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Chapter 7: Problem 41

In Exercises 41-48, graph each horizontal or vertical line. \(y=4\)

Short Answer

Expert verified
The graphical representation of the equation 'y = 4' is a horizontal line that intersects the y-axis at point 4.

Step by step solution

01

Understand the given equation.

Examine the equation which is 'y = 4'. This is a straightforward linear equation in which y is isolated. In such equations, the value of y is always constant and it is determined by the number on the right side of the equation. In this case, it is 4.
02

Interpreting the equation graphically

In the equation 'y = 4', there is no 'x' variable involved. This suggests that the line doesn't slope but it is rather horizontal. The number indicates the position of the line in relation to the y-axis. Since it is 4 in this case, the line will sit on the position 4 on the y-axis.
03

Plot the line on the graph

Take a graph paper or use a graphing tool of your choice. On the y-axis, find the position that corresponds to the number 4. Draw a horizontal line across the graph at this point. This is the line which the equation 'y = 4' represents.

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Key Concepts

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

Horizontal Lines
Horizontal lines are one of the simplest types of lines in the coordinate plane. They run straight across from left to right, parallel to the x-axis. But what makes a line horizontal? It's all about the y-value.
  • For any horizontal line, the y-coordinate remains constant across all its points.
  • The equation for a horizontal line is in the form of: \(y = c\), where \(c\) is a constant value.
  • This means no matter which point you choose on the line, the y-value will always be the same.
To visualize this on a graph, just think of a horizon鈥攁n even, flat line that doesn't go up or down. The line will pass through the y-axis at the given constant y-value and extend infinitely in both directions.
Coordinate Plane
The coordinate plane is a fundamental concept in graphing linear equations. It consists of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four quadrants.
  • The intersection point of the x-axis and y-axis is called the origin, represented by the coordinates (0, 0).
  • The x-values increase to the right of the origin and decrease to the left.
  • The y-values increase as you move up from the origin and decrease as you move down.
When graphing an equation, points are plotted according to their x and y values, creating a visual representation. For the line \(y = 4\), you would move 4 units up from the origin on the y-axis and draw a line parallel to the x-axis without changing the y position.
Equations of Lines
Equations of lines are mathematical expressions that describe the relation between the x-values and y-values of any point on the line. They can be simple or complex depending on the type of line described.
  • The most common form of the equation of a line is the slope-intercept form: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
  • Horizontal lines are a special case where the slope \(m = 0\), so the equation simplifies to \(y = b\).
  • In these cases, as we saw with \(y = 4\), the y-value is constant and solely depends on \(b\), the y-intercept.
Understanding these equations helps in easily identifying the nature of the line, such as its slope and where it crosses the y-axis. For horizontal lines, it鈥檚 all about recognizing that the slope is zero, which simplifies interpretation and graphing.

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Most popular questions from this chapter

The value of \(a\) in \(y=a x^{2}+b x+c\) and the vertex of the parabola are given. How many \(x\)-intercepts does the parabola have? Explain how you arrived at this number. \(a=1 ;\) vertex at \((2,0)\)

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function $$ z=6 x+10 y $$ Constraints $$ \left\\{\begin{array}{l} x+y \leq 12 \\ x+2 y \leq 20 \\ x \geq 0 \\ y \geq 0 \end{array}\right\\} \begin{aligned} &\text { Quadrant I and } \\ &\text { its boundary } \end{aligned} $$

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}x \geq 4 \\ y \leq 2\end{array}\right.\)

In Exercises 7-8, a. Rewrite each equation in exponential form. b. Use a table of coordinates and the exponential form from part (a) to graph each logarithmic function. Begin by selecting \(-2,-1,0,1\), and 2 for \(y\). \(y=\log _{4} x\)

a. Determine if the parabola whose equation is given opens upward or downward. b. Find the vertex. c. Find the \(x\)-intercepts. d. Find the y-intercept. e. Use (a)-(d) to graph the quadratic function. \(y=-x^{2}+4 x-3\)
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