Question

Graph the function. $$ f(x)=3-7 x $$

Short answer

The graph of \(f(x) = 3 - 7x\) is a straight line with a slope of -7 and y-intercept of 3.

Step-by-step solution

Identifying the slope

By recognizing that the function is presented in slope-intercept form, \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The slope of the line \(f(x) = 3 - 7x\) is -7.

Identifying the y-intercept

The y-intercept is the value of the function when \(x = 0\). In the equation, \(f(x) = 3 - 7x\), the y-intercept can be derived as \(f(0) = 3 - 7(0) = 3\). So, the y-intercept is 3.

Graphing the line

Start by drawing the y-intercept, which is the point (0, 3) on the y-axis. Then, because the slope is -7, from the y-intercept, move 7 units down and 1 unit to the right. Draw a straight line through these two points, extending in both directions. This line represents the graph of the function \(f(x) = 3 - 7x\).

Key concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is crucial for graphing linear functions. It is expressed as \( y = mx + b \), where \( m \) represents the slope, and \( b \) is the y-intercept. This form quickly tells us how a line behaves by providing information about its steepness and where it crosses the y-axis.

For the equation \( f(x) = 3 - 7x \), the function is already in slope-intercept form, with the slope, \( m \), being -7, and the y-intercept, \( b \), being 3. This concise form allows us to identify these values at a glance, facilitating the graphing process. The negative slope indicates the line will be descending as it moves from left to right.

Slope of a Line

The slope of a line measures its steepness or incline and is often represented by the letter \( m \). It's calculated as the 'rise over run' between two points on the line. In the context of the equation \( f(x) = 3 - 7x \), the slope is -7. This means for each unit the line moves horizontally to the right (the run), it will move 7 units down (the rise), because the slope is negative.

Understanding the concept of slope is fundamental for predicting the direction and angle of a line on a graph. A positive slope means the line will rise as it moves right, while a negative slope, as seen in this case, indicates a downward trend.

Y-Intercept

The y-intercept is the point where a line crosses the y-axis, and it occurs when the input value, or \( x \), is zero. It is an essential anchor for graphing the line because it is the starting point from which we begin to apply the slope to find other points on the line.

In our equation \( f(x) = 3 - 7x \), the y-intercept is 3, denoted by the coordinates \( (0, 3) \). This means when graphing the line, you'll always begin at this point on the y-axis. By locating the y-intercept and applying the slope, we can draw an accurate representation of the linear function on a coordinate plane.