Problem 46 - Algebra 1: Concepts and Skills

Question

Graph the function. $$ g(x)=-x+2 $$

Step-by-Step Solution

Answer

To graph the function \(g(x) = -x + 2\), start by plotting the y-intercept at point (0,2). Then use the slope (-1) to plot a second point by moving one unit to the right and one unit down from the y-intercept, resulting in point (1,1). Finally, draw a straight line through these two points.

Step 1: Identify the Slope and Y-Intercept

To start, identify the slope (m) and y-intercept (b) from the given function. In the function \(g(x) = -x + 2\), the slope (m) is -1 and the y-intercept (b) is 2.

Step 2: Plot the Y-Intercept

Next, plot the y-intercept on the graph. The y-intercept is the point where the line crosses the y-axis. Since the y-intercept is 2, plot a point at (0,2).

Step 3: Use the Slope to Plot a Second Point

The slope of a line is the amount that y changes for each unit change in x. Since the slope of this function is -1, that means for every one unit increase in x, y will decrease by one unit. Starting from the y-intercept point (0,2), move one unit to the right and one unit down to place a second point at (1,1).

Step 4: Draw the Line

Finally, connect the two plotted points with a straight line, extending the line past both points. This line represents the function \(g(x) = -x + 2\).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept. This form is a convenient way to quickly identify two key characteristics of a linear function:

• Slope (m): This is the steepness of the line, determining its angle of inclination. A positive slope means the line rises as you move from left to right, while a negative slope means it falls.
• Y-Intercept (b): This is the point where the line crosses the y-axis. It tells you the value of y when x is zero.

In the equation \( g(x) = -x + 2 \), comparing it to the slope-intercept form, it's evident that the slope \( m \) is -1 and the y-intercept \( b \) is 2. Knowing these two numbers allows you to draw the basic framework of the line on a graph.

Plotting Points

After determining the slope and y-intercept, the next step is plotting points on a graph to mark the line's path. Start by locating the y-intercept, which is a straightforward step:

• Locate the Y-Intercept: Plot this point on the y-axis. For \( g(x) = -x + 2 \), the y-intercept is 2, so plot the point (0, 2).
• Use the Slope: The slope of -1 tells us that as you increase x by 1, y decreases by 1. Starting from (0, 2), if you move 1 unit right (x increases by 1), move 1 unit down (y decreases by 1) to reach the next point (1, 1).

By understanding these plotted points, you can create a visual representation of the linear equation. This approach ensures you have at least two points through which to draw your line.

Linear Equations

Linear equations are mathematical sentences that represent straight lines on a graph. Their general form can be written as \( ax + by = c \), but the slope-intercept form \( y = mx + b \) is also common since it straightforwardly describes the line for easy graphing.
Some features of linear equations include:

• Constant Rate of Change: The slope is the rate at which y changes per unit increase in x. For \( g(x) = -x + 2 \), the slope of -1 indicates a constant decrease in y as x increases.
• Graphing the Line: Once two points are plotted, drawing a straight line through them produces the graph of the equation. This line extends in both directions infinitely.

Understanding linear equations is essential for graphing functions and interpreting relationships between variables in algebra and real-world contexts. They represent scenarios with a consistent rate of growth or decline and are foundational to more complex mathematical concepts.