Question

Graph the function. $$ f(x)=-5 x-7 $$

Short answer

The graph of the function \(f(x) = -5x - 7\) is a straight line with a negative slope of 5 that crosses the y-axis at -7.

Step-by-step solution

Identify the slope and y-intercept

The equation of the function is in the form \(y = mx + b\), where \(m = -5\) is the slope and \(b = -7\) is the y-intercept. So, the slope of the line is -5, and the line crosses the y-axis at -7.

Plot the y-intercept

Since the y-intercept is -7, place a point on the y-axis at \(y = -7\). This point represents where the function crosses the y-axis.

Use the slope to find another point and draw the line

The slope is -5, which means for every 1 unit increase in x, y decreases by 5 units. From the y-intercept point, count 1 unit to the right and 5 units down to find another point. Draw a line through these two points to represent the function.

Key concepts

Slope-Intercept Form

The slope-intercept form is one of the simplest and most commonly used ways to write the equation of a line. It is expressed as \( y = mx + b \). In this equation, \( m \) represents the slope of the line, and \( b \) is the y-intercept, which is where the line crosses the y-axis.

By expressing a linear function in this form, it becomes very straightforward to graph it. The slope, \( m \), tells us the steepness or incline of the line and in which direction it goes. A positive slope indicates the line rises as it moves from left to right, while a negative slope means it falls. In the case of \( f(x) = -5x - 7 \), the slope is -5, indicating a steep downward line.

To give you a clear picture, if you were walking along the graph of this line, you would step down five units vertically for every one unit you move to the right horizontally. This characteristic of the slope is very helpful in graphing the line, as you'll see in the plotting points section.

Y-intercept

The y-intercept of a line is the point where the line crosses the y-axis on a graph. This is characterized by the \( x \) value being zero. In the slope-intercept equation \( y = mx + b \), the y-intercept is represented by \( b \).

For our example function, \( f(x) = -5x - 7 \), the y-intercept is -7. This tells us that the point (0, -7) is on the line, which becomes our starting point for plotting. When you're graphing a line, always begin by placing a dot at the y-intercept. It's a fixed point that doesn't depend on the slope and provides a concrete place to start drawing your line.

Remember, the y-intercept is a critical marker in your graph; it's where your journey of plotting the line begins. Visualizing where the line crosses the y-axis can instantly give you a sense of the line's position in relation to the origin.

Plotting Points

After determining the slope and the y-intercept, plotting points is the next step to graph the line on the coordinate plane. Start by plotting the y-intercept, which we've identified as (0, -7). This single point gives us a fixed location to work from. Next, use the slope to find another point.

For our equation \( f(x) = -5x - 7 \), the slope tells us to go down 5 units for every unit we go to the right. So starting from (0, -7), move 1 unit to the right (increasing the x-value by 1) and then 5 units down (decreasing the y-value by 5). You would now have the point (1, -12).

Drawing The Line

Once you have two points, you can draw a line through them, which extends infinitely in both directions. Graphing additional points using the slope can help ensure accuracy, especially in more complex functions or when precision is paramount. However, with linear functions, two points are sufficient to determine the line. It is the pattern of using the slope from a known point (like the y-intercept) that makes graphing linear functions manageable and systematic.