Problem 25 - College Algebra with Corequisite Support

Question

For the following exercises, find the intercepts of the functions. $$ f(t)=2(t-1)(t+2)(t-3) $$

Step-by-Step Solution

Answer

The x-intercepts are (1, 0), (-2, 0), and (3, 0). The y-intercept is (0, 12).

Step 1: Understanding Intercepts

Intercepts are points where a graph crosses the axes. There are two types, x-intercepts and y-intercepts. An x-intercept is found where the function equals zero, \(f(t) = 0\). A y-intercept is found where the function crosses the y-axis, \(t = 0\).

Step 2: Finding the x-intercepts

To find x-intercepts, set the function equal to zero and solve for \(t\): \[ f(t) = 2(t-1)(t+2)(t-3) = 0 \]The function equals zero when any factor equals zero. Set each factor equal to zero: 1. \(t - 1 = 0 \Rightarrow t = 1\)2. \(t + 2 = 0 \Rightarrow t = -2\)3. \(t - 3 = 0 \Rightarrow t = 3\)Thus, the x-intercepts are \((1, 0), (-2, 0),\) and \((3, 0)\).

Step 3: Finding the y-intercept

To find the y-intercept, evaluate the function at \(t = 0\): \[ f(0) = 2(0-1)(0+2)(0-3) \]Calculating this gives: \[ f(0) = 2(-1)(2)(-3) = 2 \times (-1) \times 2 \times (-3) = 12 \]Thus, the y-intercept is \((0, 12)\).

Step 4: Summary of Intercepts

The x-intercepts of the function \(f(t) = 2(t-1)(t+2)(t-3)\) are \((1, 0), (-2, 0),\) and \((3, 0)\). The y-intercept is \((0, 12)\).

Key Concepts

x-intercepts

To find the x-intercepts of a function, we need to determine the points where the function crosses the x-axis. At these points, the output of the function, or in mathematical terms, the value of the function, is zero.

To locate these points, we set the function equal to zero. For our function, \(f(t) = 2(t-1)(t+2)(t-3)\), we can express it as \(f(t) = 0\). This tells us that the product of the factors must be zero, implying that at least one of the factors must be zero for the entire expression to hold true.

The factors are \((t-1)\), \((t+2)\), and \((t-3)\).

• For \((t-1) = 0\), we solve: \(t = 1\).
• For \((t+2) = 0\), we solve: \(t = -2\).
• For \((t-3) = 0\), we solve: \(t = 3\).

Thus, the x-intercepts are the points where these factors equal zero. The values of \(t\) where this happens are \(t = 1\), \(t = -2\), and \(t = 3\). Therefore, the x-intercepts are \((1, 0)\), \((-2, 0)\), and \((3, 0)\).

y-intercepts

The y-intercept is the point at which the function intersects the y-axis. This occurs when the input \(t\) is zero. Essentially, we evaluate the function for \(t = 0\) to directly find this intercept.

Starting with our function \(f(t) = 2(t-1)(t+2)(t-3)\), we substitute zero for \(t\): \[ f(0) = 2(0-1)(0+2)(0-3) \]
This simplifies to: \[ f(0) = 2 \times (-1) \times 2 \times (-3) \]This results in \(f(0) = 12\), confirming that the y-intercept is at the point \((0, 12)\).

The y-intercept not only tells us how the graph behaves initially as it crosses the y-axis but also provides valuable information about the function's initial value when time or the independent variable is zero.

factoring polynomials

Factoring polynomials is a crucial step in solving for intercepts and other properties of polynomial functions. It simplifies complex expressions into products of simpler expressions called factors. These factors make finding intercepts straightforward because setting them to zero reveals the roots or solutions of the polynomial.

Our function, \(f(t) = 2(t-1)(t+2)(t-3)\), is already in factored form. This factorization helps us immediately see potential values of \(t\) that will result in the entire function equating to zero, thus revealing the x-intercepts. Each factor corresponds to an x-intercept when that factor equals zero.

Here’s a quick approach to polynomial factorization:

• Look for common factors across all terms.
• Use techniques like grouping, simplification, or special identity formulas (such as the difference of squares).
• Recognize simple factors that indicate easy zero points, such as linear factors \((t-1)\) which directly yield \(t = 1\) as a root.

Polynomials in a simple factored form simplify the task of finding intercepts, solving equations, and graphing the function to better understand its behavior across inputs. Understanding factoring is key to exploring how these equations represent curves and shapes on graphs.