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Chapter 1: Problem 19

Find an equation of the given line. Slope is \(-2 ; x\) -intercept is \(-2\).

Short Answer

Expert verified
The equation of the line is y = -2x - 4.

Step by step solution

01

Understand the given parameters

The problem provides two pieces of information: the slope of the line, which is -2, and the x-intercept, which is the point where the line crosses the x-axis, given as -2.
02

Use the slope-intercept form of a line

The slope-intercept form of a line is given by the equation y = mx + b, where m is the slope and b is the y-intercept. In this problem, we must find y-intercept (b) using the given x-intercept.
03

Determine the y-intercept (b)

Since the line crosses the x-axis at the x-intercept, the coordinates of this point can be written as (-2, 0). Substitute this point and the slope into the slope-intercept form of the line: 0 = (-2)(-2) + b.
04

Solve for b

Calculate as follows: 0 = 4 + b So, b = -4.
05

Write the final equation of the line

Substitute the value of the slope (-2) and y-intercept (-4) back into the slope-intercept form of the equation: y = -2x - 4.

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Key Concepts

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

slope-intercept form

Let's start by understanding the slope-intercept form of a line, which is fundamental for finding the equation of a line. The slope-intercept form of a line is given by:
\( y = mx + b \).

This form is especially useful because it directly shows the slope and y-intercept of the line, making it easier to graph.

In the equation, the term \( m \) represents the slope of the line.
  • This indicates how steep the line is and in which direction it slants.

Meanwhile, the term \( b \) represents the y-intercept.
  • This is the point where the line crosses the y-axis.

So, if you know both the slope and the y-intercept, you can easily write the equation of a line in slope-intercept form, making it a critical concept to master.
slope

The slope is a measure of how steep a line is and the direction it goes.

In our problem, the slope is given as \( -2 \).

  • If the slope is positive, the line rises as it moves from left to right.
  • If the slope is negative, the line falls as it moves from left to right.

To calculate the slope, we use the formula: \(\frac{y_2 - y_1}{x_2 - x_1}\).

So, why is knowing the slope important?
  • It allows you to understand the rate of change between the two variables in the equation.
  • It also helps you determine the steepness and direction of the line.
intercept

In this exercise, the intercepts play a significant role as they help in determining the exact position of the line on a graph.

There are two types of intercepts:
  • The y-intercept, which we denote as \( b \), is where the line crosses the y-axis.

The y-intercept is essential because it acts as a starting point when graphing the line, and shows the value of the function when \( x \) is 0.

The exercise gives us the x-intercept as \( -2 \), which is the point where the line crosses the x-axis.

Given this x-intercept and the slope, we can find the y-intercept by using the slope-intercept form of a line and substituting the known values into it.
This step-by-step approach makes it easier to visualize and understand the line's position and trajectory on the graph.

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Most popular questions from this chapter

Use limits to compute the following derivatives. \(f^{\prime}(2)\), where \(f(x)=x^{3}\)

Suppose that \(f(t)=t^{2}+3 t-7\) (a) What is the average rate of change of \(f(t)\) over the interval 5 to \(6 ?\) (b) What is the (instantaneous) rate of change of \(f(t)\)

Compute the limits that exist, given that $$ \lim _{x \rightarrow 0} f(x)=-\frac{1}{2} \text { and } \lim _{x \rightarrow 0} g(x)=\frac{1}{2} \text { . } $$ (a) \(\lim _{x \rightarrow 0}(f(x)+g(x))\) (b) \(\lim _{x \rightarrow 0}(f(x)-2 g(x))\) (c) \(\lim _{x \rightarrow 0} f(x) \cdot g(x)\) (d) \(\lim _{x \rightarrow 0} \frac{f(x)}{g(x)}\)

Determine whether each of the following functions is continuous and/or differentiable at \(x=1\). \(f(x)=x^{2}\)

Use limits to compute \(f^{\prime}(x)\). [Hint: In Exercises \(45-48\), use the rationalization trick of Example \(8 .]\) \(f(x)=-1+\frac{2}{x-2}\)
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