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Chapter 3: Problem 1

Find the \(C\) and \(t\) intercepts of each function \(C(t)=2(t-4)(t+1)(t-6)\)

Short Answer

Expert verified
The C-intercept is 48; the t-intercepts are 4, -1, and 6.

Step by step solution

01

Identify the Function

The function you need to work with is given as \[ C(t) = 2(t-4)(t+1)(t-6) \] This function is expressed in factored form.
02

Find the C-intercept

To find the \(C\)-intercept, set \(t = 0\) and solve for \(C(0)\):\[ C(0) = 2(0-4)(0+1)(0-6) \]\[ C(0) = 2(-4)(1)(-6) \]\[ C(0) = 2 \times -4 \times 1 \times -6 \]\[ C(0) = 48 \]Thus, the \(C\)-intercept is 48.
03

Find the t-intercepts

To find the \(t\)-intercepts, set \(C(t) = 0\) and solve for \(t\):The equation becomes \[ 2(t-4)(t+1)(t-6) = 0 \]This factored form will equal zero if any one of the factors is zero. Therefore, solve:1. \(t-4 = 0\) \rightarrow \(t = 4\)2. \(t+1 = 0\) \rightarrow \(t = -1\)3. \(t-6 = 0\) \rightarrow \(t = 6\)The \(t\)-intercepts are 4, -1, and 6.

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Key Concepts

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

Factored Form
In mathematics, expressing a function in its factored form simplifies the process of finding roots or intercepts. The factored form of a function is written as a product of its factors, such as \[ C(t) = 2(t-4)(t+1)(t-6) \]Here, each part in the parentheses, like \((t-4)\), \((t+1)\), and \((t-6)\),is a factor of the function. This format is extremely useful because it reveals the roots of the equation quickly, aiding us in finding intercepts.
  • Reduces complexity when solving equations.
  • Makes it easier to identify solutions to the equation.
  • Helps in visualizing the graph of the function.
Using factored form, you can determine various properties of a function, beneficial especially in graph-related problems.
C-intercept
The C-intercept of a function is where the graph of the equation crosses the C-axis on a graph. This is done by setting \(t = 0\)in the function and solving for \(C\). For the given function \(C(t)=2(t-4)(t+1)(t-6)\),by setting all \(t\) values to zero, we calculate \[ C(0) = 2(0-4)(0+1)(0-6) \]Which simplifies to\[ C(0) = 48 \]This means the C-intercept is 48, indicating that the graph passes through (0,48) on the C-axis.
  • Helps in plotting graphs accurately.
  • Particularly useful for determining initial values in real-world applications.
t-intercepts
The \(t\)-intercepts of a function are where the graph crosses the \(t\)-axis. To find \(t\)-intercepts, set\(C(t) = 0\)and solve for \(t\). The function \[ C(t) = 2(t-4)(t+1)(t-6) = 0 \]indicates that any of the factors being zero makes the entire expression zero, due to multiplication. Apply this understanding for solving:
  • \(t-4 = 0\) gives \(t = 4\)
  • \(t+1 = 0\) gives \(t = -1\)
  • \(t-6 = 0\) gives \(t = 6\)
Thus, the t-intercepts are at 4, -1, and 6. Each of these points corresponds to where the curve touches or passes through the \(t\)-axis.
Zero Product Property
The Zero Product Property is an extremely important concept in algebra, particularly useful for solving equations in factored form. It states that if a product of multiple factors equals zero, then at least one of the factors must also be zero.
  • Essential for finding roots or solutions to polynomial equations.
  • Simplifies checking multiple solutions quickly.
  • Reduces complex problems into smaller, manageable parts.
In our function \[ C(t) = 2(t-4)(t+1)(t-6) = 0 \],the Zero Product Property helps by breaking down the problem into smaller equations:\((t-4) = 0\),\((t+1) = 0\), and \((t-6) = 0\).Solving these individual equations gives us the \(t\)-intercepts quickly and efficiently.

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