/*! 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 6 Sketch a graph of the line. $$... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 6

Sketch a graph of the line. $$g(x)=-5$$

Short Answer

Expert verified
The graph is a straight line, lying horizontally and crossing the y-axis at -5.

Step by step solution

01

Understand the Line Equation

The equation for this graph is \( g(x) = -5 \). This is a simple line equation, and it tells us that the line is horizontal across the x-axis at y = -5. This means that no matter what x is, y will always be -5.
02

Set Up the Graph

Draw an X - Y plane graph. The x-axis represents the x values, and the y-axis represents the g(x) or y values.
03

Draw the Line

Since \( g(x) = -5 \) regardless of the x value, the line will be a straight line at -5 on the y-axis. Draw a horizontal line at -5 across the x-axis. The line represents all points where the y value is -5, so it goes on indefinitely in both left and right directions

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Key Concepts

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

Understanding Graphing Linear Equations
Graphing linear equations involves plotting straight lines on a coordinate plane. Amazingly, with just an equation in the form of a line, you can discover all its points. Linear equations can be varied, such as vertical, horizontal, or diagonal lines. To graph these lines effectively:
  • Identify the equation's form. Simple ones might look like \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.
  • Determine if it's a special case like \( y = c \), which is a horizontal line, or \( x = c \), which is a vertical line.
This understanding helps us to easily visualize how lines will position themselves on a graph.
Navigating the Coordinate Plane
A coordinate plane is a two-dimensional surface with two axes: the horizontal x-axis and the vertical y-axis. It is like a map that helps in locating points using ordered pairs \( (x, y) \). Each axis is marked with numbers:
  • x-axis: Horizontal guide where values increase to the right and decrease to the left.
  • y-axis: Vertical guide where values rise upwards and fall downwards.
When plotting a graph, the origin \((0, 0)\) is the starting point where both axes meet. It divides the plane into four quadrants, providing an easy setup for drawing lines and understanding linear relationships.
Decoding the Y-Intercept
The y-intercept is a crucial point where a line crosses the y-axis. Knowing this point is key to graphing a line accurately. For the equation \( y = mx + c \), the y-intercept is the constant term \( c \). This indicates that when \( x = 0 \), the value of \( y \) is \( c \). Here's how you can find and use the y-intercept:
  • Directly identify it in simple horizontal lines like \( y = -5 \), where \( c = -5 \).
  • Use it with the slope (if available) for determining line direction and slope orientation.
For horizontal lines like \( y = -5 \), the y-intercept at \( (-5) \) shows that every point along this line will have a y-coordinate of \(-5\), simplifying your graphing tasks.

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Most popular questions from this chapter

Graph the function by hand. $$f(x)=\left\\{\begin{array}{ll} 0, & x \leq-1 \\ 4, & -12 \end{array}\right.$$

Solve the inequality. Express your answer in interval notation. $$-\frac{x}{2}>\frac{3 x}{2}+3$$

The piecewise-defined function given below is known as the characteristic function, \(C(x) .\) It plays an important role in advanced mathematics. $$C(x)=\left\\{\begin{array}{ll}0, & \text { if } x \leq 0 \\\1, & \text { if } 0

Graph the piecewise-defined function using a graphing utility. The display should be in DOT mode. $$f(x)=\left\\{\begin{array}{ll} x^{2}, & \text { if }-2 \leq x<0 \\ -x+1, & \text { if } 0 \leq x<2.5 \\ x-3.5, & \text { if } x \geq 2.5 \end{array}\right.$$

Solve the inequality algebraically and graphically. Express your anstoer in intereal notation. $$-3 x+2 \leq 5 x+10$$
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