Problem 48 - College Algebra and Calculus: An Applied Approach

Question

Analyzing a Graph In Exercises \(47-58\), analyze the graph of the function algebraically and use the results to sketch the graph by hand. Then use a graphing utility to confirm your sketch. $$h(x)=-\frac{3}{4} x+2$$

Step-by-Step Solution

Answer

The slope of the function \(h(x)=-\frac{3}{4} x+2\) is -3/4 and the y-intercept is 2. Therefore, the line will decrease by 3 units vertically for every 4 units it moves to the right horizontally, starting from the point (0, 2) on the y-axis.

Step 1: Identify the slope and the y-intercept

The equation \(h(x)=-\frac{3}{4} x+2\) is in the form of a linear equation also known as slope-intercept form which is \(y = mx + b\), where m represents the slope of the line and b represents the y-intercept. Here, the slope \(m=-\frac{3}{4}\) and the y-intercept \(b=2\). The slope indicates that for every increase of 1 in x, the value of h(x) decreases by \(-\frac{3}{4}\). The y-intercept is the value of h(x) when x = 0, so the line crosses the y-axis at (0,2).

Step 2: Sketch the graph

To sketch the graph, start from the y-intercept at point (0,2) on the coordinate plane. Then, use the slope to find more points on the line. Given the slope is -3/4, move down 3 units (because of the negative sign) and right 4 units for each subsequent point. Draw the line joining these points.

Step 3: Verify with a graphing utility

Now, use a graphing calculator or an online graphing tool to plot the function \(h(x)=-\frac{3}{4} x+2\). The graph should match with the earlier hand sketch.

Key Concepts

Graphing

Graphing is a visual way to represent linear equations like the one in the exercise, which is a powerful tool in understanding how the variables interact. When we plot a linear equation, we get a straight line which gives us a lot of insights into the function's behavior. For the equation \( h(x) = -\frac{3}{4}x + 2 \), graphing allows us to visually see:

• The direction and steepness of the line (which is determined by the slope).
• Where the line crosses the axes, primarily the y-axis in a line equation.
• How the line moves negatively or positively along the graph.

By plotting the function on a coordinate plane, you can see exactly how \( x \) and \( y \) values correspond. Starting from a known point, usually the y-intercept, and using the slope, multiple points can be identified which helps to form the overall line.

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most common forms and is expressed as \( y = mx + b \). Here, \( m \) represents the slope, and \( b \) is the y-intercept. In the equation \( h(x) = -\frac{3}{4}x + 2 \), the slope is \( -\frac{3}{4} \) and the y-intercept is 2.
This form is incredibly insightful because:

• The slope \( m \) tells us how steep the line is and the direction it takes. A negative slope like \( -\frac{3}{4} \) means that as \( x \) increases, \( h(x) \) decreases.
• The y-intercept \( b \) indicates where the line crosses the y-axis. For our line, it crosses at (0,2).

This form is not only simple but highly effective for quickly sketching a graph because it directly tells you where to begin (at the y-intercept) and how to proceed (using the slope).

Y-Intercept

The y-intercept is a significant point in understanding a linear graph. It is where the line crosses the y-axis, denoted by \( b \) in the slope-intercept form \( y = mx + b \). For \( h(x) = -\frac{3}{4}x + 2 \), the y-intercept is 2.
Getting the y-intercept from a graph is simple:

• Look at where the line meets the y-axis.
• This intersection point tells you the output value when \( x \) is zero.

This is the easiest point to plot because you already know one coordinate: \( x = 0 \). From the y-intercept, you can use the slope to determine other points on the line. This makes the y-intercept an essential starting point for sketching the graph by hand.

Graphing Utilities

Graphing utilities are tools—either physical like a graphing calculator or digital as an online graphing application—that aid in accurately plotting functions. These tools are particularly helpful for confirming your hand-drawn graphs. When working with the exercise's function \( h(x) = -\frac{3}{4}x + 2 \):

• Enter the equation into a graphing calculator or software.
• The utility will render the line on the screen, allowing you to compare it with your own sketch.

These utilities often provide additional features such as zooming and adjusting scales, which can be beneficial for a deeper analysis of the graph. Such confirmation helps you check your steps and ensure the graph is accurate, enhancing your understanding of the relationship expressed by the equation.