/*! 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 55 Sketch the graph of each polynom... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 55

Sketch the graph of each polynomial function. First graph the function on a calculator and use the calculator graph as a guide. $$f(x)=x-20$$

Short Answer

Expert verified
Plot points (0, -20) and (20, 0), then draw a straight line through them.

Step by step solution

01

Identify the Function

The given function is a linear polynomial function: \[f(x) = x - 20\]
02

Determine Key Points

For a linear function, the key points are the y-intercept and some additional points. Here, the y-intercept is found by setting \(x = 0\), which gives:\[f(0) = 0 - 20 = -20\]Another simple point can be found by setting \(x = 20\):\[f(20) = 20 - 20 = 0\]
03

Plot the Points

Plot the two key points on the graph. They are (0, -20) and (20, 0).
04

Draw the Line

Using the two points, draw a straight line through them. This line represents the graph of the function \(f(x) = x - 20\).
05

Verify with Calculator

Graph the function using a graphing calculator to ensure accuracy. The calculator should show a straight line passing through the points (0, -20) and (20, 0). Use this as a guide to adjust your sketch if necessary.

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Key Concepts

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

linear polynomial
A linear polynomial is one of the simplest forms of polynomials. It is a polynomial of degree one. This means it has the form \[f(x) = ax + b\] where \(a\) and \(b\) are constants and \(x\) is the variable.
For our specific example, \[f(x) = x - 20\] we see that \(a = 1\) and \(b = -20\). Since the highest exponent of \(x\) is 1, this is called a linear function.
Linear functions graph as straight lines. They are easy to work with, which makes them a great starting point when studying polynomials.
y-intercept
The y-intercept is a very important part of any graph. It is where the graph crosses the y-axis. To find the y-intercept, you set \(x\) to 0 and solve for \(y\).
In our function, \[f(x) = x - 20\], we set \(x = 0\).
So, \[f(0) = 0 - 20 = -20\].
That means the y-intercept is at the point (0, -20). This point is crucial as it gives you one of the reference points you need to draw the graph.
When plotting, always start with the y-intercept. Since it is easy to find, it gives you a strong footing for sketching the graph accurately.
plotting points
Plotting points correctly is essential when graphing any function. For linear functions, you need at least two points to draw a straight line.
In the provided example, we found two key points: (0, -20) and (20, 0).
Here are the steps to plot these points:
  • First, place a dot at (0, -20) on your graph. This is the y-intercept.
  • Next, place another dot at (20, 0). This is another critical point.
After plotting these points, you can draw a straight line through them. This line represents the graph of your function. Always double-check to ensure that the points are plotted accurately; any mistake can throw off your graph.
graphing calculator
A graphing calculator is an invaluable tool when studying polynomial functions. It helps you visualize the function and verify your calculations.
Here鈥檚 how you can use a graphing calculator to graph a function like \[f(x) = x - 20\]:
  • First, turn on the calculator and enter the function into the 鈥榶=鈥� menu.
  • Next, press the 鈥楪raph鈥� button. The calculator will then display the graph of the function on the screen.
  • To zoom in or out, use the zoom buttons to see the graph more clearly.
  • Use the computed graph to verify the accuracy of your hand-drawn graph.
The graphing calculator shows a line passing through the points you identified, which helps you ensure your sketch is correct. It鈥檚 a great way to double-check your work and build confidence in your graphing skills.

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Most popular questions from this chapter

Let \(f(x)=x^{4}-1, g(x)=x^{3}-3 x^{2}+5,\) and \(h(x)=4 x^{4}-\) \(3 x^{2}+3 x-1 .\) Find the following function values by using two different methods. See Example \(I\) $$ f(5) $$

In each case, find a polynomial function whose graph behaves in the required manner. Answers may vary. The graph has only two \(x\) -intercepts at \((5,0)\) and \((-6,0)\) It does not cross the \(x\) -axis at either \(x\) -intercept.

Establish the best integral bounds for the roots of each equation according to the theorem on bounds. $$x^{2}-7 x-16=0$$

Discuss the possibilities for the roots to each equation. Do not solve the equation. $$-x^{4}+3 x^{3}+5 x+5=0$$

Sketch the graph of each polynomial function. First graph the function on a calculator and use the calculator graph as a guide. $$f(x)=(x-20)^{2}(x+30)^{2} x^{2}$$
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