To graph the function [tex]\( f(x) = |\sqrt[3]{x} + 2x| + 12 \)[/tex], let's analyze it step by step.
### Vertex of the Graph
1. Understanding the Function Structure: The function consists of an absolute value term and a constant. The general form is [tex]\( f(x) = |g(x)| + c \)[/tex], where [tex]\( g(x) = \sqrt[3]{x} + 2x \)[/tex] and [tex]\( c = 12 \)[/tex].
2. Finding the Vertex: The vertex of such a function occurs at the point where the expression inside the absolute value [tex]\( g(x) \)[/tex] is zero, since this is where the minimum value of the absolute value function occurs.
3. Solving [tex]\( g(x) = 0 \)[/tex]:
- We solve [tex]\( \sqrt[3]{x} + 2x = 0 \)[/tex].
- Solving [tex]\( \sqrt[3]{x} = -2x \)[/tex].
- Raising both sides to the power of 3: [tex]\( x = -8x^3 \)[/tex].
- Rearranging the equation: [tex]\( x^3 + 8x = 0 \)[/tex], which simplifies to [tex]\( x(x^2 + 8) = 0 \)[/tex].
This equation has solutions:
- [tex]\( x = 0 \)[/tex] because [tex]\( x^2 + 8 \)[/tex] has no real roots.
4. Vertex Coordinates: Substituting [tex]\( x = 0 \)[/tex] back into the original function:
- [tex]\( f(0) = | \sqrt[3]{0} + 2 \cdot 0 | + 12 \)[/tex].
- This simplifies to [tex]\( f(0) = |0| + 12 = 12 \)[/tex].
So, the vertex is at [tex]\( (0, 12) \)[/tex].
### Direction in which the Graph Opens
The graph opens upwards because the absolute value function results in a non-negative output which increases as we move away from the vertex along both directions of the x-axis. The constant part [tex]\( 12 \)[/tex] ensures that all values of [tex]\( f(x) \)[/tex] are greater than or equal to 12.
Thus, the answers are:
- The vertex of the graph is at [tex]\( (0, 12) \)[/tex].
- The direction in which the graph opens is upwards.
Selections:
- [tex]\(\square\)[/tex] = (0, 12)
- [tex]\(\square\)[/tex] = upwards
