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Chapter 0: Problem 5

Graph the following equations. $$y=-2 x+3$$

Short Answer

Expert verified
Plot the y-intercept at (0, 3). Use the slope to find another point (1, 1), and draw a line through the points.

Step by step solution

01

Identify the slope and y-intercept

The given equation is in the slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. For the equation y = -2x + 3, the slope (m) is -2 and the y-intercept (b) is 3.
02

Plot the y-intercept

Start by plotting the y-intercept on the graph. The y-intercept is the point where the line crosses the y-axis. In this case, the y-intercept is 3, which corresponds to the point (0, 3) on the graph.
03

Use the slope to find another point

The slope of -2 means that for every 1 unit increase in x, y decreases by 2 units. Starting from the y-intercept (0, 3), move 1 unit to the right (positive x direction), and then 2 units down (negative y direction) to find the next point, which is (1, 1).
04

Draw the line

With the points (0, 3) and (1, 1) marked on the graph, draw a straight line through these points. This line represents the equation y = -2x + 3.

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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 linear equation. This is a straight line equation and is typically written as:
\[y = mx + b\]
Here,
  • m represents the slope of the line.
  • b stands for the y-intercept of the line.

The slope-intercept form is useful because it quickly tells us two key things about the line: the slope and the point where the line crosses the y-axis. Knowing these helps in sketching the graph efficiently.
For example, in the equation \[y = -2x + 3\], we can see that the slope (m) is -2 and the y-intercept (b) is 3.
slope
The slope of a line is a measure of its steepness. It tells us how much the y-value (vertical change) changes for a given change in the x-value (horizontal change).
Mathematically, the slope (m) is calculated by:
\[ m = \frac{\text{rise}}{\text{run}} = \frac{\text{change in y}}{\text{change in x}} \]
In the equation \[y = -2x + 3\], the slope is -2. This means:
  • For every 1 unit increase in x, the value of y decreases by 2 units.
  • A negative slope indicates the line is descending as it moves from left to right.

Understanding the slope makes it easier to find points on the line and graph it accurately.
y-intercept
The y-intercept is the point where the line crosses the y-axis. This point occurs when x is zero. In the slope-intercept form \[y = mx + b\], the y-intercept is represented by .
For example, in \[y = -2x + 3\], the y-intercept is 3.
This means the line will cross the y-axis at the point (0, 3).
  • The y-intercept gives us a starting point for graphing the line.

Once we plot the y-intercept, we use the slope to find other points on the line.
plotting points
Plotting points is critical for accurately drawing the graph of the linear equation. We start by identifying the y-intercept and then use the slope to find other points.
Let's see this step-by-step for the equation \[y = -2x + 3\]:
  • Step 1: Plot the y-intercept (0, 3) on the y-axis.
  • Step 2: Use the slope of -2 to find another point. From (0, 3), move 1 unit to the right (positive x-direction) and 2 units down (negative y-direction) to get the point (1, 1).

We now have two points: (0, 3) and (1, 1). Connect these points with a straight line to graph the equation. By repeating the process, you can plot additional points to ensure the accuracy of your line.

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Most popular questions from this chapter

Let \(f(x)=\frac{x}{x-2}, g(x)=\frac{5-x}{5+x},\) and \(h(x)=\frac{x+1}{3 x-1} .\) Express the following as rational functions. $$g\left(\frac{1}{u}\right)$$

Let \(f(x)=\sqrt[3]{x}\) and \(g(x)=\frac{1}{x^{2}} .\) Calculate the following functions. Take \(x > 0\). $$f(g(x))$$

Find the zeros of the function. (Use the specified viewing window.) $$f(x)=x^{3}-3 x+2 ;[-3,3][-10,10]$$

Let \(f(x)=x^{2}+3 x+1\) and \(\operatorname{let} g(x)=x^{2}-3 x-1 .\) Graph the two functions \(f(g(x))\) and \(g(f(x))\) together in the window [-4,4] by [-10,10] and determine if they are the same function.

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. $$\left(x^{3} \cdot y^{6}\right)^{1 / 3}$$
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