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Chapter 39: Problem 847

Discuss the graph of the function \(\mathrm{y}=-3 \mathrm{x}+4\).

Short Answer

Expert verified
The graph of the function \(y = -3x + 4\) is a straight line with a negative slope (-3), indicating a steep downward slant from left to right. It intersects the y-axis at point (0, 4) and the x-axis at point \((\frac{4}{3}, 0)\).

Step by step solution

01

Identify the slope and y-intercept from the equation

The given equation is in the slope-intercept form: \(y = -3x + 4\). From the equation, we can see that the slope (m) is -3, and the y-intercept (b) is 4.
02

Calculate the x-intercept by setting y = 0

To find the x-intercept, we set y = 0 and solve for x: \(0 = -3x + 4\) Now, add 3x to both sides of the equation: \(3x = 4\) Next, divide both sides by 3 to isolate x: \(x = \frac{4}{3}\) So, the x-intercept is \(\frac{4}{3}\).
03

Discuss the characteristics of the graph

1. Slope: The slope of the function is -3, which indicates that the line is slanting downwards from left to right, and it is steep due to the value of the slope being far away from 0 (large magnitude). 2. y-intercept: The y-intercept is 4, meaning the line intersects the y-axis at the point (0, 4). 3. x-intercept: The x-intercept is \(\frac{4}{3}\), which means the line intersects the x-axis at the point \((\frac{4}{3}, 0)\). In summary, the graph of the function \(y = -3x + 4\) is a straight line with a negative slope (-3) that intersects the y-axis at the point (0, 4) and the x-axis at the point \((\frac{4}{3}, 0)\). The line will slant downwards from left to right and will be steep due to the large value of the slope (-3).

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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 very useful way to express linear equations. It's given as \(y = mx + b\), where \(m\) represents the slope, and \(b\) is the y-intercept. This form allows you to quickly identify both the slope and the y-intercept by merely inspecting the equation. For instance, in the equation \(y = -3x + 4\), the slope \(m\) is \(-3\) and the y-intercept \(b\) is \(4\).

When working with slope-intercept form, it is easy to determine how changes in \(m\) and \(b\) affect the graph:
  • The value of \(m\) influences the direction and steepness of the line. A negative slope like \(-3\) means the line will go downwards as you move to the right.
  • The y-intercept \(b\) is the point where the line crosses the y-axis, which is \(4\) in this equation.
Graphing Linear Equations
Graphing linear equations involves plotting the line described by the equation on a coordinate plane. Using the slope-intercept form, you can graph a line easily and accurately.

Here's how to do it:
  • Start by plotting the y-intercept \((0, b)\) on the y-axis. For the equation \(y = -3x + 4\), begin by marking the point \((0, 4)\).
  • Use the slope \(m\) to determine the rise over run from the y-intercept. For a slope of \(-3\), move down 3 units and one unit to the right to find another point on the line.
  • Draw a straight line through these points extending in both directions, and you have your graph of the linear equation.
This process provides a visual representation of the equation and aids in understanding its behavior.
X-Intercept
The x-intercept is a key point on the graph of a linear equation where the line crosses the x-axis. It is found by setting \(y\) equal to zero in the equation and solving for \(x\).

To find the x-intercept in the equation \(y = -3x + 4\), set \(y = 0\), and follow these steps:
  • Replace \(y\) with 0: \(0 = -3x + 4\).
  • Add \(3x\) to each side to simplify the equation: \(3x = 4\).
  • Divide both sides by 3 to isolate \(x\): \(x = \frac{4}{3}\).
The x-intercept is at \(\left(\frac{4}{3}, 0\right)\), indicating exactly where the line cuts through the x-axis. Knowing the x-intercept is valuable when analyzing the point framework of the linear equation.
Y-Intercept
The y-intercept is the point at which a line crosses the y-axis of a graph. It is a crucial element, as it allows you to easily commence the graphing of a line. In the slope-intercept form equation \(y = mx + b\), the y-intercept is represented by \(b\).

For \(y = -3x + 4\), the y-intercept is \(4\). This tells us the line will cross the y-axis at the point \((0, 4)\). To find any y-intercept:
  • Locate \(b\) in the equation, which directly indicates the y-intercept.
Your graph will always start at this intersection, making it a reliable starting point for sketching a line accurately on a graph.

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Most popular questions from this chapter

Find the equation for the line passing through \((3,5)\) and \((-1,2)\).

(a) Find the equation of the line passing through \((2,5)\) with slope 3 . (b) Suppose a line passes through the \(\mathrm{y}\) -axis at \((0, \mathrm{~b})\). How can we write the equation if the point-slope form is used?

Determine whether there is a point of intersection of the graphs of \(2 \mathrm{x}-3 \mathrm{y}=5\) and \(6 \mathrm{x}-9 \mathrm{y}=10\).

Find the equation of the family of lines that satisfy the conditions (1) parallel to \(1_{0}: x+2 y+4=0\) (2) passing through the point \(\mathrm{P}_{0}(4,9)\).

Find the point of intersection of the graphs of \(3 \mathrm{x}-\mathrm{y}=5\) and \(9 x-3 y=15\).
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