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Chapter 0: Problem 1

Graph. (Unless directed otherwise, assume that "Graph" means "Graph by hand.") \(y=x-1\)

Short Answer

Expert verified
The line crosses the y-axis at (0, -1) and has a slope of 1.

Step by step solution

01

Identify the Type of Equation

The equation given is in the form \(y = mx + b\), which is known as the slope-intercept form of a line. Here, \(m\) is the slope, and \(b\) is the y-intercept.
02

Determine the Slope and the Y-Intercept

From the equation \(y = x - 1\), we can see that the slope \(m = 1\) and the y-intercept \(b = -1\). This means the line crosses the y-axis at \( (0, -1) \).
03

Plot the Y-Intercept

Start by plotting the y-intercept on the graph at the point \((0, -1)\). This is the point where the line will cross the y-axis.
04

Use the Slope to Find Another Point

The slope of 1 means that for every unit you move to the right along the x-axis, you move up 1 unit along the y-axis. From the point \((0, -1)\), move right 1 unit to \((1, -1)\) and then move up 1 unit to \((1, 0)\). Plot this second point \((1, 0)\).
05

Draw the Line

Using a ruler, draw a straight line through the points \((0, -1)\) and \((1, 0)\). Extend the line in both directions, adding arrows at either end to indicate that it continues indefinitely.

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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 a way of writing the equation of a line. It is an especially useful format because it provides important information about the line at a glance.
In the equation format of the slope-intercept form, represented as \( y = mx + b \):
  • \( y \) is the dependent variable or the vertical value on a graph.
  • \( m \) represents the slope of the line. This determines the line's steepness and direction.
  • \( x \) is the independent variable or the horizontal value.
  • \( b \) is the y-intercept, indicating where the line crosses the y-axis.
This form makes it straightforward to identify both the slope and the y-intercept, allowing you to easily sketch the graph of the line or understand its behavior.
For example, in the equation \( y = x - 1 \), it's clear that the slope \( m \) is 1, and the y-intercept \( b \) is -1.
Slope
The slope is a crucial concept in understanding how a line behaves on a graph. It describes how steep the line is. The line's slope is calculated using the ratio of the rise (vertical change) over the run (horizontal change).
In the slope-intercept form \( y = mx + b \), \( m \) is the slope value. A positive slope means the line slopes upward as you move to the right, while a negative slope means it slopes downward.
  • If \( m = 1 \), it means for every 1 unit you move right, you move up 1 unit.
  • If \( m = 2 \), for every 1 unit right, you move up 2 units.
In our example equation, \( y = x - 1 \), since \( m = 1 \), the slope means for each step to the right on the x-axis, there is a corresponding upward step on the y-axis of the same size.
Y-Intercept
The y-intercept is the point where the line crosses the y-axis. It tells you the starting point of the line when the x-value is 0.
In any equation of the form \( y = mx + b \), \( b \) signifies the y-intercept.
  • For the equation \( y = x - 1 \), the y-intercept \( b \) is -1.
  • This means the line cuts the y-axis at the point \((0, -1)\).
The y-intercept serves as a great starting point to begin graphing a line. Once you have this point, you can apply the slope to find more points, helping you sketch the entire line.

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Most popular questions from this chapter

$$ \begin{aligned} &\text { A function } f \text { is given by }\\\ &f(x)=|x-2|+|x+1|-5\\\ &\text { Find } f(-3), f(-2), f(0), \text { and } f(4) \end{aligned} $$

Use the ZERO feature or the INTERSECT feature to approximate the zeros of each function to three decimal places. $$ f(x)=x^{4}+4 x^{3}-36 x^{2}-160 x+300 $$

Determine the domain of each function. $$ f(x)=\frac{x^{2}-25}{x-5} $$

Use the ZERO feature or the INTERSECT feature to approximate the zeros of each function to three decimal places. \(f(x)=x^{3}-x\) (Also use algebra to find the zeros of this function.)

In computing the dosage for chemotherapy, the measure of a patient's body surface area is needed. A good approximation of this area s, in square meters \(\left(m^{2}\right),\) is given by $$ s=\sqrt{\frac{h w}{3600}} $$ where \(w\) is the patient's weight in kilograms \((\mathrm{kg})\) and \(h\) is the patient's height in centimeters (cm). Assume that a patient's height is \(170 \mathrm{~cm} .\) Find the patient's approximate surface area assuming that: a) The patient's weight is \(70 \mathrm{~kg}\). b) The patient's weight is \(100 \mathrm{~kg}\). c) The patient's weight is \(50 \mathrm{~kg}\).
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