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Chapter 5: Problem 42

Identify the slope and \(y\) -intercept, then graph the line. $$y=\frac{1}{4} x-2$$

Short Answer

Expert verified
The slope is \(\frac{1}{4}\) and the y-intercept is \(-2\). To graph the line, plot the y-intercept at \((0, -2)\) and then use the slope to find another point on the line by moving 4 units to the right and 1 unit up to reach \((4, -1)\). Connect these two points with a straight line to represent the graph of the equation \(y = \frac{1}{4}x - 2\).

Step by step solution

01

Identify the slope and y-intercept

The given equation is \(y = \frac{1}{4}x - 2\), which is already in the slope-intercept form. Here, the slope (m) is \(\frac{1}{4}\) and the y-intercept (b) is \(-2\).
02

Plot the y-intercept

Start by plotting the y-intercept on the graph. The y-intercept is the point where the line crosses the y-axis, so it has coordinates \((0, -2)\). Place a point at this coordinate on the y-axis.
03

Use the slope to find another point on the line

The slope of the line is the ratio of the change in y (rise) to the change in x (run). In this case, the slope is \(\frac{1}{4}\), which means for every 4 units we move in the positive x direction (right), we move 1 unit in the positive y direction (up). Starting at the y-intercept that we plotted in Step 2, \((0, -2)\), move 4 units to the right and 1 unit up. This will give us the point \((4, -1)\). Place a point at this coordinate on the graph.
04

Draw the line

Now that you have two points on the graph, the y-intercept \((0, -2)\) and another point \((4, -1)\), simply connect these points using a straight line. This line represents the graph of the given equation, \(y = \frac{1}{4}x - 2\).

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Key Concepts

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

graphing linear equations
Graphing linear equations involves translating an equation into a visual line on a coordinate plane. This line represents all possible solutions to the equation. For linear equations, which have the standard form of **y = mx + b**, the graph will always be a straight line. The **m** in this formula is the slope, while **b** is the y-intercept. To graph a linear equation:
  • Write the equation in slope-intercept form (if needed).
  • Identify the slope (**m**) and y-intercept (**b**).
  • Begin by plotting the y-intercept on the graph.
  • Utilize the slope to determine the direction and steepness of the line.
  • Draw a line passing through the plotted points, which extends infinitely in both directions.
By following these steps, you'll transform this algebraic statement into a graphical representation, helping visualize its behavior and solutions.
identifying slope
Identifying the slope is an essential step when working with linear equations in slope-intercept form, which is given by **y = mx + b**. Here, the slope **m** demonstrates how steep the line is and which direction it goes.The slope is usually expressed as a fraction where:
  • The numerator indicates the vertical change, known as the "rise."
  • The denominator indicates the horizontal change, known as the "run."
In the equation **y = \( \frac{1}{4} \)x - 2**, the slope **m** is \( \frac{1}{4} \). This means as you move 4 units horizontally (to the right), the line moves 1 unit vertically (upwards). Therefore, this slope not only tells us the angle of the line but also guides us on how to extend it across the graph.
y-intercept
The y-intercept is another crucial component of the slope-intercept form equation, located at the point where the line crosses the y-axis. It's simply the **b** value in the formula **y = mx + b**. For a given line, the y-intercept tells you where your graph starts vertically. It has coordinates of **(0, b)**, matching where **x = 0**.In the linear equation **y = \( \frac{1}{4} \)x - 2**, the y-intercept **b** is \(-2\). This tells us that when x is zero, y is **-2**, placing our initial point at the vertical position **(0, -2)**.This starting point acts as an anchor for your line, providing a reference from which to apply the slope, further mapping out the line onto the plane.
linear equations graphing steps
Graphing a linear equation involves a series of straightforward steps. These steps ensure that the equation is correctly represented on a graph, providing a visual insight into its solutions and behavior.
  • **Step 1**: Express the equation in slope-intercept form, ensuring its clear use of **y = mx + b**. If needed, manipulate the equation algebraically to reach this form.
  • **Step 2**: Locate and plot the y-intercept (b), placing a point on the y-axis where **x = 0**.
  • **Step 3**: Use the slope (m) to find another point, moving from your y-intercept according to the rise/run of the slope.
  • **Step 4**: Once a second point is determined, draw a straight line through both plotted points. Extend this line across the graph in both directions.
By repeating these steps for different equations, you gain a reliable method to chart and understand linear relationships visually and effectively.

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Most popular questions from this chapter

Solve each systen \(3 x+4 y=-6\) \(-x+3 z=1\) \(2 y+3 z=-1\)

Write an equation of the line perpendicular to the given line and containing the given point. Write the answer in slope intercept form or in standard form, as indicated. \(y=-x+11 ;(-10,-8) ;\) slope-intercept form

Write the slope-intercept form of the equation of the line, if possible, given the following information. \(m=7\) and contains \((2,5)\)

Ticket prices to a Cubs game at Wrigley Ficld vary depending on whether they are on a value date, a regular date, or a prime date. At the beginning of the 2005 season, Bill, Corrinne, and Jason bought tickets in the bleachers for several games. Bill spent \(\$ 286\) on four value dates, four regular dates, and three prime dates. Corrinne bought tickets for four value dates, three regular dates, and two prime dates for \(\$ 220 .\) Jason spent \(\$ 167\) on three value dates, three regular dates, and one prime date. How much did it cost to sit in the bleachers at Wrigley Field on a value date, regular date, and prime date in \(2005 ?\) (Source: http \(J /\) chicago cubs. mlb.com)

Solve each system \(\begin{aligned} 2 x-y+4 z &=-1 \\ x+3 y+z &=-5 \\\\-3 x+2 y &=7 \end{aligned}\)
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