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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 are at (0,0) and (2,0) and the y-intercept is at (0,0)

Step by step solution

01

Finding the y-intercept

To find y-intercepts we set \(x=0\) in the equation: \(y = 2(0)^{3} - 4(0)^{2}\) which simplifies to \(y=0\). Therefore, the y-intercept is (0,0)
02

Finding the x-intercepts

To find x-intercepts, set \(y=0\) and solve the equation. The equation becomes: \(0 = 2x^{3} - 4x^{2}\). We can factor out the common factor of \(2x^{2}\) which gives us \(0 = 2x^{2}(x - 2)\). Setting this equal to zero gives us the solutions \(x = 0\) and \(x = 2\). Thus, the x-intercepts are (0,0) and (2,0)

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Key Concepts

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

Understanding x-intercepts
The x-intercept of a graph represents the point(s) where the graph crosses the x-axis. At these points, the value of y is zero, meaning we are looking for the values of x that satisfy the equation when y equals zero. In mathematical notation, this involves solving the equation by setting y to zero and finding the values of x.
For the equation given, \( y = 2x^3 - 4x^2 \), we find the x-intercepts by solving the equation \( 0 = 2x^3 - 4x^2 \). This was done by factoring the equation.
This approach shows that the x-intercepts for the graph are at the points (0,0) and (2,0). These are the points where the graph touches or crosses the x-axis.
Exploring y-intercepts
The y-intercept of a graph is the point where it crosses the y-axis. At this point, the x value is zero because it lies on the y-axis. To find it, substitute zero for x in the equation.
For the given equation \( y = 2x^3 - 4x^2 \), we set \( x = 0 \). Substituting this into the equation, we compute \( y = 2(0)^3 - 4(0)^2 \) which simplifies to \( y = 0 \).
Thus, the y-intercept of this equation is at the point (0,0). This tells us where the graph intersects the y-axis.
Graphing polynomials
Graphing involves plotting points on a Cartesian plane based on the equation given. For polynomial equations, this generally creates curves that may have multiple intercepts, peaks, or valleys.
The equation \( y = 2x^3 - 4x^2 \) is a cubic polynomial. Its graph can be obtained by plotting various points obtained from the equation, including the intercepts.
  • First, mark the x-intercepts (0,0) and (2,0) where the graph crosses the x-axis.
  • Next, mark the y-intercept (0,0) which is the same as the x-intercept in this case.
Using additional points that may be derived from choosing various x values, you can plot out the shape of the curve, which generally forms a symmetric pattern with respect to the origin for this polynomial.
Factoring polynomials
Factoring is a key technique in simplifying polynomial equations and is crucial in finding intercepts. For the equation \( y = 2x^3 - 4x^2 \), factoring involves identifying common factors.
Notice the term \( 2x^2 \) is common in both parts of the expression, allowing us to factor it out. The equation then becomes \( 0 = 2x^2(x - 2) \).
  • By setting \( 2x^2 = 0 \), we find one solution \( x = 0 \).
  • Similarly, setting \( (x - 2) = 0 \), provides another solution \( x = 2 \).
Factoring makes solving complex equations simpler by breaking them down into manageable parts. This facilitated the identification of x-intercepts for the given polynomial equation.

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