Problem 52
Question
Graph each equation. $$y=-1 \text { (Let } x=-3,-2,-1,0,1,2, \text { and } 3 .)$$
Step-by-Step Solution
Verified Answer
The graph of y = -1 is a horizontal line passing through the point (0, -1).
1Step 1: Identify the equation type
This is a simple linear equation in the form y = c, where c is a constant. Such equations graph to a horizontal line at y = c, in this case, y = -1.
2Step 2: Evaluate the equation for various x values
Next, substitute different values of x into the equation and determine the corresponding y values. For this equation, no matter what value of x is plugged into the equation, the value of y is always -1 which is our constant.
3Step 3: Plot the points
For every x value chosen (-3, -2, -1, 0, 1, 2, 3), the y-value is always -1. When these points are plotted, they all fall on the line y = -1.
4Step 4: Draw the line
Join the plotted points together with a straight line. That's the graph of y = -1.
Key Concepts
Linear EquationHorizontal Line GraphPlotting Points
Linear Equation
Understanding the concept of a linear equation is essential in the realm of algebra. It forms the foundation of graphing and interpreting straight lines on a coordinate plane. A linear equation is generally an algebraic equation in which each term is either a constant or the product of a constant and a single variable.
Consider the simple equation provided in the exercise, y = -1. In this case, the equation represents a line where the variable y is always equal to -1, independent of the value of x. A critical characteristic of a linear equation in one variable, like this one, is that it describes a horizontal line on a graph. The line signifies that no matter what value the x variable takes, the y variable remains constant. This is a hallmark of the type of linear equation known as y equals a constant.
Consider the simple equation provided in the exercise, y = -1. In this case, the equation represents a line where the variable y is always equal to -1, independent of the value of x. A critical characteristic of a linear equation in one variable, like this one, is that it describes a horizontal line on a graph. The line signifies that no matter what value the x variable takes, the y variable remains constant. This is a hallmark of the type of linear equation known as y equals a constant.
Horizontal Line Graph
When graphing a horizontal line graph, it's crucial to identify that the 'y' value remains constant despite variations in 'x'. Such an equation is an example of a horizontal line because it has the form y = c, where c is a constant. In the context of the exercise we're looking at, the equation y = -1 directs us to draw a line where the y coordinate is always -1.
This line runs parallel to the x-axis and reflects the situation where all points have the same y value, no matter their x value. The graph is incredibly straightforward as it only conveys one piece of information: the y value, which does not change. In the exercise, the horizontal line encountered at y = -1 symbolizes an unvarying relationship in the vertical direction.
This line runs parallel to the x-axis and reflects the situation where all points have the same y value, no matter their x value. The graph is incredibly straightforward as it only conveys one piece of information: the y value, which does not change. In the exercise, the horizontal line encountered at y = -1 symbolizes an unvarying relationship in the vertical direction.
Importance of Horizontal Lines
- Reflect consistency or constancy in data.
- Easily depict invariable relationships in equations.
- Denote zero slope, meaning no change as one moves along the line.
Plotting Points
The method of plotting points comes into play when converting an equation into a visual graph. It involves placing dots at coordinates that satisfy the equation onto a grid called the Cartesian coordinate system. In this particular exercise, plotting points is the step where the abstract equation y = -1 becomes tangible on paper.
To start, you select values for x, as suggested: -3, -2, -1, 0, 1, 2, and 3. Although in this specific equation, the x values do not affect the outcome for y, it's good practice to choose a range of values to confirm the consistency of the horizontal line. Each selected x maintains the y value at -1, resulting in seven points: (-3, -1), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), and (3, -1).
After plotting these on the graph, they clearly align to form our horizontal line at y equals negative one.
To start, you select values for x, as suggested: -3, -2, -1, 0, 1, 2, and 3. Although in this specific equation, the x values do not affect the outcome for y, it's good practice to choose a range of values to confirm the consistency of the horizontal line. Each selected x maintains the y value at -1, resulting in seven points: (-3, -1), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), and (3, -1).
After plotting these on the graph, they clearly align to form our horizontal line at y equals negative one.
Benefits of Plotting Points:
- Helps visualize mathematical concepts.
- Ensures accuracy in the representation of equations.
- Facilitates understanding of the relationship between variables.
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