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Chapter 3: Problem 180

For the following exercises, find the intercepts of the functions. $$f(x)=x^{3}+27$$

Short Answer

Expert verified
Y-intercept is (0, 27); X-intercept is (-3, 0).

Step by step solution

01

Understand Intercepts

Intercepts are the points where the graph of a function intersects the axes. In this exercise, we will focus on the y-intercept and x-intercepts of the function. - **Y-intercept**: This is where the graph intersects the y-axis, which happens when \(x = 0\).- **X-intercept(s)**: This is where the graph intersects the x-axis, which happens when \(f(x) = 0\).
02

Find the Y-intercept

To find the y-intercept, substitute \(x = 0\) into the function \(f(x) = x^3 + 27\).\[f(0) = 0^3 + 27 = 27\]The y-intercept is therefore at the point \((0, 27)\).
03

Find the X-intercept(s)

To find the x-intercept(s), set the function equal to zero and solve for \(x\):\[0 = x^3 + 27\]We can rewrite the equation by moving \(27\) to the other side:\[x^3 = -27\]Taking the cube root of both sides gives:\[x = \sqrt[3]{-27} = -3\]Thus, the x-intercept is at the point \((-3, 0)\).
04

Summarize the Intercepts

From the calculations, the y-intercept is at \((0, 27)\) and the x-intercept is at \((-3, 0)\). These points represent where the function \(f(x) = x^3 + 27\) crosses the y-axis and x-axis respectively.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

y-intercept
The y-intercept of a function is a point where the graph intersects the y-axis. This point occurs when the value of the independent variable, often denoted as \(x\), is zero. Thus, to find the y-intercept, we simply substitute \(x = 0\) into the given function and solve for \(f(x)\) as follows:
  • For a polynomial function like \(f(x) = x^3 + 27\), plug \(x = 0\) to get \(f(0) = 0^3 + 27 = 27\). This calculation shows that the graph crosses the y-axis at the point \((0, 27)\).
Understanding the y-intercept is crucial because it provides information about the vertical position of the graph at \(x = 0\).
In real-world contexts, this point can represent the initial value or starting point of a situation being modeled by the function.
x-intercepts
The x-intercepts of a function are points where its graph crosses the x-axis. This occurs when the output of the function, \(f(x)\), is zero. To find x-intercepts, set \(f(x) = 0\) and solve for \(x\).
  • For the polynomial function \(f(x) = x^3 + 27\), we set up the equation as \(0 = x^3 + 27\). Simplifying, we get \(x^3 = -27\).
  • The solution to this equation is found by taking the cube root of both sides: \(x = \sqrt[3]{-27} = -3\).
Thus, the x-intercept is at the point \((-3, 0)\). This tells us where the function value is zero, which is important for understanding where the function might change behavior or cross over the x-axis. These intercepts can represent solutions to equations or show significant changes in applied contexts.
polynomial functions
Polynomial functions are expressions that involve sums of powers of a variable with coefficients. They take the general form \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\), where each \(a_i\) is a constant coefficient, and \(x\) is raised to whole number powers.
  • In our example, \(f(x) = x^3 + 27\), it's a polynomial function of degree 3 because the highest power of \(x\) is 3.
    • This means the graph of this polynomial will have a characteristic shape that can include up to three x-intercepts, depending on the particular coefficients and constants.
  • The function \(f(x)\) will also exhibit a smooth and continuous curve, dictated by the degree and the specific terms involved.
Polynomial functions are fundamental in algebra because they are versatile and appear frequently in various mathematical models. Understanding them helps in predicting and analyzing complex behaviors modeled by such functions.

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