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Chapter 5: Problem 21

The form of the expression for the function tells you a point on the graph and the slope of the graph. What are they? Sketch the graph. $$ g(s)=(s-1) / 2+3 $$

Short Answer

Expert verified
Answer: The slope of the given function is 1/2, and one point on the graph is (0, 5/2).

Step by step solution

01

Recognize the slope-intercept form of the linear equation

In order to analyze the given function, we first need to recognize that it is already in the slope-intercept form of a linear equation, which is given by: $$ y = mx + b $$ where m is the slope and b is the y-intercept. Our given function is: $$ g(s) = (s-1)/2 + 3 $$ which can be written as: $$ g(s) = \frac{1}{2}s - \frac{1}{2} +3 $$
02

Identify the slope and y-intercept

Now that we have the function in the form of a linear equation, we can identify the slope and y-intercept. Comparing with the general form (y = mx + b), we find that: Slope (m) = \(\frac{1}{2}\) Y-intercept (b) = -\(\frac{1}{2}\) + 3 = \(\frac{5}{2}\) Therefore, our slope is \(\frac{1}{2}\) and the y-intercept is \(\frac{5}{2}\).
03

Find a point on the graph

Since we now have the y-intercept value, we can easily find a point on the graph. The y-intercept is the point at which the line intersects the y-axis (s = 0). When s = 0, g(s) = \(\frac{5}{2}\). Therefore, the point on the graph corresponding to the y-intercept is (0, \(\frac{5}{2}\)).
04

Sketch the graph

To sketch the graph of our function, we follow these steps: 1. Plot the y-intercept point on the graph, (0, \(\frac{5}{2}\)). 2. Since the slope is \(\frac{1}{2}\), for every unit increase in the s-coordinate, the g(s)-coordinate will increase by \(\frac{1}{2}\). So starting from the y-intercept point, move one unit to the right and \(\frac{1}{2}\) unit up and draw a second point. 3. Connect these two points to draw the graph of our function. In conclusion, the slope of our function is \(\frac{1}{2}\), and the y-intercept is \(\frac{5}{2}\). One of the points on the graph is (0, \(\frac{5}{2}\)).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope-Intercept Form
Understanding the slope-intercept form is essential for graphing linear equations. The slope-intercept form of a linear equation is given by the formula:\[ y = mx + b \] where:
  • \( m \) represents the slope of the line.
  • \( b \) is the y-intercept, which is the point where the line crosses the y-axis.
This form makes it easy to quickly identify and interpret the key components of a linear function.
The slope (\( m \)) tells us how steep the line is, while the y-intercept (\( b \)) provides a specific point on the graph.By converting any given linear equation into the slope-intercept form, we gain a clearer understanding of the line's behavior on the graph.
Graphing Linear Functions
To graph a linear function, you can follow a straightforward process using its slope-intercept form. Start with identifying the y-intercept, which is the point where the graph touches the y-axis. For example, if the intercept is \( \frac{5}{2} \), the first point will be \( (0, \frac{5}{2}) \).
  • After plotting the y-intercept, the next step involves using the slope (\( m \)). In our case, it is \( \frac{1}{2} \).
  • This means for every 1 unit you move right along the s-axis, you move \( \frac{1}{2} \) unit up along the g(s)-axis.
Draw a second point by moving from the y-intercept using the slope as a guide, then connect these points.
This will give you a straight line, representing the linear function's graph. It's as simple as connecting the dots once you know your y-intercept and slope.
Y-Intercept
The y-intercept plays a pivotal role in graphing and understanding linear equations. It is the point where the line crosses the y-axis, which occurs when the independent variable (often \( x \) or \( s \)) is zero. In the context of the function \( g(s) = \frac{1}{2}s + \frac{5}{2} \), the y-intercept can be calculated by setting \( s = 0 \). This calculation gives us \( g(0) = \frac{5}{2} \), so the y-intercept is \( (0, \frac{5}{2}) \).
  • The y-intercept provides a starting point or a reference to plot the graph of a linear function.
  • Knowing this point helps visualize the function's path on the graph efficiently.
By identifying the y-intercept, you can easily begin the graph of a linear equation and use it in conjunction with the slope to plot other points on the graph.

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