Problem 18 - Algebra Form and Function

Question

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

Step-by-Step Solution

Answer

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 1: 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. $$ j(t) = 1.2 + 0.4t - 2 $$

Step 2: Simplify the equation

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

Step 3: 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.

Key Concepts

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.