Problem 62 - Algebra 1: Concepts and Skills

Question

Graph the function. (Lesson 4.8) $$ g(x)=-x-7 $$

Step-by-Step Solution

Answer

The graph of the function \( g(x)=-x-7 \) is a straight line that intersects the y-axis at -7 and descends at a slope of -1.

Step 1: Identify the slope and the y-intercept

The general form for a linear equation is \( y = mx + c \), where m is the slope and c is the y-intercept. Here, in \( g(x) = -x - 7 \), the slope \( m = -1 \) and the y-intercept \( c = -7 \).

Step 2: Plot the y-intercept

Start with the y-intercept that is at the point (0,-7) since the y-intercept is the point where the line crosses the y-axis. This can be done by marking a point on the vertical axis, also known as the y-axis, at -7.

Step 3: Use the slope to find the next point

The slope of -1 means that for every 1 unit move to the right on the x-axis, move 1 unit down on the y-axis. This will give a point at (1, -8). Then, draw a line through the two points, extending it past both points.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is crucial for graphing. In its simplest terms, the slope-intercept form is expressed as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) indicates the y-intercept—the point at which the line intersects the y-axis. The beauty of this form lies in its direct representation of how a line behaves on a two-dimensional graph.

For the given function \( g(x)=-x-7 \), it follows the slope-intercept structure with \( m = -1 \) and \( b = -7 \). This instantly tells us two things: the steepness of the line (slope) and where it crosses the y-axis (y-intercept). The negative slope indicates that as we move to the right along the x-axis, the line will descend, creating a downward slope.

Y-Intercept

The y-intercept is a foundational concept in graphing linear equations. It specifically refers to the point where the line crosses the y-axis. This can be seen as the line's 'starting point' when graphing. You can identify the y-intercept in the equation \( y = mx + b \) by looking for \( b \).

In the example \( g(x)=-x-7 \), the y-intercept is \( -7 \), which means the line will intersect the y-axis at the point (0, -7). When plotting this point, you find the -7 on the vertical axis and make a mark. It is the first concrete step in drawing the entire line on a graph and heavily influences the line's placement.

Plotting Points

Plotting points is where the action takes place in graphing. Once you have the y-intercept, plotting additional points using the slope will help you visualize the linear equation. The slope, often written as \( m = \frac{rise}{run} \), instructs you on how to move from one point to another along the line.

In our function \( g(x)=-x-7 \), a slope of \( m = -1 \) means you move 1 unit down for every 1 unit you move to the right. Starting from the y-intercept (0, -7), moving one unit right to the point (1, -7) and then one unit down reaches the point (1, -8). Plotting this second point, along with the y-intercept, and drawing a line through them gives you the graph of the function. It's like connecting the dots to reveal the path of the equation.