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Chapter 0: Problem 46

Use transformations to graph each function. $$ y=-4 x+200 $$

Short Answer

Expert verified
Plot (0, 200), then use the slope -4 to plot more points and draw the line.

Step by step solution

01

Identify the Parent Function

The parent function here is a linear function, which is expressed as y = x.
02

Determine the Slope and Y-Intercept

The given function is y = -4x + 200. In this form, the slope is m = -4 and the y-intercept is b = 200.
03

Apply the Slope

Starting from the y-intercept of 200, use the slope to determine another point. The slope m = -4 means for every unit increase in n in the x-direction, the y value decreases by 4n. For example, (1, 196) can be used.
04

Plot the Y-intercept

On the graph, plot the point where y = 200 (i.e., the y-intercept). This will be the point (0, 200).
05

Plot Additional Points Using the Slope

Use the slope to find additional points. From the point (0, 200), move 1 unit to the right (to x = 1) and 4 units down. This will give another point at (1, 196). Repeat this step for more points if necessary.
06

Draw the Line

Connect the points plotted in steps 4 and 5 with a straight line. This will be the graph of the function y = -4x + 200.

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Key Concepts

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

linear transformations
Linear transformations are operations that take a linear function and change its position or orientation on the graph. These changes can be done by modifying the slope and y-intercept of the function. The original parent function for a linear equation is usually expressed as y = x. In our example, we start with the parent function and apply transformations to graph the given linear function y = -4x + 200.
slope and y-intercept
The slope and y-intercept are crucial when graphing linear functions. The slope, denoted by m, indicates the steepness and direction of the line. In y = -4x + 200, the slope is -4, meaning the line moves downward as we go from left to right.
The y-intercept, denoted by b, is where the line crosses the y-axis. In our function, the y-intercept is 200. This is the starting point for plotting the graph.
Simply put, without the slope and y-intercept, it would be impossible to understand how to draw the line representing the linear function.
plotting points
Plotting points involves using the y-intercept and slope to determine the coordinates to place on the graph. We start at the y-intercept (0, 200) and use the slope to find more points.
  • From (0, 200), move 1 unit to the right and 4 units down to get (1, 196).
  • Continue plotting points in this manner to ensure accuracy.
Remember to connect these points with a straight line to represent the linear function. This step-by-step plotting helps visualize how the function behaves across different values of x, bringing the function to life on the graph.

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Most popular questions from this chapter

Each registered nurse at Blue Hills Memorial Hospital is first given a \(5 \%\) cost-of-living raise and then a \(\$ 3000\) merit raise. Write a function that transforms each old salary \(x\) into a new salary \(N(x)\). Does it make any difference in which order these raises are given? Explain.

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