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Chapter 2: Problem 30

Give the slope and y-intercept of each line, and graph it. $$y=-2 x+7$$

Short Answer

Expert verified
Slope (m) = -2, y-intercept (b) = 7. Plot (0, 7) and another point using the slope -2.

Step by step solution

01

Identify the Equation Format

Identify that the equation is in the slope-intercept form, which is given by the formula: y = mx + bwhere m is the slope and b is the y-intercept.
02

Find the Slope

In the equation y = -2x + 7,compare the provided equation to the slope-intercept form. Here, the coefficient of x is the slope (m).So, m = -2.
03

Determine the y-Intercept

In the same equation, y = -2x + 7,the constant term is the y-intercept (b).Thus, b = 7.
04

Graph the Equation

To graph the equation, start by plotting the y-intercept (0, 7) on the Cartesian plane. Then use the slope to find another point on the line. Since the slope is -2, from the y-intercept move down 2 units and right 1 unit to plot a second point. Draw a straight line through these points to represent the equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

slope-intercept form
When looking at linear equations, one of the most common formats is the slope-intercept form. This format is written as y = mx + b. Here, m represents the slope of the line, and b represents the y-intercept.

The slope-intercept form is helpful because it provides quick information about how a line behaves on a graph. You can easily identify the slope, which tells you how steep the line is, and the y-intercept, which tells you where the line crosses the y-axis.

For example, in the equation y = -2x + 7, the slope m is -2, and the y-intercept b is 7. This means the line descends as it moves from left to right, and it cuts through the y-axis at the point (0, 7).
graphing linear equations
Graphing linear equations is an important skill that helps you visualize mathematical relationships. To graph an equation in slope-intercept form, follow these steps:

  • Identify the y-intercept: This is the point where the line crosses the y-axis. For the equation y = -2x + 7, the y-intercept is 7, so you plot the point (0, 7).

  • Use the slope: The slope tells you how to move from the y-intercept to another point on the line. A slope of -2 means you go down 2 units for every 1 unit you go to the right.

  • Plot another point: Starting from the y-intercept, move down 2 units and right 1 unit to find another point on the line. Plot this point.

  • Draw the line: Finally, draw a straight line through the points you鈥檝e plotted to complete the graph. This line represents your equation y = -2x + 7.
Following these steps will help you accurately graph any linear equation in the slope-intercept form.
Cartesian plane
The Cartesian plane is a two-dimensional surface where you can graph equations and analyze their relationships. It is made up of two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical).

Each point on the Cartesian plane is represented by a pair of coordinates \(x, y\). The first number is the position on the x-axis, and the second number is the position on the y-axis.

Here are some key features:

  • The point (0, 0) is called the origin. It's where the x-axis and y-axis intersect.

  • Positive x values are to the right of the origin, while negative x values are to the left.

  • Positive y values are above the origin, while negative y values are below.
Websites where you can plot any point and visualize the equation as a line. Understanding the Cartesian plane is essential as it is the foundation for graphing any type of equation accurately.

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Most popular questions from this chapter

Find the coordinates of the points of intersection of the line \(y=1\) and the circle centered at \((3,0)\) with radius 2.

The tables give some selected ordered pairs for functions \(f\) and \(g\). $$\begin{array}{|c|c|c|c|}\hline x & 3 & 4 & 6 \\\\\hline f(x) & 1 & 3 & 9 \\\\\hline\end{array}$$ $$\begin{array}{|c|c|c|c|c|}\hline x & 2 & 7 & 1 & 9 \\\\\hline g(x) & 3 & 6 & 9 & 12 \\\\\hline\end{array}$$ Find each of the following. $$(f \circ g)(2)$$

Give the slope and y-intercept of each line, and graph it. $$2 x+3 y=16$$

Find the function \(g(x)=a x+b\) whose graph can be obtained by translating the graph of \(f(x)=2 x+5\) up 2 units and to the left 3 units.

For certain pairs of functions \(f\) and \(g,(f \circ g)(x)=x\) and \((g \circ f)(x)=x .\) Show that this is true for each pair. $$f(x)=\sqrt[3]{5 x+4}, g(x)=\frac{1}{5} x^{3}-\frac{4}{5}$$
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