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Cubic functions are polynomial functions of degree three, typically written in the standard form \( f(x) = ax^3 + bx^2 + cx + d \) with \( a \) not equal to zero. These functions are known for their \( S \) shaped curves and their distinctive features such as turning points, where the graph changes its direction, and inflection points, where the concavity of the graph changes.

In our exercise, the cubic function is \( y = 4x^3 + 6x^2 \). To graph it, we can calculate values for \( y \) by substituting various \( x \) values into the equation. For instance, when \( x = 0 \) the \( y \) value is \( 0 \) as well. Generally, points where \( x \) or \( y \) equals zero are known as intercepts - important features when graphing. To see the 'S' curve, it's crucial to calculate points on both sides of the intercepts, choosing negative and positive \( x \) values.

Understanding the end behavior of cubic functions helps us as well - as \( x \) approaches infinity, if \( a > 0 \) the function will rise to infinity and if \( x \) approaches negative infinity, the function will fall into negative infinity. The reverse holds true if \( a \) is less than zero.

Starting with the Basic Shape

When graphing polynomials, start by considering the leading term, as it influences the graph’s end behavior. For instance, the leading term in our cubic function dictates that the ends of the graph will either rise or fall off to infinity, depending on the sign of the coefficient.

Plotting Key Points

Next, plot key points such as intercepts and calculate the value of the polynomial at relevant \( x \) values. This step is critical in our exercise for plotting the intersection points with horizontal lines \( y = t \).

Considering Symmetry

Certain polynomials display symmetry, which can simplify the graphing process. Even-powered polynomials show \( y \) axis symmetry, while odd-powered polynomials may have origin symmetry.

Finally, to enhance understanding, revise the plotted points and the curve by using a graphing tool, slowly tracing over the conceptual graph with actual values. This iterative process helps consolidate the understanding of graphing polynomial equations.

Symmetry in algebraic curves simplifies understanding complex graphs. An axis of symmetry is a line that divides a figure or a graph into two mirrored halves. For polynomial functions, the presence of an \( x^2 \) or \( y^2 \) term often indicates symmetrical properties about the \( x \) or \( y \) axis respectively.

In the original exercise, we encounter the equation \( y^2 = 4x^3 + 6x^2 - t \). The squared \( y \) term transforms the cubic equation into one that's symmetrical about the \( x \) axis because for every \( (x, y) \) point on the curve, there exists a \( (x, -y) \) point. The horizontal lines \( y = t \) intersect the curve at points that, when mirrored across the \( x \) axis due to the \( y^2 \) term, will fall on the modified cubic curve.

This symmetry helps us graph the modified cubic equation efficiently, as we only need to find points for \( y \) in the positive or negative region and then mirror them across the \( x \) axis to complete the graph. Symmetrical features are not just visually pleasing – they're fundamental in tackling more advanced mathematical topics such as integration and solving equations analytically.