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Chapter 1: Problem 7

Sketch a graph of the line. $$f(x)=x+3$$

Short Answer

Expert verified
The graph of the line \(f(x) = x + 3\) is a straight line that crosses the y-axis at (0,3) and has a slope of 1.

Step by step solution

01

Identify the Slope and y-intercept

Identify the slope and y-intercept from the equation. Here, the slope, \(m\), is 1 and the y-intercept, \(b\), is 3.
02

Plot the y-intercept

On a graph paper, begin by plotting the y-intercept. The y-intercept is the point where the line crosses the y-axis. This will be the point (0,3).
03

Plot Another Point Using the Slope

Use the slope to find another point. The slope is the rise over the run, that is, how much \(y\) changes for every change in \(x\). Since the slope here is 1, there is an increase of 1 in the y-direction for every increase of 1 in the x-direction. So, from the y-intercept, count 1 unit up and 1 unit to the right to plot a second point at (1,4).
04

Draw the Line

Draw a straight line through the points plotted on the graph.

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Key Concepts

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

Understanding Slope
The slope of a line represents how steep the line is and the direction it moves. It tells us how much the line goes up or down as we move across the graph. In mathematical terms, the slope is the ratio of the change in the vertical direction (y-axis) to the change in the horizontal direction (x-axis). This is often described as 'rise over run'.
If you picture walking on a hill, the slope is like the steepness of that hill.
  • If the slope is positive, the line rises as you move from left to right.
  • If the slope is negative, the line falls as you move from left to right.
  • A slope of zero means the line is flat and horizontal.
In the equation of the line given as \( f(x) = x + 3 \), the slope is 1. This means as you move 1 unit to the right along the x-axis, the line goes up by 1 unit on the y-axis. The line thus moves diagonally upwards at a 45-degree angle.
Decoding the y-intercept
The y-intercept is the point where the line crosses the y-axis. It is represented by the \( b \) in the line equation \( y = mx + b \). Understanding the y-intercept is crucial because it helps us know where to begin drawing our line on a graph.
Think of it as the starting point from which the line starts its path.
  • The y-intercept is always at the point where \( x = 0 \).
  • In the equation \( f(x) = x + 3 \), the y-intercept is 3, which means the line crosses the y-axis at the point (0, 3).
To graph a line, you would start by plotting this point on the graph. This gives you a solid beginning to then use the slope to find additional points and draw a steady line.
Graphing Lines With Ease
Graphing lines can seem tricky at first, but once you understand the basics of slope and y-intercept, it becomes a simple task. Here's how you can graph any line step by step:
  • Start by identifying the y-intercept in the line equation and plot this point on the y-axis.
  • Next, use the slope to find another point on the line. From the y-intercept, "rise" and "run" according to the slope value.
  • For instance, with a slope of 1, move 1 unit up and 1 unit to the right to locate another point.
  • Connect these points with a straight line that extends in both directions beyond the plotted points.
In doing so, you're effectively sketching the graph of a line, a fundamental skill in handling linear equations. The equation \( f(x) = x + 3 \) results in a line that can keep growing indefinitely on the graph paper due to this simple process.

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Most popular questions from this chapter

The piecewise-defined function given below is known as the characteristic function, \(C(x) .\) It plays an important role in advanced mathematics. $$C(x)=\left\\{\begin{array}{ll}0, & \text { if } x \leq 0 \\\1, & \text { if } 0

Solve the inequality. Express your answer in interval notation. $$0<\frac{x+3}{2}<3$$

Solve the inequality. Express your answer in interval notation. $$2-2 x \geq x-1$$

Graph the function by hand. $$F(x)=\left\\{\begin{array}{ll} 0, & x \leq 1 \\ 2, & x>1 \end{array}\right.$$

Solve the inequality. Express your answer in interval notation. $$\frac{x}{3} \leq \frac{2 x}{3}-1$$
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