Problem 41 - College Algebra

Question

Give the slope and \(y\) -intercept of each line Whose equation is given. Then graph the r function. \(f(x)--2 x+1\)

Step-by-Step Solution

Answer

The slope of the function \(f(x) = -2x + 1\) is -2 and the y-intercept is 1. The line graph of the function can be drawn by starting at y-intercept and using the slope to find other points on the graph.

Step 1: Identify the Slope

In this function \(f(x) = -2x + 1\), the coefficient of x, which is -2, is the slope. Therefore, the slope of the function is -2.

Step 2: Identify the Y-intercept

In this function \(f(x) = -2x + 1\), the constant term, which is 1, is the y-intercept. Therefore, the y-intercept of the function is 1.

Step 3: Graph the function

To draw the graph of the function, first plot the y-intercept on the y-axis. Since the intercept is 1, put a point at (0,1). Since the slope is -2, which means for every 1 unit change in x, y will decrease by 2 units. Thus, starting from point (0,1), for every 1 unit increase in x coordinate, decrease the y coordinate by 2 units and plot the points. Connect these points and extend in both directions to plot the line graph of the function. This line graph represents the function \(f(x) = -2x + 1\).

Key Concepts

Understanding Slope

The slope of a line is a measure of its steepness and direction. In math, the slope is often denoted by the letter "m". It indicates how much the y-value of a point on the line changes for every increase in the x-value. For the function given in the exercise, \(f(x) = -2x + 1\), the slope is identified as -2. This negative sign tells us that the line is decreasing, meaning as we move from left to right, the line goes downwards.

There's a simple way to remember how slope works:

• A positive slope means the line rises as it goes from left to right.
• A negative slope means the line falls as it goes from left to right.
• If the slope is zero, the line is flat and horizontal.
• An undefined slope means the line is vertical.

The actual numerical value of the slope, 2 in this case, tells us how steep the line is. A slope of -2 means that for each unit increase in x, y decreases by 2 units.

This makes it easy to graph the line once we know the y-intercept!

Y-Intercept in Linear Equations

The y-intercept is a key feature of a linear equation. It tells us where the line crosses the y-axis. For any linear function written in the form \(f(x) = mx + b\), "b" is the y-intercept. In the function \(f(x) = -2x + 1\), the y-intercept is 1. This means that the point (0,1) lies on the graph of the function.

Being able to identify the y-intercept is extremely helpful:

• You know that the line will always pass through this point (0, b).
• It helps you to start drawing the graph accurately.

Drawing the line begins by plotting the y-intercept. After that, using the slope makes it possible to find other points on the line.

Graphing Functions Using Slope and Y-Intercept

Graphing linear functions is straightforward once you grasp how the slope and y-intercept work together. To graph the function \(f(x) = -2x + 1\):

• Start by plotting the y-intercept on the y-axis at point (0,1) since that is where the line crosses the y-axis.
• Use the slope to determine the direction and steepness of the line. Remember, with a slope of -2, for each step of 1 unit to the right (positive x direction), you move 2 units down.
• You can plot multiple points this way, maintaining the same slope.

Connect all these points to form a straight line. This line effectively represents the function \(f(x) = -2x + 1\).

Always ensure the line extends in both directions, as linear functions continue indefinitely. Once plotted, you will have a visual representation of how changes in x affect y based on the defined relationship of slope and y-intercept.