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Chapter 3: Problem 7

Graph the given functions. $$y=2 x-4$$

Short Answer

Expert verified
Plot the y-intercept at (0, -4) and use the slope 2 to plot another point, like (1, -2), then connect these points with a straight line.

Step by step solution

01

Identify the Type of Function

The given function is in the form of a linear equation: \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. In this case, \( m = 2 \) and \( c = -4 \).
02

Determine the Y-Intercept

The y-intercept is the point where the graph crosses the y-axis. Since \( c = -4 \), the y-intercept is at the point (0, -4).
03

Determine the Slope

The slope \( m \) of the line is 2, which means for every 1 unit increase in \( x \), the value of \( y \) increases by 2 units. This tells us the steepness and direction of the line.
04

Plot the Y-Intercept

Draw a point at the y-intercept on the graph at (0, -4). This is the starting point for our line.
05

Use the Slope to Find Another Point

From the point (0, -4), use the slope to find another point on the line. Since the slope is 2, move 1 unit to the right (along the x-axis) and 2 units up to find the next point at (1, -2).
06

Draw the Line

Connect the y-intercept (0, -4) and the point (1, -2) with a straight line, extending in both directions. This line represents the graph of the equation \( y = 2x - 4 \).
07

Verify Correctness

Choose another value for \( x \) to ensure the line is correct. If \( x = 2 \), then \( y = 2(2) - 4 = 0 \). Plot (2, 0) and see if it lies on the line. It should match the line drawn.

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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 drawing a visual representation of the relationship defined by a linear equation. Linear equations have the general form \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.To graph a linear equation like \( y = 2x - 4 \):
  • Identify the y-intercept. This is where the graph crosses the y-axis. For this equation, it's at point (0, -4).
  • Determine the slope. A slope of 2 means the line rises 2 units for every 1 unit it moves to the right.
  • Use the slope to find another point. From (0, -4), move 1 unit right and 2 units up to find (1, -2).
  • Draw a straight line through the points to extend the graph in both directions.
Graphing gives a visual understanding of how changes in variables relate to each other.
Slope-Intercept Form
The slope-intercept form \( y = mx + c \) is a powerful way to express linear equations. It clearly shows both the slope and the y-intercept, making it easy to graph:
  • The slope \( m \) indicates the line's steepness and direction. A positive slope means the line rises as you move right, while a negative slope means it falls.
  • The y-intercept \( c \) represents where the line crosses the y-axis. For example, in \( y = 2x - 4 \), the y-intercept is -4, at point (0, -4).
Using slope-intercept form simplifies the process of prediction and analysis in various fields such as physics, economics, and statistics.
Plotting Points
Plotting points involves marking specific coordinates on a graph to visually represent data or functions. It starts with understanding the Cartesian plane, where:
  • Each point is represented by an \( (x, y) \) coordinate.
  • The x-coordinate shows the horizontal position, while the y-coordinate shows the vertical position.
For instance, in the equation \( y = 2x - 4 \), the point (0, -4) is the y-intercept. To plot another point using the slope, start from (0, -4), then move one step in the x-direction and two steps in the y-direction to reach (1, -2). Plotting points accurately ensures the drawn line truly represents the equation.
Linear Equation Graphing
Linear equation graphing transforms theoretical equations into concrete visual forms. This technique allows us to observe patterns and predict trends between variables:
  • Start by plotting known points, usually the y-intercept, and then any additional points found using the slope.
  • Connect these points with a straight line that stretches infinitely in both directions, symbolizing the equation's continuous nature.
For example, drawing the line for \( y = 2x - 4 \) involves connecting (0, -4) and (1, -2) and extending the line. This approach is crucial for a deeper comprehension of real-world relationships described by linear functions, such as speed versus time or cost versus quantity.

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Most popular questions from this chapter

In Exercises \(37-66,\) graph the indicated functions. Plot the graphs of (a) \(y=x+2\) and (b) \(y=\frac{x^{2}-4}{x-2} .\) Explain the difference between the graphs.

Solve the given problems. The sales tax in a city is \(7.00 \% .\) The price tag on an item is \(D\) dollars. If the store has a \(19 \%\) off price tag sale, express the total cost of the item as a function of \(D\)

Graph the given functions. $$y=6-2 x^{2}$$

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In Exercises \(37-66,\) graph the indicated functions. Plot the graphs of (a) \(y=x^{2}-x+1\) and \((b) y=\frac{x^{3}+1}{x+1}\)
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