Problem 43 - College Algebra

Question

Give the slope and \(y\) -intercept of each line Whose equation is given. Then graph the r function. \(f(x)-\frac{3}{4} x-2\)

Step-by-Step Solution

Answer

The slope of the function is \(\frac{3}{4}\) and the y-intercept is -2. The graph of the function is a straight line which crosses the y-axis at -2 and rises with a slope of \(\frac{3}{4}\).

Step 1: Identifying the slope and y-intercept

From the given function \(f(x) = \frac{3}{4} x - 2\), it's clear that the slope \(m = \frac{3}{4}\) and the y-intercept \(b = -2\). In a linear equation, the coefficient of x is the slope and the constant term is the y-intercept.

Step 2: Plotting the function

To graph the line, start by plotting the y-intercept, which in this case is -2. This is the point where the line crosses the y-axis. Next, using the slope \(\frac{3}{4}\) , which means for every 3 steps up (positive slope) along the y direction, it implies 4 steps to the right along the x direction. Keep moving like this and add several points. Finally, connect all the points with a straight line. This line represents the graph of given function.

Key Concepts

Slope

The slope of a line is a crucial concept in understanding linear equations. It describes how steep a line is and represents the rate at which the y-coordinate changes with respect to the x-coordinate. Mathematically, the slope is often denoted as "m" and is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

• For a positive slope, the line rises from left to right, indicating an upward direction.
• A negative slope means the line falls from left to right, indicating a downward direction.
• If the slope is zero, the line is perfectly horizontal, showing no change in y with respect to x.

In the given function, the slope is \( \frac{3}{4} \). This means for every 4 units you move to the right on the x-axis, the line moves 3 units up on the y-axis. Being able to interpret the slope helps you predict the behavior of the graph and understand the relationship between x and y values in a linear function.

Y-intercept

The y-intercept is the point where a line crosses the y-axis. This is a key feature of a linear equation in the slope-intercept form, which is expressed as \( y = mx + b \). Here, "b" represents the y-intercept.
This point is significant because it provides an initial value of the linear function when \( x = 0 \). The y-intercept is the value of the function at this specific point.

• For positive y-intercepts, the line crosses above the origin.
• Negative y-intercepts indicate the line crosses below the origin.

In this instance, the given equation \( f(x) = \frac{3}{4}x - 2 \) has a y-intercept of \( -2 \). This tells us that the line will cross the y-axis at -2, providing a visual starting point when graphing the line. Recognizing the y-intercept allows you to graph the linear function with accuracy, starting from where the line naturally intersects the y-axis.

Graphing Linear Functions

Graphing a linear function involves plotting points on a coordinate plane to visually represent the equation. The process starts by using the y-intercept and slope to establish the line.
Here’s a simplified process to graph a linear function like \( f(x) = \frac{3}{4}x - 2 \):

• Identify the y-intercept: Start by marking the y-intercept on the graph. For this function, you will plot a point at \( (0, -2) \) on the y-axis.
• Use the slope: From the y-intercept, apply the slope to find the next point. With a slope of \( \frac{3}{4} \), move 4 units to the right and 3 units up to locate another point.
• Plot more points: Continue using the slope to plot additional points. This helps ensure accuracy when drawing the line.
• Draw the line: Use a ruler or a straight edge to connect the points with a straight line, extending in both directions.

By following these steps, you can accurately graph the linear function and better visualize the relationship between x and y values present in the linear equation.