Problem 12 - College Algebra with Modeling and Visualization

Question

Graph \(y=f(x)\) by hand by first plotting points to determine the shape of the graph. $$ f(x)=2-2 x $$

Step-by-Step Solution

Answer

Plot the points (0,2), (1,0), (-1,4) and draw a straight line through them.

Step 1: Identify the Equation

The given equation is a linear function: \( y = f(x) = 2 - 2x \). This equation is in the slope-intercept form \( y = mx + b \), where \( m = -2 \) is the slope and \( b = 2 \) is the y-intercept.

Step 2: Find the Y-Intercept

To find the y-intercept, set \( x = 0 \) in the equation and solve for \( y \). \[ y = 2 - 2(0) = 2\]Thus, the y-intercept is the point \((0, 2)\).

Step 3: Calculate Additional Points

Choose a few values for \( x \) to determine corresponding \( y \) values. For instance, choose \( x = 1 \) and \( x = -1 \):1. If \( x = 1 \), \[ y = 2 - 2(1) = 0\]Thus, the point is \((1, 0)\).2. If \( x = -1 \), \[ y = 2 - 2(-1) = 4\]Thus, the point is \((-1, 4)\).

Step 4: Plot Points on a Graph

On a coordinate plane, plot the points \((0, 2)\), \((1, 0)\), and \((-1, 4)\). These are the points derived from the calculations.

Step 5: Draw the Line

Using a ruler, draw a straight line through the points \((0, 2)\), \((1, 0)\), and \((-1, 4)\). Since it is a linear equation, the graph should be a straight line that continues infinitely in both directions.

Step 6: Verify the Slope

Check the slope of the line by ensuring the rise over run is \(-2\), consistent with the equation. Between the points \((0, 2)\) and \((1, 0)\), the line descends 2 units and moves 1 unit to the right. Therefore, the slope \( \frac{-2}{1} = -2 \) is correct.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a convenient way of writing the equation of a line. It is usually expressed as \( y = mx + b \). Here, \( m \) stands for the slope and \( b \) is the y-intercept. This form is widely used because it makes it easy to identify two critical components of a line: how steep it is and where it crosses the y-axis.

When you look at an equation like \( y = 2 - 2x \), you can immediately tell that \( m = -2 \) and \( b = 2 \). This means the line slopes downward (since the slope is negative) and crosses the y-axis at the point where the y-coordinate is 2.

Plotting Points

Plotting points is a method used to visualize linear equations on a graph. To plot points, you determine the values of \( y \) for various \( x \)-values and mark these coordinates on a Cartesian plane. This helps in determining the shape and position of the graph.

For example, given the linear function \( y = 2 - 2x \), you can choose values like \( x = 1 \), \( x = 0 \), and \( x = -1 \). Calculating their corresponding \( y \)-values, you find points like \((1, 0)\), \((0, 2)\), and \((-1, 4)\). Plotting these points on a graph allows you to see exactly where the line will be.

By marking these points accurately and connecting them with a straight line, you visualize the linear function."

Linear Equations

Linear equations are algebraic expressions that represent a straight line on a graph. They are called "linear" because their graph forms a line. The most common form for linear equations is the slope-intercept form, \( y = mx + b \), which provides straight-forward information about the line.

The equation \( y = 2 - 2x \) is a simple example of a linear equation. Here, it signifies a straight line with a slope of \(-2\). This means for every unit increase in \( x \), the value of \( y \) decreases by 2 units. Linear equations are popular in algebra because they establish a direct relationship between two variables. This makes them easy to work with and to graph.

Y-Intercept

The y-intercept is the point where a line crosses the y-axis of a coordinate plane. It indicates where the graph meets the vertical axis, and it's an essential feature in understanding linear equations.

In the slope-intercept form \( y = mx + b \), the \( b \) value is the y-intercept. In the equation \( y = 2 - 2x \), the y-intercept is 2. This means the line crosses the y-axis at the point \((0, 2)\).

Understanding the y-intercept helps in plotting the initial point of the line on the graph. From there, you can use the slope to determine the direction and steepness, allowing you to draw the entire line accurately. Being clear about the y-intercept provides a solid foundation for graphing linear functions.