/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 244 Discuss the graph of the functio... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 244

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 slope of -3, indicating a downward direction from left to right, and a y-intercept of (0, 4). The line decreases at a rate of 3 units vertically for every unit horizontally to the right. The x-intercept is located at \(\left(\frac{4}{3}, 0\right)\).

Step by step solution

01

Identify the slope and y-intercept

The given function is y = -3x + 4, which is in the slope-intercept form (y = mx + b). In this form, m represents the slope of the line, and b represents the y-intercept. From the equation, we can identify that the slope (m) is -3 and the y-intercept (b) is 4.
02

Interpret the slope

The slope represents the rate of change in the function. Since the slope is -3, it means that the line is decreasing at a rate of 3 units vertically down for every unit it moves horizontally to the right. So, for every increase of 1 in x, the y-value decreases by 3. The negative value indicates the line's direction: a negative slope means the line goes downward from left to right, whereas a positive slope means the line goes upward from left to right.
03

Identify the y-intercept

The y-intercept is the point at which the graph intersects the y-axis, given by the coordinate (0, b). In this case, our y-intercept (b) is 4, so the coordinate is (0, 4). To plot this point on the graph, start at the origin (0, 0) and move 4 units up along the y-axis. This will be the first point on the line.
04

Find the x-intercept

The x-intercept is the point at which the graph intersects the x-axis, and it occurs when y = 0. To find the x-intercept, substitute y = 0 into the equation and solve for x: \(0 = -3x + 4 \) Add 3x to both sides: \(3x = 4 \) Divide both sides by 3: \(x = \frac{4}{3} \) So, the x-intercept of the line is at the coordinate \(\left(\frac{4}{3}, 0\right)\). To plot this point on the graph, start at the origin (0, 0) and move \(\frac{4}{3}\) units to the right along the x-axis. This will be the second point on the line.
05

Plot and discuss the graph

Now that we have the slope, y-intercept, and x-intercept of the function, we are ready to plot the graph. Begin at the y-intercept (0, 4), then use the slope (-3) to find another point by going down 3 units and to the right 1 unit from the y-intercept. Connect these two points, and your line should also pass through the x-intercept at the coordinate \(\left(\frac{4}{3}, 0\right)\). In conclusion, the graph of the function y = -3x + 4 displays a decreasing, straight line due to its negative slope, intersecting the y-axis at (0, 4) and the x-axis at \(\left(\frac{4}{3}, 0\right)\).

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Construct the graph of the function defined by \(\mathrm{y}=3 \mathrm{x}-9\).

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