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Chapter 1: Problem 32

For each equation, find the slope \(m\) and \(y\) intercept \((0, b)\) (when they exist) and draw the graph. \(\frac{x+1}{2}+\frac{y+1}{2}=1\)

Short Answer

Expert verified
Slope: -1, Y-intercept: (0, 0), Graph: line through origin with slope -1.

Step by step solution

01

Simplify the equation

We start with the given equation: \( \frac{x+1}{2} + \frac{y+1}{2} = 1 \). To simplify, we eliminate the fractions by multiplying the whole equation by 2. This gives us: \( x+1 + y+1 = 2 \). Simplifying further, we obtain \( x + y + 2 = 2 \).
02

Rearrange to find y

Now, we rearrange the equation \( x + y + 2 = 2 \) to solve for \( y \). We subtract \( x \) and 2 from both sides of the equation, resulting in: \( y = -x \).
03

Identify the slope and y-intercept

In the equation \( y = -x \), we can compare this with the slope-intercept form \( y = mx + b \). Here, the slope \( m \) is \( -1 \), and the y-intercept \( b \) is \( 0 \) (as there is no constant term). Thus, the y-intercept point is \((0, 0)\).
04

Graph the equation

To graph the equation \( y = -x \), start by plotting the y-intercept \((0, 0)\) on the coordinate plane. Using the slope \( -1 \) (which means that for every unit increase in \( x \), \( y \) decreases by one unit), plot another point鈥攍ike \((1, -1)\), and draw a line through these points. The graph is a straight line passing through the origin with a slope of \(-1\).

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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 common way to represent a linear equation. It looks like this: \( y = mx + b \). Here, \( m \) represents the slope, which tells us how steep the line is. The slope indicates how much \( y \) changes for a one-unit increase in \( x \). The \( b \) is the y-intercept, which is the point where the line crosses the y-axis. This form is very useful when you want to quickly graph a line or understand its behavior.
- If \( m \) is positive, the line ascends from left to right.- If \( m \) is negative, the line descends from left to right.- If \( m \) is zero, the line is horizontal, meaning it has no slope and stays flat.Understanding and using the slope-intercept form allows us to easily identify both the direction and starting point of a line on a graph.
Identifying slope and intercepts
Identifying the slope and intercepts from a linear equation can help us graph the equation and understand its properties. Let's break it down:
- **Slope (\( m \))**: This is the number in front of \( x \) in the formula \( y = mx + b \). In the equation \( y = -x \), the slope is \(-1\). This tells us the line moves downwards by one unit for every unit it moves to the right. Keep an eye on the sign: a positive slope rises, and a negative slope falls.- **Y-intercept (\( b \))**: This is the number added or subtracted after \( mx \). In \( y = -x \), there is no additional number. This means \( b = 0 \). The line passes through the origin (\(0, 0\)) where it crosses the y-axis.
Summing up, to find the slope and intercepts, identify \( m \) and \( b \) in the equation and use these to sketch or analyze the line's graph.
Linear function
A linear function describes a relationship between two variables, typically \( x \) and \( y \), producing a straight line when graphed. In the form \( y = mx + b \), each value of \( x \) gives exactly one value of \( y \), showing a direct proportionality when plotted. Linear functions are simple yet powerful, forming the foundation for understanding more complex functions.

Some other key facets of linear functions include:
- **Consistency**: Unlike curves, linear functions have a constant rate of change; the slope \( m \) doesn't vary.- **Graph features**: The graph is a straight line defined thoroughly by its slope \( m \) and y-intercept \( b \).- **Use cases**: These functions model countless real-world situations, like calculating speed or tracking financial growth.

In learning to work with linear functions, identify their forms and properties, which are invaluable skills in mathematics and everyday problem-solving. Utilizing the slope and y-intercept allows easy graphing and helps in visual analysis of the relationship between variables.

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Most popular questions from this chapter

Use the TABLE feature of your graphing calculator to evaluate \(\left(1+\frac{1}{x}\right)^{x}\) for values of \(x\) such as \(100,10,000,1,000,000\), and higher values. Do the resulting numbers seem to be approaching a limiting value? Estimate the limiting value to five decimal places. The number that you have approximated is denoted \(e\), and will be used extensively in Chapter 4 .

Find, rounding to five decimal places: a. \(\left(1+\frac{1}{100}\right)^{100}\) b. \(\left(1+\frac{1}{10,000}\right)^{10,000}\) c. \(\left(1+\frac{1}{1,000,000}\right)^{1,000,000}\) d. Do the resulting numbers seem to be approaching a limiting value? Estimate the limiting value to five decimal places. The number that you have approximated is denoted \(e\), and will be used extensively in Chapter 4 .

How do two graphs differ if their functions are the same except that the domain of one excludes some \(x\) -values from the domain of the other?

For each function, find and simplify \(\frac{f(x+h)-f(x)}{h}\). (Assume \(h \neq 0 .\) ) $$ \begin{array}{l} f(x)=x^{3} \\ \text { [Hint: Use } \left.(x+h)^{3}=x^{3}+3 x^{2} h+3 x h^{2}+h^{3} .\right] \end{array} $$

An insurance company keeps reserves (money to pay claims) of \(R(v)=2 v^{0.3}\), where \(v\) is the value of all of its policies, and the value of its policies is predicted to be \(v(t)=60+3 t\), where \(t\) is the number of years from now. (Both \(R\) and \(v\) are in millions of dollars.) Express the reserves \(R\) as a function of \(t\), and evaluate the function at \(t=10\).
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