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Chapter 1: Problem 10

Graph. Find the slope and the \(y\) -intercept. $$ y=-x+4 $$

Short Answer

Expert verified
Slope = -1, y-intercept = 4.

Step by step solution

01

Identify the Slope-Intercept Form

The equation of the line is given in the slope-intercept form: \( y = mx + b \). In this form, \( m \) represents the slope, and \( b \) represents the \( y \)-intercept.
02

Extract the Slope

Compare the given equation \( y = -x + 4 \) with the slope-intercept form \( y = mx + b \). The coefficient of \( x \) in this equation is \( -1 \). Thus, the slope \( m \) is \( -1 \).
03

Find the Y-Intercept

In the equation \( y = -x + 4 \), the constant term is \( 4 \). Thus, the \( y \)-intercept \( b \) is \( 4 \).

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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 is one of the most common ways to write the equation of a straight line. It looks like this:
y = mx + b
In this formula:
  • y is the value on the y-axis.
  • m is the slope of the line, representing the steepness and direction.
  • x is the value on the x-axis.
  • b is the y-intercept, which is where the line crosses the y-axis.
The slope-intercept form makes it easy to identify both the slope and the y -intercept directly from the equation. This way, you can quickly plot the line on a graph.
graphing linear equations
To graph linear equations, you typically start by finding the y -intercept and the slope. Using the equation y = -x + 4 as an example:
  • First, identify the y-intercept (b). In our equation, b = 4, which tells us that the line crosses the y -axis at y = 4. Plot this point on the graph.
  • Next, use the slope (m) to determine the direction and steepness of the line. Our slope is -1. This means for every step you take to the right on the x-axis (positive x direction), you take one step down on the y -axis (negative y direction). Plot another point using this rule.
  • Connect the points with a straight line, extending it in both directions. Now the linear equation is graphed!
It's a good practice to check your work by picking a value for x and substituting it back into your equation to find y. Plot this point and see if it falls on your line.
extracting slope
Extracting the slope from a linear equation in slope-intercept form is straightforward. Let's look at our example equation y = -x + 4:
  • First, identify the part of the equation in the form y = mx + b.
  • In our example, compare it with y = mx + b. The slope (m) is the coefficient of x.
  • So here, the slope m is -1.
The slope tells us how steep the line is and in which direction it goes. A negative slope means the line goes down as it moves from left to right. Conversely, a positive slope means the line goes up as it moves from left to right.
finding y-intercept
Finding the y -intercept in a linear equation written in slope-intercept form is also simple. Let's revisit our example y = -x + 4:
  • The y-intercept is represented by the constant term in the equation (b).
  • In our equation, the y -intercept b is 4.
  • This means the line crosses the y -axis at y = 4.
  • To graph this, place a point on the y -axis where y equals 4.
This point is where the line hits the y-axis, indicating one of the key points needed to draw your line accurately. Understanding the y -intercept helps you position the graph of the equation correctly on a coordinate plane.

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Most popular questions from this chapter

Trec Volume. The following function approximates the proportion \(V\) of a tree's volume that lies below height \(h:\) $$\begin{aligned}V(h)=& \sin ^{p}\left(\frac{\pi h}{2}\right) \sin ^{q}\left(\frac{\pi \sqrt{h}}{2}\right) \\ & \times \sin ^{r}\left(\frac{\pi \sqrt[3]{h}}{2}\right) \sin ^{s}\left(\frac{\pi \sqrt[4]{h}}{2}\right)\end{aligned}$$ In this equation, the exponents \(p, q, r\), and \(s\) are constants and \(h\) is the proportion of the total height of the tree. For example, \(h=0\) corresponds to the bottom of the tree, \(h=1\) corresponds to the top, and \(h=1 / 2\) corresponds to halfway up the tree. \({ }^{18}\) a) Compute \(V(0)\) and \(V(1)\). b) Explain your answers to part (a) in terms of tree volume.

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