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Chapter 3: Problem 7

Sketch the line passing through the point \((-4,2)\) with slope \(-1 / 4\).

Short Answer

Expert verified
The line is \( y = -\frac{1}{4}x + 1 \), passing through points like \((0, 1)\) and \((4, 0)\).

Step by step solution

01

Understand the slope-intercept form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept.
02

Identify the slope and point

From the problem, the slope \( m = -\frac{1}{4} \). The line passes through the point \((-4, 2)\). We will use this point to find the y-intercept \( b \).
03

Substitute into the slope-intercept formula

Substitute \( m = -\frac{1}{4} \) into the equation and use the point \((-4, 2)\) to solve for \( b \). Set up the equation: \( 2 = -\frac{1}{4}(-4) + b \).
04

Solve for the y-intercept \( b \)

Compute \(-\frac{1}{4}(-4) = 1\), then substitute back to get \( 2 = 1 + b \). Solve for \( b \) by subtracting 1 from both sides: \( b = 1 \).
05

Write the equation of the line

The equation of the line using the slope \(-\frac{1}{4}\) and y-intercept \( b = 1 \) is \( y = -\frac{1}{4}x + 1 \).
06

Plot the line

To sketch the line, start by plotting the y-intercept at \( (0, 1) \). From this point, use the slope, \( -\frac{1}{4} \), to determine the direction: move 4 units to the right and 1 unit down to get another point, \( (4, 0) \). Connect these points to sketch the line.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form is one of the most straightforward ways to represent a linear equation, especially useful for graphing and understanding the behavior of the line. This form is given by:
  • \( y = mx + b \)
  • Here, \( m \) denotes the slope of the line, indicating both its steepness and direction.
  • The \( b \) represents the y-intercept, which is where the line crosses the y-axis.
Understanding this formula allows us to quickly determine how a line will appear on a graph. The slope \( m \) affects whether the line ascends, descends, or remains flat. Meanwhile, the y-intercept \( b \) provides a starting point for graphing. It's crucial to remember that the slope-intercept form is versatile, suitable for various types of problems involving linear equations.
Linear Equations
Linear equations are equations that create straight lines when graphed in a coordinate plane. They play a fundamental role in algebra and analytic geometry. Here's why they're important:
  • They have a constant rate of change, represented by the slope.
  • They only include variables raised to the first power (no exponents).
  • Their graph is always a straight line, which can be vertical, horizontal, or slanting.
In our exercise, we dealt with a linear equation in the slope-intercept form. To find the specific line, we identified the slope as \(-\frac{1}{4}\) and used a point it passes through to determine the y-intercept. By using the linear equation, we could easily derive the line's equation \( y = -\frac{1}{4}x + 1 \). This line showcases a negative slope, meaning it descends from left to right.
Graphing a Line
Graphing a line involves translating the linear equation into a visual representation on the coordinate plane. Here's how you can do it in several efficient steps:
  • Start by identifying the y-intercept \( b \) from the equation, as this gives your initial point on the graph at \((0, b)\).
  • Use the slope \( m \), particularly if it is in the form of a fraction like \(-\frac{1}{4}\), to determine how the line moves across the plane. From the y-intercept, move according to the slope: go 4 units right and 1 unit down to locate a second point.
  • With these two points marked on the graph, draw a straight line through them. This line is the graphical representation of your equation.
Each graph offers an intuitive understanding of the linear relationship defined by your equation. By plotting the line, you visualize both its slope and y-intercept clearly, helping in understanding more complex analytic geometry concepts.

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Most popular questions from this chapter

Approximate the point of intersection of the graphs of \(y=x-4\) and \(y=5-x\) using the TRACE button.

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Find the equation of the line passing through the points \(P(-2,1)\) and \(Q(4,-1)\).

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