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Chapter 12: Problem 89

Graph by using the slope and \(y\) -intercept. (GRAPH CAN'T COPY) $$y=-4 x+2$$

Short Answer

Expert verified
The linear graph for the equation \(y = -4x + 2\) will intersect at the y-axis at 2 and will have a slope that steeply falls to the right because the slope is -4.

Step by step solution

01

Identify the Slope and Y-Intercept

In the equation \( y = -4x + 2 \), the slope (m) is -4 and y-intercept (b) is 2. The slope indicates that for every single unit increase in x, y will decrease by 4 units. The y-intercept tells us that when x equals zero, y will be 2.
02

Plot the Y-Intercept

Start by plotting the y-intercept (b) on the y-axis. For this equation, plot a point at y=2.
03

Use the Slope to Find Another Point and Draw the Line

Since the slope is -4, this means that for every step to the right on the x-axis, go down by 4 steps on the y-axis. Plot that point. Now, draw a straight line through the two points to graph the equation.

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Key Concepts

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

Slope
The slope of a line is a measure of its steepness and direction. When graphing linear equations, the slope is often represented as 'm'. A positive slope means the line is rising as it moves from left to right, while a negative slope means the line is falling.
In our example, the equation is given by \( y = -4x + 2 \). Here, the slope is \(-4\), which means the line descends steeply to the right. Think of it as a rate of change, where for every 1 unit increase in \(x\), \(y\) decreases by 4 units.
Understanding slope lets you predict how a line behaves across the coordinate plane. Always look for slope as a starting point when plotting linear graphs.
Y-Intercept
The y-intercept is the point where the line crosses the y-axis. It sets the starting point on the graph. In the equation \(y = -4x + 2\), the y-intercept (b) is 2.
Visually, this tells us that when \(x = 0\), the value of \(y\) is 2. You begin graphing by plotting this y-intercept on the y-axis at the coordinate point (0,2).
Y-intercepts are handy because they give an immediate way to start drawing your line and understanding the function's initial point on a coordinate plane.
Coordinate Plane
The coordinate plane is a two-dimensional surface on which we can graph points, lines, and curves. It consists of two axes: the horizontal x-axis and the vertical y-axis. These axes intersect at the origin, which is designated as (0,0).
Each point on the plane is identified by an ordered pair \((x, y)\). In our example of graphing the equation \(y = -4x + 2\), the important points to plot start with the y-intercept and proceed according to the slope.
  • The y-intercept informs us where to place an initial point.
  • The slope directs how to move from that point to plot subsequent points on the line.
This grid system helps visualize how linear functions behave and allows us to see their basic structure.
Linear Functions
Linear functions are among the simplest types of functions you will encounter. They are expressed in the form \(y = mx + b\), combining a slope (m) with a y-intercept (b). This kind of equation produces a straight line when plotted on a coordinate plane.
The function we dealt with, \(y = -4x + 2\), is a prime example of a linear function. Its simplicity makes it a great starting point for understanding more complex relationships and functions in mathematics.
Important aspects of linear functions include:
  • Their graph is always a straight line.
  • The slope impacts the angle and direction of the line.
  • The y-intercept provides a specific starting location on the y-axis.
Linear functions demonstrate a constant rate of change, which is a fundamental concept in both algebra and calculus.

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Most popular questions from this chapter

Use the function \(f(x)=x^{2}-4 .\) For the given condition on \(a\) determine whether \(f(a)\) must be positive, must be negative, or could be either positive or negative. $$a<-2$$

Find the slope and \(y\) -intercept of the graph of the equation. $$y=-x+7$$

Determine whether the line through \(P_{1}\) and \(P_{2}\) is parallel, perpendicular, or neither parallel nor perpendicular to the line through \(Q_{1}\) and \(Q_{2}\). $$P_{1}(-2,4), P_{2}(2,4) ; Q_{1}(-3,6), Q_{2}(4,6)$$

Graph by using the slope and \(y\) -intercept. (GRAPH CAN'T COPY) $$x+3 y=6$$

Graph by using the slope and \(y\) -intercept. (GRAPH CAN'T COPY) $$y=-x+1$$
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