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Chapter 5: Problem 18

Rewrite the function in slope-intercept form. $$ j(t)=1.2+0.4(t-5) $$

Short Answer

Expert verified
Question: Rewrite the given function j(t) = 1.2 + 0.4(t - 5) in slope-intercept form and identify its slope and y-intercept. Answer: The rewritten function in slope-intercept form is j(t) = 0.4t - 0.8, with a slope of 0.4 and a y-intercept of -0.8.

Step by step solution

01

Distribute the slope term to the parentheses

In this step, we will distribute the value of 0.4 to the expression (t - 5). This will give us the equation in the form of j(t) = mt + c. $$ j(t) = 1.2 + 0.4t - 2 $$
02

Simplify the equation

In this step, we will simplify the equation by combining like terms. Combining 1.2 and -2 gives us: $$ j(t) = 0.4t - 0.8 $$
03

Write the equation in slope-intercept form

Now that the equation is simplified, it is already in slope-intercept form. We can write it as: $$ j(t) = 0.4t - 0.8 $$ The given function has been rewritten in slope-intercept form, where the slope (m) is 0.4 and the y-intercept (b) is -0.8.

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Key Concepts

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

Linear Function
A linear function is a mathematical expression that describes a line on a graph. It has a constant rate of change and is written in the form of \( y = mx + b \) where:
  • \( y \) is the dependent variable.
  • \( m \) is the slope of the line, representing how steep the line is.
  • \( x \) is the independent variable.
  • \( b \) is the y-intercept, the point where the line crosses the y-axis.
This type of function is essential for modeling relationships that have a consistent rate of increase or decrease. In our exercise, the linear function is represented by \( j(t) = 0.4t - 0.8 \). Here, the slope \( m \) is 0.4, indicating that for every unit increase in \( t \), the value of \( j(t) \) increases by 0.4. Simultaneously, the y-intercept \( b \) is -0.8, which means the line crosses the y-axis at -0.8.
Distributive Property
The distributive property is a fundamental algebraic concept used to simplify expressions and equations. It states that a single term can be multiplied by each term inside the parentheses. The rule can be expressed as:\[ a(b + c) = ab + ac \]In the context of the exercise, the distributive property is used to expand the function \( j(t) = 1.2 + 0.4(t-5) \). Here, multiplying 0.4 by both \( t \) and \( -5 \) gives:
  • 0.4 times \( t \) equals 0.4t.
  • 0.4 times \(-5\) equals -2.
Thus, applying the distributive property transforms the expression into \( j(t) = 1.2 + 0.4t - 2 \). This simplification lays the groundwork for rewriting the function in slope-intercept form.
Simplifying Equations
Simplifying equations is the process of making an equation easier to read and solve. This often involves combining like terms, which are terms with the same variable raised to the same power. In our problem, after applying the distributive property, the function was \[ j(t) = 1.2 + 0.4t - 2 \]Here, the like terms are constants \(1.2\) and \(-2\). Combining them gives \(-0.8\), which simplifies our function to\[ j(t) = 0.4t - 0.8 \]This step of simplification is crucial as it aligns our function with the slope-intercept form \( y = mx + b \). It helps us clearly identify the slope and y-intercept of the function, making it easier to understand and graph.

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Most popular questions from this chapter

Why do we expect the situation to be modeled by a linear function? Give an expression for the function. The profit from making \(q\) widgets is the revenue minus the cost, where the revenue is the selling price, \(\$ 27\), times the number of widgets, and the cost is \(\$ 1000\) for setting up a production line plus \(\$ 15\) per widget.

Are the lines parallel? $$ y=2+3(x+5) ; y=2+4(x+5) $$

Solve the systems of equations. $$ \left\\{\begin{array}{l} 5 x-7 y=31 \\ 2 x+3 y=-5 \end{array}\right. $$

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 \(x\). $$ 3 x=A $$

Write an expression in \(x\) representing the result of the given operations on \(x\). Is the expression linear in \(x ?\) Add \(5,\) multiply by \(x,\) subtract \(2 .\)
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