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Chapter 0: Problem 57

Find the \(x\) - and \(y\)-intercepts of the graph of the equation. \(y=2 x^3-4 x^2\)

Short Answer

Expert verified
The x-intercepts are 0 and 2, and the y-intercept is 0.

Step by step solution

01

Find the x-intercepts

To find the x-intercepts, replace y in the equation with 0, and solve for x. This will give the equation \(0=2 x^3-4 x^2\). The solution to this equation will give the x-intercepts.
02

Solve for x

The quadratic equation can be solved by factoring. First factor out \(x^2\), that gives \(x^2(2x-4)=0\). Now use the zero product property which states that if (a 脳 b)=0, then a=0 or b=0. Applying zero product property we get two solutions, x=0 or (2x-4)=0. On further solving (2x-4)=0, we get x=2. Therefore the x-intercepts are 0 and 2.
03

Find the y-intercept

The y-intercept is found by setting x to 0 in the equation and solving for y. So, substitute x=0 into the equation, which gives \(y=2*(0)^3-4*(0)^2 = 0\). Therefore, the y-intercept is 0.

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Key Concepts

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

Factoring
Factoring is a crucial skill in algebra, especially when dealing with equations. It involves breaking down an expression into a product of simpler expressions. For example, suppose you have the equation \(0 = 2x^3 - 4x^2\). The first step is to look for a common factor in all the terms. Here, both terms share \(x^2\), which can be factored out to get \(x^2(2x - 4) = 0\).
  • This simplification creates an easier-to-solve equation because each factor can be manipulated individually.
  • By doing this, you reduce the complexity of the problem.
Factoring is a building block for solving many algebraic equations, including quadratic and cubic equations. Practicing factoring can help improve your problem-solving skills.
Zero Product Property
The zero product property is a fundamental concept in algebra used to solve equations. It states that if the product of two (or more) factors is zero, then at least one of the factors must be zero.
For example, after factoring the original equation to \(x^2(2x - 4) = 0\), you can set each factor to zero:
  • \(x^2 = 0\) leads to \(x = 0\).
  • \(2x - 4 = 0\) simplifies to \(x = 2\).
These solutions, \(x = 0\) and \(x = 2\), are where the graph crosses the x-axis. Remember, this property is particularly helpful when your equation has been factored, as it allows you to solve for the unknowns effortlessly by considering each factor independently.
Cubic Equations
Cubic equations are polynomials where the highest power of the variable is three. They have the general form \(ax^3 + bx^2 + cx + d = 0\).
In our example, the equation is \(y = 2x^3 - 4x^2\), which is a simplified version as the linear and constant terms are missing.
  • Cubic equations can have up to three real roots, but not all roots have to be distinct.
  • Factoring is a common method to solve cubic equations when possible.
Unlike linear or quadratic equations, cubic equations often require more steps to solve, but the approach remains systematic. Start by factoring, simplify the equation, and use properties like the zero product property to find the solutions. Understanding how to handle cubic equations opens you up to solving more complex algebraic problems.

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Most popular questions from this chapter

(a) use the position equation \(s=-16 t^{2}+v_{0} t+s_{0}\) to write a function that represents the situation, (b) use a graphing utility to graph the function, (c) find the average rate of change of the function from \(t_{1}\) to \(t_{2}\), (d) interpret your answer to part (c) in the context of the problem, (e) find the equation of the secant line through \(t_{1}\) and \(t_{2}\), and (f) graph the secant line in the same viewing window as your position function. An object is thrown upward from a height of 6 feet at a velocity of 64 feet per second. $$ t_{1}=0, t_{2}=3 $$

In Exercises 55-68, determine whether the function has an inverse function. If it does, find the inverse function. $$ q(x)=(x-5)^{2} $$

In Exercises 39-54, (a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=\sqrt[3]{x-1} $$

True or False? In Exercises 85 and 86, determine whether the statement is true or false. Justify your answer. $$ \text { If } f \text { is an even function, } f^{-1} \text { exists. } $$

In Exercises 39-54, (a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=x^{3 / 5} $$
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