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Chapter 2: Problem 2

Sketch the graph of the equation, and label the \(x\) - and \(y\) -intercepts. $$y=4 x+2$$

Short Answer

Expert verified
The x-intercept is \(-\frac{1}{2}\) and the y-intercept is 2.

Step by step solution

01

Identify the Slope and Y-intercept

The equation is given in the slope-intercept form, which is \( y = mx + b \). Here, \( m = 4 \) and \( b = 2 \). The slope \( m \) is 4, and the y-intercept \( b \) is 2.
02

Determine the Y-intercept

The y-intercept is the point on the graph where \( x = 0 \). Substitute \( x = 0 \) into the equation: \( y = 4(0) + 2 = 2 \). Thus, the y-intercept is (0, 2).
03

Find the X-intercept

The x-intercept is the point where \( y = 0 \). Set \( y = 0 \) in the equation: \( 0 = 4x + 2 \). Solve for \( x \): \( 4x = -2 \), so \( x = -\frac{1}{2} \). Thus, the x-intercept is \( \left(-\frac{1}{2}, 0\right) \).
04

Plot the Intercepts and Draw the Line

On the coordinate plane, plot the y-intercept (0, 2) and the x-intercept \( \left(-\frac{1}{2}, 0\right) \). Draw a straight line through these two points to represent the equation \( y = 4x + 2 \).
05

Conclusion: Label the Intercepts on the Graph

Ensure the graph is labeled with the points: the x-intercept \( \left(-\frac{1}{2}, 0\right) \) and the y-intercept (0, 2). This fully characterizes the line in the graph.

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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 of a linear equation is one of the most commonly used formats to represent a straight line. It's expressed as \( y = mx + b \), where:
  • \( m \) represents the slope of the line.
  • \( b \) represents the y-intercept, which is the point where the line crosses the y-axis.
When an equation is written in this form, it's straightforward to identify both the slope and the y-intercept without having to manipulate the equation further. The slope \( m \) describes how steep the line is: a larger absolute value of \( m \) indicates a steeper line. If \( m \) is positive, the line rises as we move from left to right. Conversely, if \( m \) is negative, the line falls. The y-intercept \( b \) helps in positioning the line on the graph. Knowing both values makes sketching a graph quick and easy.
X-Intercept
The x-intercept is the point where a graph crosses the x-axis. At this point, the y-coordinate is always zero. To find the x-intercept of an equation, you set \( y = 0 \) and solve for \( x \). For example, in the equation \( y = 4x + 2 \), setting \( y = 0 \) gives us the equation:\[0 = 4x + 2\]To find the value of \( x \), rearrange this equation:
  • Subtract 2 from both sides: \( 4x = -2 \).
  • Divide by 4: \( x = -\frac{1}{2} \).
Thus, the x-intercept is \( \left(-\frac{1}{2}, 0\right) \). Spotting the x-intercept is crucial for graphing because it tells us where the line will cross the horizontal axis, anchoring one end of our line.
Y-Intercept
The y-intercept is where a graph intersects the y-axis. At this point, the x-coordinate is always zero. Calculating the y-intercept from an equation is straightforward when it is in the slope-intercept form \( y = mx + b \). In this form, the y-intercept \( b \) is already provided as part of the equation, which for \( y = 4x + 2 \), is 2. By substituting \( x = 0 \) into the equation, we confirm this calculation:\[y = 4(0) + 2 = 2\]Thus, the point where the line crosses the y-axis in this case is (0, 2). The y-intercept serves as a starting point for sketching the graph, and knowing it helps you draw the line more accurately.

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