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Chapter 2: Problem 120

For the following exercises, sketch the graph of each equation. $$f(x)=-2 x-1$$

Short Answer

Expert verified
Sketch the line passing through points (0, -1) and (1, -3) with slope -2.

Step by step solution

01

Understand the Equation Form

The equation given is a linear equation in the slope-intercept form: \( f(x) = -2x - 1 \). Here, \(-2\) is the slope (m) and \(-1\) is the y-intercept (b). This means the graph is a straight line.
02

Identify the Y-Intercept

The y-intercept of the line is the point where the line crosses the y-axis. From the equation \( f(x) = -2x - 1 \), the y-intercept is \(-1\). Thus, the line passes through the point \((0, -1)\). Plot this point on the graph.
03

Use the Slope to Find Another Point

The slope of \(-2\) means that for every 1 unit increase in \(x\), \(f(x)\) decreases by 2 units. Starting from the y-intercept \((0, -1)\), move 1 unit to the right (to \(x=1\)) and 2 units down to \(f(x)=-3\). This gives the next point \((1, -3)\). Plot this point.
04

Draw the Line

Use a ruler to draw a straight line through the points \((0, -1)\) and \((1, -3)\). Extend the line in both directions, adding arrows on either end to indicate it continues infinitely. This line represents the graph of \(f(x) = -2x - 1\).

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Key Concepts

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

Graphing functions
Graphing a function is like painting a picture on a sheet of paper. You're essentially turning the equation into a visual image. The equation we have is in the form of a linear function, which means that its graph will be a straight line. When graphing any function, you first need to identify key information, such as the slope and y-intercept. Once you gather this information, you can start plotting the function.

Here's a simple plan to graph functions effectively:
  • Identify the type of equation: Is it linear, quadratic, or another form?
  • Find the y-intercept: This is where you'll start on the graph.
  • Next, use the slope to determine the direction and steepness of the line.
  • Plot another point using the slope, then draw a line through your plotted points.
  • Ensure the graph is extended using arrows to show it continues indefinitely.
Once you get the hang of these basics, graphing becomes more intuitive and less of a chore.
Slope-intercept form
The slope-intercept form is a straightforward way to express a line's equation. It is written as \( y = mx + b \), where:
  • \( m \) is the slope, showing how steep the line is.
  • \( b \) is the y-intercept, indicating where the line crosses the y-axis.
Knowing the slope-intercept form allows you to quickly graph linear equations and understand the line's characteristics.

The slope \( m \) reflects the change in the y-coordinate (vertical change) over the change in the x-coordinate (horizontal change). If \( m = -2 \), it means that for each step to the right, you move 2 steps down. This helps in plotting additional points on the line.

This form is user-friendly because you can immediately see the slope and y-intercept, making it easy to graph or analyze the line without additional calculation.
Y-intercept
The y-intercept is a crucial point in graphing a line because it marks where the line intersects the y-axis. In our equation \( f(x) = -2x - 1 \), the y-intercept is \(-1\). This means when the x-value is 0, the y-value is -1, giving you the point \((0, -1)\) on the graph.

Understanding the y-intercept helps in establishing a starting point for drawing the line on the graph. By knowing where the line crosses the y-axis, you can accurately represent the function on a coordinate plane.
  • First, plot the y-intercept on the y-axis.
  • This provides a reference point for further plotting using the slope.
The y-intercept simplifies the initial setup of a graph, ensuring you begin with an accurate position before extending the line across the plane.

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Most popular questions from this chapter

For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular: $$2 x-6 y=12$$ $$-x+3 y=1$$

Write an equation for a line perpendicular to \(f(x)=5 x-1\) and passing through the point \((5,20)\).

Find the \(x\) - and \(y\) -intercepts of the equation \(2 x+7 y=-14\)

For the following exercises, consider this scenario: In 2000, the moose population in a park was measured to be 6,500. By 2010, the population was measured to be 12,500. Assume the population continues to change linearly. What does your model predict the moose population to be in 2020?

Write an equation for a line perpendicular to \(f(x)=4 x+3\) and passing through the point \((8,10)\).
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