Problem 27 - College Algebra

Question

Find the intercepts of the functions. $$f(x)=x^{4}-16$$

Step-by-Step Solution

Answer

Y-intercept: (0, -16); X-intercepts: (-2, 0) and (2, 0).

Step 1: Identify the Y-intercept

To find the y-intercept, set the input variable (x) to zero and solve for f(x). This gives us the point where the graph intercepts the y-axis.For the function given, substitute x = 0:\[ f(0) = 0^4 - 16 = -16 \]Thus, the y-intercept is (0, -16).

Step 2: Find the X-intercepts

To find the x-intercepts, set the output variable (f(x)) to zero and solve the equation for x. This gives us the points where the graph intercepts the x-axis.0 = x^4 - 16.Add 16 to both sides:\[ x^4 = 16 \]Take the fourth root of both sides to solve for x:\[ x = \pm \sqrt[4]{16} \]\[ x = \pm 2 \]Thus, the x-intercepts are (-2, 0) and (2, 0).

Key Concepts

y-intercept

The y-intercept is the point where a graph intersects the y-axis. Imagine standing at this point on the Cartesian plane; you're at the location where the x-value is zero. This spot tells you what output the function provides when there's no input. For polynomials like our example, finding the y-intercept is straightforward. Just set all x values to zero and evaluate. This tells you how high or low the curve starts on the y-axis. In our function, when x equals 0, the y-value is -16. Thus, the y-intercept is located at (0, -16). Here’s the beauty of this concept: it's always just one quick substitution away from being solved!

x-intercept

The x-intercept is where a graph crosses or touches the x-axis. At this spot, the output of the function, or the f(x) value, is zero. To find x-intercepts, we set the function's entire expression equal to zero. In solving for this function, you solve the equation \( x^4 - 16 = 0 \). By manipulating the equation, you discover that the x-intercepts are the points where f(x) is zero, which occur at \( x = -2 \) and \( x = 2 \). Thus, these points are \((-2, 0)\) and \((2, 0)\), where the steady pavement of the graph meets the horizontal x-axis! This concept is essential because it guarantees we can locate horizontal interactions effectively.

solving equations

Solving equations is like cracking a code. The goal is to find which input values result in the desired output. In our example, the task is to find when \(f(x)\) becomes zero. This requires a deep look into the relationship between input x-values and their effects on the output. To solve, manipulate the equation: add, subtract, multiply, divide or apply any function mangling technique at your disposal. For \( x^4 - 16 = 0 \), adding 16 simplifies to \( x^4 = 16 \). Sneak in the appropriate root extraction, here the fourth root, to unravel the x-values. This gives \( x = \pm 2 \). Solve patiently step by step, ensuring each move simplifies the equation, bringing you closer to understanding. Solving is the core artillery of mathematics, enabling you to conquer various questions across different functions!