Question

Write a cubic function with \(x\) -intercepts of \(\sqrt{3},-\sqrt{3},\) and 1 and a \(y\) -intercept of -1.

Short answer

The function is \[ f(x) = -\frac{1}{3}x^3 + \frac{1}{3}x^2 + x - 1 \].

Step-by-step solution

Identify the General Form

A cubic function with given x-intercepts can be written in the form \[ f(x) = a (x - r_1)(x - r_2)(x - r_3) \]. The given x-intercepts are \( \sqrt{3} \), \( -\sqrt{3} \), and 1.

Substitute the x-intercepts

Using the x-intercepts, the function can be written as \[ f(x) = a (x - \sqrt{3})(x + \sqrt{3})(x - 1) \].

Simplify the Function

First, simplify the terms involving square roots: \[ (x - \sqrt{3})(x + \sqrt{3}) = x^2 - (\sqrt{3})^2 = x^2 - 3 \]. Now the function reads \[ f(x) = a (x^2 - 3)(x - 1) \].

Expand the Polynomial

Expand the simplified form by distributing \[ f(x) = a (x^3 - x^2 - 3x + 3) \].

Find a using the y-intercept

Given the y-intercept is -1, substitute \( x = 0 \) and solve for \( a \): \[ -1 = a (0^3 - 0^2 - 3(0) + 3) \]. Simplify to find \[ -1 = 3a \] and solve for \( a \): \[ a = -\frac{1}{3} \].

Formulate the Final Function

Substitute \( a \) back into the expanded polynomial: \[ f(x) = -\frac{1}{3} (x^3 - x^2 - 3x + 3) = -\frac{1}{3}x^3 + \frac{1}{3}x^2 + x - 1 \].

Key concepts

cubic polynomials

A cubic polynomial is a type of polynomial function characterized by the highest degree of 3 for its variable. The general form of a cubic function is given by: \(f(x) = ax^3 + bx^2 + cx + d\). Here, \(a\), \(b\), \(c\), and \(d\) are constants, and \(a eq 0\).

The degree of 3 indicates that the graph of a cubic polynomial will always have an inflection point, meaning the curve changes direction. Cubic polynomials can have up to three real roots (x-intercepts), but they may also have fewer.

Understanding cubic polynomials is essential for solving many types of algebraic problems. They appear in various fields, including physics, engineering, and economics. For instance, they are used to model the behavior of physical phenomena and to find optimization points.

x-intercepts

The x-intercepts of a function are the points where the graph crosses the x-axis. In mathematical terms, these are the solutions to \(f(x) = 0\).

To find the x-intercepts of a cubic function, like the one given in the exercise, we set the function equal to zero: \( f(x) = a (x - r_1)(x - r_2)(x - r_3) \). Here, \(r_1\), \(r_2\), and \(r_3\) are the x-intercepts.

In our exercise, the given x-intercepts are \(\frac{\root{3}}\forall\) , \(-\frac{\root{3}}\), and 1. This tells us the function crosses the x-axis at these points. Identifying x-intercepts is crucial as they provide roots of the polynomial, which assist in graphing and understanding the behavior of the function.

y-intercepts

The y-intercept of a function is the point where the graph crosses the y-axis. This happens when the value of \(x\) is zero. To find the y-intercept, we substitute \(x = 0\) into the function and solve for \(y\).

For our cubic function \(f(x)\), substituting \(x = 0\) gives us: \( f(0) = -\frac{1}{3} (0^3 - 0^2 - 3*0 + 3) = -\frac{1}{3} * 3 = -1\). Hence, the y-intercept is -1.

Knowing the y-intercept helps us quickly assess where the graph of the function will intersect the y-axis. It's another critical point for sketching the graph of the polynomial and understanding its behavior.

function expansion

Expanding a function means expressing it as a polynomial in its standard form. For the cubic function in the exercise, it begins in factored form: \( f(x) = a(x - \frac{\root{3}})(x + \frac{\root{3}})(x - 1)\).

To expand, we first simplify terms involving square roots: \( (x - \frac{\root{3}})(x + \frac{\root{3}}) = x^2 - (\frac{\root{3}})^2 = x^2 - 3\).

This gives us: \( f(x) = a (x^2 - 3)(x - 1)\). Next, we distribute to expand fully: \( f(x) = a(x^3 - x^2 - 3x + 3)\).

Expanding functions is crucial because it transforms them into a form that's easier to differentiate, integrate, and analyze. It reveals all the coefficients, making it simpler to understand the polynomial's impact on the graph.