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Chapter 4: Problem 40

Graph each function using the slope and \(y\) -intercept. $$g(x)=-\frac{1}{4} x-2$$

Short Answer

Expert verified
The slope of the given function \(g(x)=-\frac{1}{4}x-2\) is \(-\frac{1}{4}\) and the y-intercept is \(-2\). We plot the y-intercept (0, -2) on the graph and then use the slope to find another point by moving down 1 unit and right 4 units to get the point (4, -3). Finally, draw a line through these two points to graph the function.

Step by step solution

01

Identify the slope and y-intercept

The given equation is in the form \(g(x) = mx + b\). Comparing it with the given function, we have: - Slope (m) = \(-\frac{1}{4}\) - Y-intercept (b) = \(-2\)
02

Plot the y-intercept

The y-intercept is the point where the function crosses the y-axis. Since our y-intercept is -2, we plot the point (0, -2) on the y-axis.
03

Use the slope to find another point

We now use the slope to find another point on the function. The slope is the ratio of the change in y to the change in x, or "rise over run." Since the slope is \(-\frac{1}{4}\), we can say the rise is -1 and the run is 4. Starting from the y-intercept (0, -2), we can find a new point by: - Moving down 1 unit (rise = -1) - Moving right 4 units (run = 4) This brings us to the point (4, -3). Plot this point on the graph.
04

Draw the line

Now that we have two points on the function, we can draw a straight line through these points. The line will represent the graph of the function \(g(x)=-\frac{1}{4}x-2\). And with that, we have graphed the function using the slope and y-intercept.

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Key Concepts

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

Slope and y-intercept
Understanding the slope and y-intercept of a line is crucial in graphing linear functions. These components come from the equation of a line which is often expressed in slope-intercept form: \[ y = mx + b \]
  • Slope (m): The slope measures how steep the line is and in which direction it tilts. If the slope is positive, the line rises from left to right; if it's negative, the line falls from left to right. In the equation \(g(x) = -\frac{1}{4}x - 2\), the slope is \(-\frac{1}{4}\), meaning the line falls with a gentle decline as you move along the x-axis.
  • Y-intercept (b): This is the point where the line crosses the y-axis. It represents the value of \(y\) when \(x = 0\). In our equation, the y-intercept is -2, which means the line crosses the y-axis at the point (0, -2).
Equation of a line
The equation of a line in slope-intercept form is pivotal for understanding linear functions. The general form is \[ y = mx + b \]where \(m\) represents the slope and \(b\) is the y-intercept.Graphing a line using its equation helps in determining its direction and starting point on the graph:
  • Given equation \(g(x) = -\frac{1}{4}x - 2\), tells us that every time \(x\) increases by 4, \(y\) decreases by 1.
  • The combination of these numbers allows us to systematically plot the line on the log of coordinates.
This form of the equation creates a straightforward process to analyze linear relationships, such as how variables interact with one another.
Plotting points
Plotting points is the backbone of drawing graphs, especially for linear equations. To effectively graph a linear function like \(g(x) = -\frac{1}{4}x - 2\), you need to follow these simple steps:
  • Start with the y-intercept: Always begin plotting your graph from the y-intercept, which in this case is -2. This point is placed on the y-axis because it's where \(x = 0\).
  • Utilize the slope: The slope \(-\frac{1}{4}\) provides a guideline on how to move to your next point. Begin from the y-intercept, move down 1 unit (since the rise is -1), then to the right 4 units (since the run is 4). This brings us to point (4, -3).
  • Draw the line: With at least two points, draw a straight line through these plots.
Accurate plotting of points and drawing the line correctly ensures a precise representation of the linear function on the graph.

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Most popular questions from this chapter

Determine the domain of each relation, and determine whether each relation describes \(y\) as a function of \(x .\) $$y=-\frac{4}{9 x+8}$$

Graph each function using the slope and \(y\) -intercept. $$h(x)=-\frac{5}{2} x+4$$

Determine the domain of each relation, and determine whether each relation describes \(y\) as a function of \(x .\) $$y=\frac{3}{x-5}$$

Lines \(L_{1}\) and \(L_{2}\) contain the given points. Determine if lines \(L_{1}\) and \(L_{2}\) are parallel, perpendicular, or neither. $$\begin{aligned}&L_{1}:(-3,9),(4,2)\\\&L_{2}:(6,-8),(-10,8)\end{aligned}$$

Write the slope-intercept form of the equation of the line, if possible, given the following information. contains \((0,2)\) and \((6,0)\)
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