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Chapter 1: Problem 4

Change the equation \(y-2 x=2\) to the form \(y=m x+b,\) graph the line, and find the \(y\) -intercept and \(x\) -intercept. \(\Rightarrow\)

Short Answer

Expert verified
Equation: \( y = 2x + 2 \). y-intercept: 2, x-intercept: -1.

Step by step solution

01

Rearrange the Equation

Start with the given equation: \( y - 2x = 2 \). To express it in the form \( y = mx + b \), add \( 2x \) to both sides of the equation to get \( y = 2x + 2 \). Now, the equation is in the desired form where \( m \) is the slope and \( b \) is the y-intercept.
02

Identify the y-intercept

In the equation \( y = 2x + 2 \), the term \( b \) represents the y-intercept. Here, \( b = 2 \), so the y-intercept is the point \( (0, 2) \).
03

Identify the x-intercept

To find the x-intercept, set \( y = 0 \) in the equation \( y = 2x + 2 \). Solving for \( x \), we have \( 0 = 2x + 2 \), which simplifies to \( 2x = -2 \), and dividing by 2 gives \( x = -1 \). Hence, the x-intercept is the point \( (-1, 0) \).
04

Graph the Equation

Begin by plotting the y-intercept \( (0, 2) \) on the graph. Then plot the x-intercept \( (-1, 0) \). Draw a line through these two points to represent the equation \( y = 2x + 2 \). Ensure your line extends across the entire graph, indicating both intercepts.

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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 a simple and efficient way to express straight lines. In this form, the equation is written as \( y = mx + b \). Here, \( m \) represents the slope of the line and \( b \) represents the y-intercept.
  • The slope \( m \) indicates how steep the line is. It tells you the change in y for a one-unit change in x.
  • The y-intercept \( b \) shows where the line crosses the y-axis. It is the value of y when x is 0.
This format is convenient for quickly assessing the line's properties, like steepness and intersection with the axes. It is often preferred because it provides clear insight into the equation's graphical representation.
Y-Intercept
The y-intercept is the point where the line crosses the y-axis. In the slope-intercept equation \( y = mx + b \), the y-intercept is directly represented by the constant term \( b \). This is because it corresponds to the value of y when x is zero.
For example, in the equation \( y = 2x + 2 \), the y-intercept is 2, which means the line crosses the y-axis at the point \( (0, 2) \).
Finding the y-intercept involves examining the equation when x is 0. This can provide useful information for graphing the line or when analyzing how a function starts on a graph.
  • Quick and easy to determine.
  • Helps in plotting the first point on the graph.
X-Intercept
The x-intercept is the point where a graph crosses the x-axis. To find it, you need to set y equal to zero in the equation and solve for x. This gives the value of x when the line meets the x-axis.
In the example equation \( y = 2x + 2 \), setting \( y = 0 \) leads to the equation \( 0 = 2x + 2 \). By solving for \( x \), you find that \( x = -1 \). Thus, the x-intercept is at \( (-1, 0) \).
  • This point is crucial for understanding where a line will cross the x-axis.
  • Often used in conjunction with the y-intercept to easily plot the graph of the line.
The x-intercept helps graphically demonstrate where a line starts or stops along the x-axis.
Graphing Linear Equations
Graphing a linear equation means plotting it on a coordinate axis so that its characteristics become visually evident. To start, you can plot the y-intercept, as it is straightforward given by the equation \( y = mx + b \).
  • Begin by placing a point on the y-axis at the y-intercept, such as \( (0,2) \) in our example.
  • Then, find the x-intercept by setting y to zero and solving for x. Plot this point, \( (-1,0) \) here.
  • Join these two points with a straight line to reveal the full extent of the linear equation.
  • Extend the line across the graph to fully visualize the line's path.
This process not only makes the line visible but also gives important insights into its direction and intersection points. Graphing helps confirm the mathematical solution and is an essential skill in visualizing equations.

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Most popular questions from this chapter

Market research tells you that if you set the price of an item at \(\$ 1.50,\) you will be able to sell 5000 items; and for every 10 cents you lower the price below \(\$ 1.50\) you will be able to sell another 1000 items. Let \(x\) be the number of items you can sell, and let \(P\) be the price of an item. (a) Express \(P\) linearly in terms of \(x\), in other words, express \(P\) in the form \(P=m x+b\). (b) Express \(x\) linearly in terms of \(P . \Rightarrow\)

Starting with the graph of \(y=\sqrt{x},\) the graph of \(y=1 / x,\) and the graph of \(y=\sqrt{1-x^{2}}\) (the upper unit semicircle), sketch the graph of each of the following functions: $$f(x)=1+1 /(x-1)$$

Find the domain of each of the following functions: $$y=f(x)=\sqrt[4]{x} \Rightarrow$$

Graph the circle \(x^{2}-6 x+y^{2}-8 y=0\).

Suppose \(f(x)=3 x-9\) and \(g(x)=\sqrt{x}\). What is the domain of the composition \((g \circ f)(x) ?\) (Recall that composition is defined as \((g \circ f)(x)=g(f(x)) .)\) What is the domain of \((f \circ g)(x) ? \Rightarrow\)
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