Problem 25 - College Algebra

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

X-intercepts: (1, 0), (-2, 0), (3, 0). Y-intercept: (0, 12).

Step 1: Understand the Problem

To find the intercepts of the function, we will find where the graph of the function crosses the axes. There are two types of intercepts: x-intercepts and y-intercepts. The function given is \(f(t)=2(t-1)(t+2)(t-3)\).

Step 2: Find the x-intercepts

To find the x-intercepts, set the function equal to zero and solve for \(t\). This is where the product equals zero: \[ 2(t-1)(t+2)(t-3) = 0 \] Setting each factor equal to zero gives the solutions: \(t-1 = 0 \Rightarrow t = 1\)\(t+2 = 0 \Rightarrow t = -2\)\(t-3 = 0 \Rightarrow t = 3\)Thus, the x-intercepts are \((1, 0), (-2, 0),\) and \((3, 0)\).

Step 3: Find the y-intercept

To find the y-intercept, evaluate the function at \(t = 0\):\[ f(0) = 2(0-1)(0+2)(0-3) \]Calculate:\[ f(0) = 2(-1)(2)(-3) = 2 imes 6 = 12 \]Thus, the y-intercept is \((0, 12)\).

Step 4: Summarize the Intercepts

The x-intercepts, where the function crosses the x-axis, are \((1, 0), (-2, 0),\) and \((3, 0)\). The y-intercept, where the function crosses the y-axis, is \((0, 12)\).

Key Concepts

Understanding x-intercepts

To find the x-intercepts of a function is to discover where its graph crosses the x-axis. X-intercepts are points that denote where the value of the function is zero. For a given polynomial function, the process is simple: set the function equal to zero and solve for the variable. In this exercise, with the function \(f(t) = 2(t-1)(t+2)(t-3) \), identifying the x-intercepts involves solving the equation \(2(t-1)(t+2)(t-3) = 0\). This means determining when any factor in the product becomes zero, since the product of any number and zero is zero. After setting each factor equal to zero:

• \((t-1) = 0\) gives \(t = 1\)
• \((t+2) = 0\) gives \(t = -2\)
• \((t-3) = 0\) gives \(t = 3\)

Thus, the x-intercepts are point coordinates where the graph meets the x-axis: \((1, 0)\), \((-2, 0)\), and \((3, 0)\). These points are easily identified and plotted, offering clear intersections that tell us about the roots of the equation.

Understanding y-intercepts

Y-intercepts indicate where a function's graph crosses the y-axis, representing where the function evaluates at zero for the independent variable. Finding the y-intercept is simpler than finding x-intercepts, especially in polynomial functions. It's accomplished by evaluating the function at zero, as the x-coordinate of any y-intercept is always 0. For the function \(f(t) = 2(t-1)(t+2)(t-3)\), we substitute \(t = 0\): \[ f(0) = 2(0-1)(0+2)(0-3) \]Performing the calculations, \[ f(0) = 2(-1)(2)(-3) = 12 \]This means the graph of the function crosses the y-axis at the point \((0, 12)\). Y-intercepts provide crucial insights into the behavior of a function, as they show where on the y-axis the graph intersects when x equals zero.

Solving polynomial equations

Solving polynomial equations is a fundamental skill in algebra that involves finding the variable's values that satisfy the equation when set to zero. This is particularly relevant for determining x-intercepts in functions. The equation \(2(t-1)(t+2)(t-3) = 0\) provides a clear example of solving a polynomial by factoring. Here the polynomial is already factored, simplifying such task by allowing us to separately set each factor to zero.Common methods for solving polynomial equations include:

• Factoring: Breaking down the polynomial into a product of simpler polynomials.
• Using the Quadratic Formula: Suitable for second-degree polynomials when factoring is complex or non-obvious.
• Graphical Solutions: Visualizing where the function crosses the axis using graphing calculators or software.

Each of these methods aids in determining roots, which correspond to x-intercepts in a graph. Such roots or solutions are key to understanding both the algebraic and graphical representation of a polynomial function.