and the second point, \((1, -1)\), on the graph.

Step 7: Draw the Line

Connect the points \((0, 1)\) and \((1, -1)\) with a straight line. Extend the line in both directions with arrows to indicate that the line continues infinitely.

Key Concepts

Graphing Techniques

To draw the graph of the function, it's important to understand some basic graphing techniques. These techniques involve plotting points and understanding the direction and behavior of the line. Start by carefully setting up your graphing axes (coordinate plane) with the x-axis, which runs horizontally, and the y-axis, running vertically. Both axes should have numbers at regular intervals.

When you have the function, identify key points to plot. In our example, the y-intercept and another chosen point are vital. Plot the y-intercept \(0, c\) on the y-axis first.
Then, find another point by choosing a suitable x-value, which gives you coordinates \(x, f(x)\).

Once you have at least two points, use a ruler to draw a line through them. Extend your line across the grid with arrows at the line's ends, indicating that it continues indefinitely both ways. Another helpful technique is verifying the line's direction using the slope to ensure it accurately represents the function's characteristics.

Slope-Intercept Form

The function \(f(x) = 1 - 2x\) can be expressed in the slope-intercept form, which is generally written as \y = mx + c\. This form is highly useful in graphing and understanding linear equations.

The slope \(m\) represents the steepness of the line, showing how much y changes when x increases by 1 unit.
If \(m\) is positive, the line slopes upward; if negative, it slopes downward, as seen with \(m = -2\) in our function.

The y-intercept \(c\) is the point where the line crosses the y-axis. It shows the value of \(y\) when \(x\) is zero, which is \(c = 1\) for our function.
This makes plotting the starting point easy as it's always on the y-axis at \((0, c)\).

Understanding this form helps to quickly draft the graph of any linear equation without needing numerous calculations or points.

Coordinate Geometry

Coordinate geometry involves using a coordinate plane to illustrate mathematical functions and points. Each point on the plane has an x and a y coordinate, known as a pair \(x, y\).
This system helps transfer abstract numerical functions into visual graphs, making them easier to analyze.

Start with drawing the coordinate axes, ensuring each axis is evenly marked with numbers to form a reference grid. For the function \(f(x) = 1 - 2x\), first plot the y-intercept \(0, 1\), and use the slope to find other points. The slope tells us to move down 2 units for every unit we move right from any point on the line.

This method is efficient because it uses only basic points and the line equation to represent the entire range of solutions visually. Coordinate geometry is an excellent tool for learners to see the progression of linear equations in a structured way by using points and lines logically to explain relationships and changes across the graph.