91Ó°ÊÓ

Graphing a cubic function involves understanding the basic shape and transformations that apply to the equation. For the function given, \[ f(x) = \frac{1}{4}x^3 - 2 \], the term \( \frac{1}{4}x^3 \) represents a standard cubic curve which is typically shaped like a stretched "S".
When you

• Multiply by \( \frac{1}{4} \), it vertically compresses the function, making it less steep compared to \( x^3 \).
• Subtract 2, the entire graph shifts downward by 2 units, moving the curve below the x-axis initially.

To graph this function, plot points for different values of \( x \) and observe the smooth curve that reflects these transformations.
Notice also the inflection point at the origin. This point indicates where the curvature of the graph changes direction. The function crosses the x-axis at the root \( x = 2 \). Study the parts where the function is above or below the x-axis to understand where it’s positive or negative.

Inequalities help us determine the intervals where a function is positive or negative. Given the cubic function \[ f(x) = \frac{1}{4}x^3 - 2 \], you will want to find where it is greater than zero, \( f(x) > 0 \), and less than zero, \( f(x) < 0 \).
To solve this:

• Set the function equal to zero and solve \( \frac{1}{4}x^3 - 2 = 0 \) to get \( x = 2 \). This tells us where the function crosses the x-axis.
• To find \( f(x) > 0 \), examine the inequality \( x^3 > 8 \), which simplifies to \( x > 2 \). This means the function is positive for any \( x \) greater than 2.
• For \( f(x) < 0 \), look at \( x^3 < 8 \), translating to \( x < 2 \). Therefore, the function is negative for any \( x \) less than 2.

By analyzing these inequalities, you identify the qualitative behavior of cubic functions in different intervals of \( x \).

Analyzing a cubic function like \[ f(x) = \frac{1}{4}x^3 - 2 \] involves several key steps to understand its overall behavior and critical points.
First, consider transformations: the term \( -2 \) causes the overall function to shift down. This means the starting point of analysis considers this vertical shift.
Then, look for roots by solving \( f(x) = 0 \). The root \( x = 2 \) is crucial as it divides the function into regions of positivity and negativity.

• Before \( x = 2 \), the function decreases and remains negative. As you move from negative infinity towards \( x = 2 \), \( f(x) \) stays below the x-axis.
• After \( x = 2 \), \( f(x) \) becomes positive, indicating an increase as \( x \) heads towards positive infinity.

Lastly, observe the behavior at extremes: As \( x \rightarrow -\infty \), \( f(x) \rightarrow -\infty \), and as \( x \rightarrow \infty \), \( f(x) \rightarrow \infty \). This indicates a typical cubic growth pattern where the ends of the graph move in opposite vertical directions.