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Chapter 5: Problem 6

Give the values for \(b\) and \(m\) for the linear functions. $$ h(n)=0.01 n+100 $$

Short Answer

Expert verified
Answer: In the linear function h(n) = 0.01n + 100, the slope (m) is 0.01 and the y-intercept (b) is 100.

Step by step solution

01

Identify Slope (m)

The slope, or \(m\), is the coefficient of the independent variable, \(n\), in the given function. In \(h(n) = 0.01n + 100\), the slope \(m\) is 0.01.
02

Identify y-intercept (b)

The y-intercept, or \(b\), is the constant term in the given function. In \(h(n) = 0.01n + 100\), the y-intercept \(b\) is 100. So, the values for \(b\) and \(m\) for the linear function are as follows: - The slope (m): \(m = 0.01\) - The y-intercept (b): \(b = 100\)

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Key Concepts

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

Understanding the Slope of a Linear Function
In the context of a linear function, the slope is a crucial concept that defines the steepness or incline of a line. When we have a linear equation in the form of \(y = mx + b\), the slope is represented by the coefficient \(m\). It tells us how the dependent variable (often \(y\)) changes with respect to the independent variable (often \(x\)). In simpler terms, it shows the rate at which \(y\) changes as \(x\) changes.

For example, if the slope \(m = 0.01\), it indicates that for every one unit increase in \(n\), the function value \(h(n)\) increases by 0.01. The concept of slope is crucial for understanding how factors are related in the graph of a linear equation.
  • A positive slope indicates an upward trend.
  • A negative slope indicates a downward trend.
  • A slope of zero indicates a horizontal line, meaning there is no change in \(y\) as \(x\) changes.
The Significance of the Y-intercept in a Linear Function
The y-intercept of a linear function is the value where the line crosses the y-axis of the graph. In the equation \(y = mx + b\), the y-intercept is given by \(b\). This value represents the point where the independent variable, which is \(x\) or in our example \(n\), is zero.

For instance, in the equation given, \(b = 100\), meaning when \(n = 0\), the function \(h(n)\) will equal 100. This helps us understand the starting value or initial condition of the function.
  • The y-intercept provides valuable context even if the graphs never actually touch the y-axis, as it helps us visualize where the line begins its journey along the number plane.
  • Knowing the y-intercept allows quick sketching of the linear graph once combined with the slope.
Understanding the Role of an Independent Variable
In any function, the independent variable is the variable that we manipulate or control to see how it affects the dependent variable. It is often denoted by \(x\) in equations, but in some instances, like our example, it uses a different letter, \(n\).

The independent variable is critical because it serves as the primary cause that brings about changes in the dependent variable. By altering the independent variable, we can observe and analyze the resulting differences in the function's output.
  • For linear functions, changes in the independent variable lead to direct (or constant) changes in the dependent variable, given the slope.
  • It is typically the input or the domain of the function.
  • Understanding which variable is independent helps in structuring experiments and analyzing graphs or data relations efficiently.

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Most popular questions from this chapter

Decide for what value(s) of the constant \(A\) (if any) the equation has (a) The solution \(x=0\) (b) A positive solution in \(x\) (c) No solution in \(x\). $$ \frac{7}{x}=A $$

Find an equation for the line parallel to the graph of (a) \(y=3+5 x\) with a \(y\) -intercept of 10 . (b) \(4 x+2 y=6\) with a \(y\) -intercept of 12 . (c) \(y=7 x+2\) and containing the point (3,22) . (d) \(9 x+y=5\) and containing the point (5,15) .

(a) Write a constraint equation. (b) Choose two solutions. (c) Graph the equation and mark your solutions. The relation between quantity of chicken and quantity of steak if chicken costs \(\$ 1.29 / \mathrm{lb}\) and steak costs \(\$ 3.49 / \mathrm{lb},\) and you have \(\$ 100\) to spend on a barbecue.

Find two numbers with sum 17 and difference 12 .

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