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Chapter 1: Problem 20

Find the slope and \(y\) -intercept. $$y=3$$

Short Answer

Expert verified
The slope of the line \(y=3\) is 0, and its y-intercept is 3.

Step by step solution

01

Determine the form of equation

The equation of the line is given as \(y=3\). This can be rewritten in the form \(y = mx + b\) where, in this case, \(m\) is the coefficient of \(x\) and \(b\) is the constant. For your equation, it means \(y=0x+3\).
02

Identify the slope (m)

The slope \(m\) is the coefficient before the \(x\). In the equation \(y=0x+3\), \(m\) corresponds to 0. So, the slope of the line \(y=3\) is 0.
03

Identify the y-intercept (b)

The y-intercept \(b\) is the constant value in the equation, which is the point where the line crosses the y-axis. In the equation \(y=0x+3\), \(b\) corresponds to 3. So, the y-intercept of the line \(y=3\) is 3.

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Key Concepts

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

Slope
The slope of a line is a measure of its steepness and direction. In mathematical terms, it is calculated as the change in the y-value divided by the change in the x-value between two distinct points on the line. This is often described as "rise over run."
When identifying the slope from an equation in the form of \( y = mx + b \), the slope \( m \) is the coefficient multiplied by \( x \).
  • If the slope is positive, the line rises as it moves from left to right.
  • If the slope is negative, the line falls as it moves from left to right.
  • A horizontal line, such as \( y = 3 \), has a slope of 0, meaning there is no rise or fall. The line remains constant across all x-values.
  • Conversely, a vertical line has an undefined slope, as the run would be zero.
Understanding the slope provides insight into how one variable changes in relation to another, making it a crucial concept in linear equations and graph interpretations.
Y-intercept
The y-intercept is an important aspect of linear equations, as it defines where the line crosses the y-axis. This is the point at which the value of \( x \) is zero.
In the equation \( y = mx + b \), the y-intercept is denoted by the constant \( b \).
  • The y-intercept represents the starting point of a line when plotted on a graph, especially in the context of linear situations.
  • For instance, in the equation \( y = 3 \), the y-intercept is 3, indicating that the line crosses the y-axis at the point (0, 3).
  • The y-value of the intercept gives a fixed point through which the line will pass, regardless of the slope.
Knowing the y-intercept is particularly useful in real-world scenarios, where initial values and starting conditions are often represented by this concept.
Equation of a Line
The equation of a line is a mathematical expression used to describe a straight line in Cartesian coordinates. It is commonly expressed in the format \( y = mx + b \), known as the slope-intercept form.
This form is especially useful because it provides both the slope and the y-intercept, two key characteristics of a line.
  • By rewriting an equation to this form, you can easily identify if a line is horizontal, vertical, or at an incline.
  • In an equation like \( y = 3 \), although it seems simplified, it actually fits the form \( y = 0x + 3 \), revealing a slope of 0 and a y-intercept of 3.
  • Understanding the equation of a line allows for the graphing of linear relationships and provides insight into how two variables are interconnected.
Equations of lines are foundational in algebra and serve as building blocks for understanding more complex mathematical concepts and real-world linear phenomena.

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Most popular questions from this chapter

(a) Use a graphing utility to graph the polynomials. $$\begin{aligned}&f(x)=x^{5}-7 x^{3}+6 x+2,\\\&g(x)=-x^{5}+5 x^{3}-3 x-3. \end{aligned}$$ (b) Based on your graphs in part (a), make a conjecture about the general shape of the graph of a polynomial of degree 5. (c) Now graph $$P(x)=x^{3}+a x^{4}+b x^{3}+c x^{2}+d x+c$$ for several choices of \(a, b, c, d, e .\) (For example, try \(a=b=c=d=e=0 .)\) How do these graphs compare with your graph of \(f\) from part (a)?

Show by example that the sum of two irrational numbers (a) can be rational; (b) can be irrational. Do the same for the product of two irrational numbers.

Find \(f \circ g\) and \(g \circ f\). $$f(x)=1-x^{2}, g(x)=\sin x$$

Use a graphing utility to draw several views of the graph of the function. Select the one that most accurately shows the important features of the graph. Give the domain and range of the function. $$f(x)=\left|x^{3}-3 x^{2}-24 x+4\right|$$

Find an equation for the line that passes through the point (2,-3) and is parallel to the line \(3 x+4 y=12\)
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