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Chapter 1: Problem 29

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 of the graph are at \(x=0\) and \(x=2\), and the \(y\)-intercept is at \(y=0\).

Step by step solution

01

Finding the \(x\)-intercepts

Set \(y=0\) in the given equation and solve for \(x\):\n0 = 2x^3 - 4x^2. Solving this equation will give the \(x\)-intercepts.
02

Simplify the equation

The equation can be simplified by factoring out \(2x^2\): \n0 = 2x^2(x - 2). Thus, the roots of the equation, and thereby the \(x\)-intercepts, are \(x=0\) and \(x=2\).
03

Finding the \(y\)-intercept

Set \(x=0\) in the given equation and solve for \(y\):\n\(y = 2(0)^3 - 4(0)^2\)\nSo, \(y=0\) which means that the \(y\)-intercept is at \(y=0\)

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Key Concepts

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

X-Intercepts
To understand how to find x-intercepts of a graph, consider them as the points where the graph crosses the x-axis. In other words, they are the values of x for which the value of y is zero. To find these points on the graph of an equation like y = 2x^3 - 4x^2, you simply set the y-value to zero and solve for x.

For the given equation, the process starts with the equation 0 = 2x^3 - 4x^2. Factoring out the greatest common factor, 2x^2, gives us 0 = 2x^2(x - 2). This indicates that the x-intercepts occur when x equals 0 or when x equals 2. So, the x-intercepts of this graph are at the points (0,0) and (2,0).
Y-Intercept
On the flip side, the y-intercept is found where the graph crosses the y-axis. This occurs when the value of x is zero. To find the y-intercept for our equation, set x to zero in the original equation and solve for y.

In the equation y = 2x^3 - 4x^2, substituting x with 0 yields y = 2(0)^3 - 4(0)^2 = 0. Therefore, the graph has a y-intercept at the point (0,0). In essence, the y-intercept is the point where the graph meets the y-axis and, for this specific function, it coincides with one of the x-intercepts.
Factoring Polynomials
To deepen your understanding of factoring polynomials, imagine breaking down a complex expression into simpler, multipliable factors, much like decomposing a number into a multiplication of smaller numbers. When you have a polynomial like 2x^3 - 4x^2, factoring simplifies it, which is crucial for finding intercepts and solving equations.

In our case, factoring the polynomial involved taking out the greatest common factor, 2x^2, leading to 2x^2(x - 2) = 0. Factoring helps in revealing the roots of the polynomial, as each factor set to zero gives a possible x-value where the graph intersects the x-axis.
Roots of the Equation
The roots of the equation, also known as zeroes, are the solutions to the equation when set equal to zero. These roots correspond to the x-intercepts of the graph. In the context of the equation y = 2x^3 - 4x^2, after factoring, we have 2x^2(x - 2) = 0. Setting each factor equal to zero, x = 0 and x - 2 = 0 (which gives x = 2), we find the roots to be 0 and 2.

Finding the roots of an equation is a vital skill in algebra that applies to graphing, calculus, and beyond, as it helps to understand the behavior of the graph and the function's relationship with the x-axis. By setting the equation to zero and factoring, or using methods like the quadratic formula for quadratic equations, we can always find these crucial points.

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Most popular questions from this chapter

It Consider the functions \(f(x)=x+2\) and \(f^{-1}(x)=x-2 .\) Evaluate \(f\left(f^{-1}(x)\right)\) and \(f^{-1}(f(x))\) for the indicated values of \(x\). What can you conclude about the functions?$$\begin{array}{|l|l|l|l|l|}\hline x & -10 & 0 & 7 & 45 \\\\\hline f\left(f^{-1}(x)\right) & & & & \\ \hline f^{-1}(f(x)) & & & & \\\\\hline\end{array}$$,

Find a mathematical model that represents the statement. (Determine the constant of proportionality.) \(A\) varies directly as \(r^{2} .(A=9 \pi \text { when } r=3 .)\)

Assume that \(y\) is directly proportional to \(x .\) Use the given \(x\) -value and \(y\) -value to find a linear model that relates \(y\) and \(x .\) $$x=4, y=8 \pi$$

Find a mathematical model for the verbal statement. \(h\) varies inversely as the square root of \(s\).

Use the given values of \(k\) and \(n\) to complete the table for the direct variation model \(y=k x^{n} .\) Plot the points in a rectangular coordinate system. $$\begin{array}{|l|l|l|l|l|l|}\hline x & 2 & 4 & 6 & 8 & 10 \\\\\hline y=k x^{n} & & & & & \\\\\hline\end{array}$$ $$k=\frac{1}{4}, n=3$$
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