Let's understand how to solve the cubic graph y equals 2 times x minus 3 cubed. When we say solve a graph, we typically mean finding where it crosses the x-axis, which happens when y equals zero. This cubic function has a characteristic S-shaped curve that passes through the point 3, 0.
The first step in solving this cubic equation is to set y equal to zero. This gives us the equation zero equals 2 times x minus 3 cubed. We can simplify by dividing both sides by 2, which gives us zero equals x minus 3 cubed. The red line highlights the x-axis where y equals zero.
Now we take the cube root of both sides. The cube root of zero is zero, and the cube root of x minus 3 cubed is simply x minus 3. This gives us zero equals x minus 3. The green vertical line shows where x equals 3, which is our solution.
Finally, we solve for x by adding 3 to both sides of the equation. Zero plus 3 equals x minus 3 plus 3, which simplifies to 3 equals x, or x equals 3. This means the cubic graph crosses the x-axis at exactly one point: 3, 0. The green box highlights our final answer.
To summarize, we solved the cubic equation y equals 2 times x minus 3 cubed by setting y equal to zero, dividing by 2, taking the cube root, and solving for x. The answer is x equals 3, which means the graph crosses the x-axis at the point 3, 0. This is the only x-intercept for this cubic function.