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Chapter 2: Problem 35

Graph the line that passes through the given point and has the given slope. (0,2) and \(m=-2\)

Short Answer

Expert verified
The equation of the line is \(y = -2x + 2\). When graphed, the line passes through the point (0, 2) and has a slope of -2.

Step by step solution

01

Identify the Given Values

First, identify the given values from the problem. The problem states the line passes through the point (0,2) and has a slope of -2. So, the point (0,2) is our y-intercept because it's the point where the line crosses the y-axis. Therefore, \(m=-2\) and \(b=2\).
02

Substitute values into the equation

Next, substitute the slope and y-intercept values into the slope-intercept equation \(y = mx + b\). Therefore, the equation of the line is \(y = -2x + 2\).
03

Graph the Line

Plot the y-intercept (0,2) as the first point on the graph. From this point, use the slope(-2) as the 'rise over run' to find the next point. Because the slope is -2, go down two units (due to the negative sign) and move one unit to the right (since the denominator is 1 in this case if we assume the slope fraction to be \(-2/1\)). Continue this until you have sufficient points and then draw the line through them.

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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 popular way for expressing the equation of a line. It's written as \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) is the y-intercept. The y-intercept is the point where the line crosses the y-axis.
For example, if we have a line passing through the point \((0,2)\) with a slope of \(-2\), the y-intercept \(b\) is 2, which is evident from the point \((0,2)\).
  • The slope, \(m\), indicates how steep the line is. A positive slope means the line goes upward to the right, and a negative slope means it goes downward.
  • To write the equation, substitute the known slope and y-intercept into the slope-intercept formula. Here, \(b=2\) and \(m=-2\), giving us the equation \(y = -2x + 2\).
This form is particularly handy as it immediately reveals both the slope and y-intercept, making plotting easier.
Plotting Points
Plotting points to create a graph involves marking coordinates on a Cartesian plane. Each point is represented by an \((x, y)\) pair.
Let's look at the line equation we formed earlier: \(y = -2x + 2\). To begin plotting:
  • Start with the y-intercept point \((0,2)\). This point is special because it's where your line will definitely cross the y-axis.
  • With the slope \(-2\), understand it as \(\frac{-2}{1}\). The "rise" is -2, and the "run" is 1, meaning from the y-intercept, move 2 units down and 1 unit to the right to find the next point.
  • Continue this pattern, marking each new point, and eventually extend a straight line through all points plotted.
This method ensures accuracy in forming a straight line. It's all about moving along the slope from one marked point to the next.
Linear Functions
Linear functions describe straight-line graphs and are one of the simplest types of mathematical functions. They're represented by the equation \(y = mx + b\), clearly showing the relation between \(x\) and \(y\). Practically, think of them as relationships where changes in \(x\) directly affect \(y\) in a consistent way.
  • The degree of a linear equation is 1, meaning it creates a 1-dimensional straight line.
  • A key property is constant slope, defined by \(m\), which stays the same regardless of which two points are considered on the line.
  • Examples include simple relations like speed (distance over time) where the slope corresponds to speed.
Understanding linear functions is fundamental for modeling and interpreting real-world situations where relationships remain consistent.

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Most popular questions from this chapter

The supply curve for a product is \(y=250 x-1,000\) and the demand curve for the same product is \(y=-350 x+8,000,\) where \(x\) is the price and \(y\) the number of items produced. Find the following. a. At a price of $$\$ 10$$, how many items will be in demand? b. At what price will 4,000 items be supplied? c. What is the equilibrium price for this product? d. How many items will be manufactured at the equilibrium price?

Find the slope of the line passing through the following pair of points. (0,4) and (3,0)

A supply curve for a commodity is the number of items of the product that can be made available at different prices. A manufacturer of toy dolls can supply 2000 dolls if the dolls are sold for $$\$ 8$$ each, but he can supply only 800 dolls if the dolls are sold for $$\$ 2$$ each. If \(x\) represents the price of dolls and \(y\) the number of items, write an equation for the supply curve.

It costs $$\$ 2,700$$ to manufacture 100 items of a product, and $$\$ 4,200$$ to manufacture 200 items. If \(x\) represents the number of items, and \(y\) the costs, find the cost equation, and use this function to predict the cost of 1,000 items.

Write an equation of the line satisfying the following conditions. Write the equation in the form \(y=\mathrm{mx}+b\). It passes through (7,-2) and its \(y\) -intercept is 5 .
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