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To find the x-intercept of a graph, you begin by setting the output variable, usually represented as \( y \), to zero in the equation. The x-intercept is the point where the graph crosses the x-axis, and at this point, the value of \( y \) is zero.

For the given equation \( y = 2x^3 - 4x^2 \), you set \( y = 0 \). Then, solve for \( x \) by factoring the resulting expression.

In this case, first factor out \( x^2 \), which results in \( x^2(2x - 4) = 0 \).

• This equation implies two possibilities for \( x \):
• \( x^2 = 0 \), which gives \( x = 0 \).
• \( 2x - 4 = 0 \), solving this results in \( x = 2 \).

Therefore, the x-intercepts are \( x = 0 \) and \( x = 2 \). This means the graph crosses the x-axis at these points.

The y-intercept is found by setting \( x = 0 \) in the equation, which identifies the point where the graph crosses the y-axis. This is because, at this intercept, the input variable \( x \) has a value of zero.

For the equation \( y = 2x^3 - 4x^2 \), substitute \( x = 0 \) to find the y-intercept. This simplifies to \( y = 2(0)^3 - 4(0)^2 \), which results in \( y = 0 \).

Therefore, the graph intersects the y-axis at point \( y = 0 \). When plotted, this intercept tells you that the graph passes through the origin \((0, 0)\), easily identified by where neither the x nor the y coordinate is altering.

Factoring polynomials is a key skill in algebra that helps simplify equations, especially when solving for intercepts or roots. This technique involves breaking down a polynomial into simpler terms, called factors, that can be multiplied together to produce the original polynomial.

In the case of \( y = 2x^3 - 4x^2 \), we factor out \( x^2 \) from each term to simplify the expression to \( x^2(2x - 4) \).

• This process transforms a complex equation into manageable parts, making it easier to solve for \( x \).
• The factors \( x^2 \) and \( 2x - 4 \) offer immediate solutions for x-intercepts when set to zero.

Factoring shortcuts the process by revealing possible zeros of the function, which correspond to the x-intercepts on the graph.

Mastering this technique is crucial for efficiently solving and graphing equations.

Graphing equations involves plotting the solutions of an equation onto a coordinate plane. This visual representation helps in understanding the behavior and characteristics of the equation over different inputs.

For the equation \( y = 2x^3 - 4x^2 \), we already know the x-intercepts (\( x = 0 \) and \( x = 2 \)) and the y-intercept (\( y = 0 \)). These points are critical in sketching the basic outline of the graph.

• The x-intercepts indicate where the curve crosses the x-axis.
• The y-intercept shows where the curve crosses the y-axis.

By plotting these intercepts, you can begin to see the shape of the cubic polynomial. Additional points can be calculated by choosing more values of \( x \) and solving for \( y \), if needed.

Understanding the intercepts and additional solutions is key to an accurate graph, assisting in visualizing how the polynomial behaves across its domain.