Problem 26 - College Algebra

Question

Find the intercepts of the functions. $$g(n)=-2(3 n-1)(2 n+1)$$

Step-by-Step Solution

Answer

X-intercepts: \(\left(\frac{1}{3}, 0\right)\), \(\left(-\frac{1}{2}, 0\right)\); Y-intercept: \((0, 2)\).

Step 1: Understand the Intercepts

To find the intercepts of the function, we need to determine where the function crosses the axes. The intercept with the y-axis occurs when \(n = 0\), and the intercepts with the x-axis occur where \(g(n) = 0\).

Step 2: Find the Y-Intercept

To find the y-intercept, substitute \(n = 0\) into the function \(g(n)\): \[g(0) = -2(3(0) - 1)(2(0) + 1) = -2(-1)(1) = 2\] Thus, the y-intercept is \((0, 2)\).

Step 3: Find the X-Intercept(s)

To find the x-intercept(s), set \(g(n) = 0\) and solve for \(n\): \[-2(3n-1)(2n+1) = 0\]Since a product is zero when any factor is zero, set each factor equal to zero:1. \(3n - 1 = 0\)2. \(2n + 1 = 0\)

Step 4: Solve for X-Intercept from First Factor

Solve \(3n - 1 = 0\):\[3n = 1\]\[n = \frac{1}{3}\]So, one x-intercept is \(\left(\frac{1}{3}, 0\right)\).

Step 5: Solve for X-Intercept from Second Factor

Solve \(2n + 1 = 0\):\[2n = -1\]\[n = -\frac{1}{2}\]So, the other x-intercept is \(\left(-\frac{1}{2}, 0\right)\).

Step 6: Verify the Solutions

Confirm each intercept by substituting back into the function and ensuring it results in zero for x-intercepts and 2 for y-intercept. The calculations confirm that both \(\left(\frac{1}{3}, 0\right)\) and \(\left(-\frac{1}{2}, 0\right)\) are valid x-intercepts, and \((0, 2)\) is the y-intercept.

Key Concepts

Y-Intercept

In mathematics, the y-intercept is a point where the graph of a function crosses the y-axis. This point is significant because it represents the value of the function when the input (in this context, n) is zero. For the function given in the exercise,

• To determine the y-intercept, we substitute 0 for n.
• The calculation becomes: \[ g(0) = -2 (3(0) - 1) (2(0) + 1) = 2 \]

Thus, the y-intercept is \(0, 2\). This means the function crosses the y-axis at the point 2 on the y-axis. This concept can be applied to any function by setting the input to zero and solving for the output.

X-Intercept

The x-intercept(s) of a function represent the point(s) where the graph crosses the x-axis. At these points, the output of the function is zero. For solving \[-2(3n-1)(2n+1)=0\], we must find the values of n that make the function equal to zero by determining the outputs of each factor separately by setting them to zero:

• First, set \(3n-1=0\) which simplifies to \(n=\frac{1}{3}\)
• Second, set \(2n+1=0\) which simplifies to \(n=-\frac{1}{2}\)

Thus, the x-intercepts are \((\frac{1}{3}, 0)\) and \((-\frac{1}{2}, 0)\). Understanding the x-intercepts helps us visualize where the graph crosses the x-axis, indicating points where the function's output is zero.

Algebraic Solutions

Finding intercepts often involves algebraic solutions, meaning we solve equations using known algebraic processes. Algebraic solutions are necessary for uncovering important points such as intercepts. In this exercise, we use algebraic techniques to

• Solve for n by isolating n in the equations \(3n-1=0\) and \(2n+1=0\).
• Simplify to find precise points where the function intersects the axes, providing clear solutions without guesswork.

Algebraic solutions are powerful because they offer a methodical approach to solving complex mathematical problems, ensuring all steps are documented and each calculation is verified, as required in the exercise to confirm the intercepts.

Quadratic Function

A quadratic function is any function that can be written in the form \(ax^2+bx+c\), where a, b, and c are constants. These functions commonly produce parabolic graphs. The exercise's function: \[ g(n)=-2(3n-1)(2n+1) \]is an example of this, though in its factored form.

• Quadratic functions always have a degree of 2, which often results in two x-intercepts, as seen in the exercise.
• They have significant properties such as a vertex (the lowest or highest point of the parabola), axis of symmetry, and roots (x-intercepts in the coordinate plane context).

Recognizing the function's type can assist in predicting its graph shape and behavior, and in understanding how to manipulate it in various mathematical contexts.