Problem 36 - Algebra 1

Question

GRAPHING FUNCTIONS Graph the function. $$f(x)=-6 x+1$$

Step-by-Step Solution

Answer

The graph of the function \(f(x) = -6x + 1\) is a straight line with a slope of -6 and y-intercept of 1. It crosses the y-axis at the point (0,1) and passes through the point (1, -5).

Step 1: Identify the Slope and Y-intercept

The slope is -6 and the y-intercept is 1. This means the line falls by 6 units for every unit it moves to the right, and it crosses the y-axis at 1.

Step 2: Plot the Y-intercept

Start by marking the y-intercept, which is the point (0, 1), on the graph. You do this by going to 0 on the x-axis and moving up 1 on the y-axis.

Step 3: Use the Slope to Find Another Point

The slope of the line is -6. Start from the y-intercept and move to the right 1 unit (because the denominator of the slope is implicitly 1), and then move down 6 units to get to the next point on the line. This should bring you to the point (1, -5). Draw a dot at this point.

Step 4: Draw a line through the points

Finally, draw a straight line passing through the two points plotted. This represents the function \(f(x) = -6x + 1\).

Key Concepts

Slope and Y-Intercept

Understanding the slope and y-intercept is crucial when it comes to graphing linear functions. The slope indicates the steepness and the direction of the line. A positive slope means the line rises as it moves from left to right, while a negative slope means it falls.

In the given exercise, the function is described by the equation: \(f(x) = -6x + 1\). The number in front of \(x\), which is -6, represents the slope. This tells us that for every step you move to the right along the x-axis, the function falls by 6 steps vertically. It's like walking down a hill that drops 6 meters for every meter you step forward.

The y-intercept, in this case, is 1. It describes where the line crosses the y-axis. This is always the point at which \(x=0\), so for this function, it's the point (0, 1). It's like starting your walk on the hill at a height of 1 meter above the ground level.

Plotting Points

Once the slope and y-intercept are known, plotting points can begin. Always start with the y-intercept. For the equation \(f(x) = -6x + 1\), you plot the point (0, 1) by starting at the origin, where the x-axis and y-axis intersect, and then moving up 1 unit since the y-coordinate is 1.

With the slope identified as -6, we look for another point that fits on this line. We generally use the rise over run method, meaning we rise (move up or down) first and then run (move left or right). Since our slope is negative, we'll move down 6 units (the rise) from our y-intercept, and since our run (the horizontal change) is implicitly 1, we move 1 unit to the right, landing on the point (1, -5). After plotting the second point, drawing a line through these points will show the complete linear function on the graph.

Linear Equation

A linear equation is an algebraic equation where each term is either a constant or the product of a constant and the first power of a single variable. Linear equations plot as straight lines when graphed, and they have the general form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

In our exercise, the linear equation is \(f(x) = -6x + 1\). Notably, the variable \(x\) is to the first power, which fits the definition of a linear equation. This simple form makes it very predictable and is the reason a straight line can represent all solutions to the equation. Every point on that line is a solution to the equation, which validates the concept of plotting points and drawing the line through them to represent the function graphically.