Problem 103 - Algebra and Trigonometry

Question

Although I have not yet learned techniques for finding the \(x\) -intercepts of \(f(x)=x^{3}+2 x^{2}-5 x 6,\) I can easily determine the \(y\) -intercept.

Step-by-Step Solution

Answer

The y-intercept of the function f(x) = x^{3} + 2x^{2} - 5x - 6 is -6.

Step 1: Correct the function

First, we need to correct the function in the problem. The function should be written as \(f(x)=x^{3}+2x^{2}-5x-6\).

Step 2: Calculate the y-intercept

To find the y-intercept, plug in \(x=0\) into the function \(f(x)\). So, calculate \(f(0)=0^{3}+2(0)^{2}-5(0)-6.\)

Step 3: Simplify the expression

Now, simplify the expression by performing the operations. \(0^{3}\) is 0, \(2(0)^{2}\) is 0, \(5(0)\) is 0, and subtract 6 gives us \(-6\). So, \(f(0)=-6\).

Key Concepts

Cubic Functions

When approaching problems involving cubic functions, one must understand their basic nature. A cubic function is a polynomial of degree three, which means it has the standard form of f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants, and a is not equal to zero. The graph of a cubic function is a curve that can have either one or three real roots (where it crosses the x-axis) and can turn at most twice.

These functions are significant in algebra for modeling various real-world processes due to their capability to represent changes in direction. For instance, the acceleration of an object as a function of time can often be represented by a cubic equation.

Understanding the shape and basic properties of cubic functions can be immensely helpful in predicting their behavior without extensive calculations. This foundation makes it easier to approach tasks such as finding intercepts or local extrema (high and low points).

Function Evaluation

Function evaluation is straightforward yet vital in understanding any function's behavior at specific points. To evaluate a function, one substitutes the given value of x into the function and simplifies. This process is used to find function values for graphing or solving equations.

When determining the y-intercept of a function, you evaluate the function at x=0, because the y-intercept is the point where the function crosses the y-axis, and at this point, the value of x is always zero. By plugging in x=0 into the cubic function provided, you get the point where the graph of the function would meet the y-axis. This point provides a reference when graphing the cubic function by hand or understanding its behavior conceptually.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that involves reducing an expression to its most basic form. To simplify an expression, you combine like terms, apply the distributive property, and perform arithmetic operations according to order of operations (PEMDAS/BODMAS).

In the step-by-step solution for finding the y-intercept of a cubic function, f(0) = 0^3 + 2(0)^2 - 5(0) - 6 is simplified by recognizing that any term with a zero multiplied is itself zero. After all the zero terms are removed, you're left with -6 as the function's y-intercept.

Being efficient at simplifying expressions can make solving algebraic problems faster and reduce the chances of making errors. For instance, realizing immediately that multiplication by zero yields zero can save you time and effort in more complex equations.