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Chapter 1: Problem 12

In Exercises 7 through 28 , draw a sketch of the graph of the equation. $$ y=5 $$

Short Answer

Expert verified
Draw a horizontal line at y = 5 on the Cartesian plane.

Step by step solution

01

Identify the Type of Equation

The given equation is of the form y = c, where c is a constant. This represents a horizontal line in the Cartesian plane.
02

Determine the Y-Coordinate

Since the equation is y = 5, this means that the value of y is always 5, regardless of the value of x.
03

Draw the Horizontal Line

To sketch the graph, plot a horizontal line at y = 5. This line will pass through all points where the y-coordinate is 5.
04

Label Key Points and Axes

Mark any points that lie on the horizontal line, such as (0, 5), (1, 5), and (-1, 5). Label the y-axis and x-axis for clarity.

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Key Concepts

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

Horizontal Line
A horizontal line is a type of line on a graph where every point on the line has the same y-coordinate. This means the line runs from left to right without ever rising or falling. In the equation y = 5, the y-coordinate is always 5 for all values of x.

Whenever you see an equation in the form y = c, where c is a constant number, you can be sure that the graph will be a horizontal line. This is because a constant y-value means the line never goes up or down.

Horizontal lines are essential to understand as they represent situations where there is no change in the dependent variable. In this case, no matter what x-value you select, y will always remain equal to 5.
Constant Function
A constant function is a function that returns the same value no matter what input it is given. In mathematical terms, for the function f(x) = c, the value of the function is always c.

In the example, y = 5 is a constant function because it always equals 5, regardless of the input value for x. Constant functions are simple to graph and understand because they do not change; they stay the same.

Constant functions are useful in many real-world applications, such as setting a fixed price or indicating a non-changing temperature over time. Graphing a constant function always produces a horizontal line in the Cartesian plane.
Cartesian Plane
The Cartesian plane is a two-dimensional plane used for graphing equations. It consists of two perpendicular lines called axes: the x-axis (horizontal) and the y-axis (vertical).

The point where these axes intersect is called the origin, marked as (0, 0). Every point on the plane is identified by an ordered pair (x, y), indicating its position relative to the origin.

The Cartesian plane allows us to visually represent equations and understand their behavior through graphs. When graphing an equation like y = 5, we draw a horizontal line at y = 5 using the Cartesian plane's coordinate system.
Plotting Points
Plotting points is the process of marking points on the Cartesian plane based on their coordinates (x, y). To graph y = 5, you plot points where the y-coordinate is always 5.

For example:
  • (0, 5)
  • (1, 5)
  • (-1, 5)

After marking these points, draw a horizontal line through them to complete the graph.

Plotting points helps in visualizing the equation better and ensures your graph is accurate. It is a straightforward method to begin understanding more complex graphing techniques.

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Most popular questions from this chapter

In Exercises 1 through 10 , find the domain and range of the given function, and draw a sketch of the graph of the function. $$ g=\left\\{(x, y) \mid y=x^{2}+2\right\\} $$

Find formulas for \((f \circ g)(x)\) if $$ f(x)=\left\\{\begin{array}{ll} 0 & \text { if } x<0 \\ 2 x & \text { if } 0 \leq x \\ 0 & \text { if } x>1 \end{array} \quad \text { and } \quad g(x)= \begin{cases}1 & \text { if } x<0 \\\ \frac{1}{2} x & \text { if } 0 \leq x \leq 1 \\ 1 & \text { if } x>1\end{cases}\right. $$

In Exercises 11 through 32 , find the solution set of the given inequality and illustrate the solution on the real number $$ 4 x^{2}+9 x<9 $$

In Exercises 1 through 10, solve for \(x\). $$ |3 x-8|=4 $$

In Exercises 11 through 14 , find the center and radius of each circle, and draw a sketch of the graph. $$ 2 x^{2}+2 y^{2}-2 x+2 y+7=0 $$
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