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Chapter 1: Problem 20

Sketch the graph of \(f\) by hand. Do not use a calculator. $$f(x)=-4$$

Short Answer

Expert verified
The graph is a horizontal line at \(y = -4\).

Step by step solution

01

Identify the Type of Function

The function given is a constant function, expressed as \(f(x) = -4\). A constant function produces the same output value for any input \(x\).
02

Determine the Graph Format

Since \(f(x) = -4\) is constant, the graph will be a horizontal line. This line is parallel to the x-axis and positioned at \(y = -4\) on the y-axis.
03

Plot Points on the Graph

To plot, mark a series of points along the line where the y-coordinate is \(-4\). So, for any chosen x-values (e.g., \(-2, 0, 2, 4\)), the corresponding y-values will be \(-4\).
04

Draw the Graph

Draw a straight horizontal line through the plotted points. This line is the graph of \(f(x) = -4\).
05

Label the Graph

Label the line with the function \(f(x) = -4\) and ensure that the y-value for each point along this line is \(-4\).

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Key Concepts

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

Graphing Functions
Graphing functions is a fundamental skill in mathematics, helping you visualize how a function behaves. For any function, the graph is an essential tool to understand its structure and properties.
The graph of a function represents all the points \(x, f(x)\) on the coordinate plane:
  • Where \(x\) is the input.
  • \(f(x)\) is the corresponding output.
This allows us to see patterns and trends in the function. Whether a straight line, a curve, or something more complex, each graph gives a unique picture of its corresponding function. Understanding how to graph various functions prepares you for more advanced topics such as calculus.
Horizontal Line
When plotting the graph of a constant function, you will always encounter a horizontal line. The reason being, a constant function like \(f(x) = -4\) always returns the same value, regardless of the input \(x\).
This means every point on the graph will have the same y-coordinate, in this case, \(-4\). Hence, we get a straight, horizontal line. Here鈥檚 how it applies to \(f(x) = -4\):
  • The line is parallel to the x-axis.
  • It intersects the y-axis at \(y = -4\).
If you imagine stacking sheets horizontally at the same height, that鈥檚 essentially what this line does. Even if the x-values extend indefinitely in both directions, the y-value remains constant.
Function Plotting
Function plotting is the practice of drawing the graph of a function on a coordinate plane.
This requires identifying some key elements:
  • The type of function鈥攕uch as linear, quadratic, or constant.
  • The path the graph takes鈥攚hich can be a line, a curve, or another shape.
For constant functions like \(f(x)=-4\), plotting involves:
  • Selecting any x-values on the x-axis.
  • Marking points where the y-coordinate is constant (\-4\ in this case).
By connecting these points, you visualize the function鈥檚 behavior accurately. Function plotting effectively communicates the relationship between variables in a function, which is crucial for analyzing and interpreting data.

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Most popular questions from this chapter

Obtain the least squares regression line and the correlation coefficient. Make \(a\) statement about the correlation. Urban Areas in the World (Population and Area) $$\begin{array}{|l|c|c|} \text { Urban Area } &\begin{array}{c} \text { Population, } \\ x \text { (in millions) } \end{array} & \begin{array}{c} \text { Area, } y \\ \text { (in square } \\ \text { miles) } \end{array} \\ \hline \text { Tokyo-Yokohama, Japan } & 34.2 & 3287 \\ \text { Delhi, India } & 23.9 & 747 \\ \text { Mumbai, India } & 23.3 & 210 \\ \text { Mexico City, Mexico } & 22.8 & 787 \\ \text { New York, United States } & 22.2 & 4477 \\ \text { S茫o Paulo, Brazil } & 20.8 & 1220 \\ \text { Shanghai, China } & 18.8 & 1345 \\ \text { Los Angeles, United States } & 17.9 & 2423 \\ \text { Kolkata (Calcutta), India } & 16.6 & 463 \\ \text { Buenos Aires, Argentina } & 13.1 & 1016 \end{array}$$

Classify each equation as a contradiction, an identity, or a conditional equation. Give the solution set. Use a graph or table to support your answer. $$5[1-(3-x)]=3(5 x+2)-7$$

Solve each equation analytically. Check it analytically, and then support the solution graphically. $$-(8+3 x)+5=2 x+3$$

Solve each equation analytically. Check it analytically, and then support the solution graphically. $$\frac{7}{3}(2 x-1)=\frac{1}{5} x+\frac{2}{5}(4-3 x)$$

Solve each equation and inequality analytically. Use interval notation to write the solution set for each inequality. (a) \(x+12=4 x\) (b) \(x+12>4 x\) (c) \(x+12<4 x\)
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