UpStudy Free Solution:

To determine which function corresponds to the given graph, we need to analyze the characteristics of each function and compare them with the graph.

The graph shows a polynomial function with the following characteristics:

- It has two local maxima and one local minimum.

- It starts from negative infinity, goes up, comes down, and then goes up again to positive infinity.

Let's analyze each function:

A. \(F( x) = x^ 2 + 5\)

- This is a quadratic function (parabola) that opens upwards.

- It has only one minimum point and no maximum points.

- This does not match the given graph.

B. \(F( x) = x^ 4 - 2x\)

- This is a quartic function.

- Quartic functions can have up to three turning points (local maxima and minima).

- This could potentially match the given graph's characteristics.

C. \(F( x) = - \frac { 1} { 2} x^ 4 - x^ 3 + x + 2\)

- This is also a quartic function but with a negative leading coefficient.

- It should open downwards at the ends, but the given graph shows it opening upwards.

- This does not match the given graph.

D. \(F( x) = x^ 3 + x^ 2 + x + 1\)

- This is a cubic function.

- Cubic functions can have up to two turning points (one maximum and one minimum).

- This does not match the given graph, which has more turning points.

Given the analysis, the function that best matches the given graph is:

B. \(F( x) = x^ 4 - 2x\)

Supplemental Knowledge

To determine which function could have created a given graph, it's important to understand the general shapes and characteristics of polynomial functions based on their degrees and leading coefficients:

- Quadratic Function (\(F( x) = ax^ 2 + bx + c\)):

- Shape: Parabola

- Opens upwards if \(a > 0\) and downwards if \(a < 0\)

- Cubic Function (\(F( x) = ax^ 3 + bx^ 2 + cx + d\)):

- Shape: S-shaped curve

- Can have one or two turning points

- Quartic Function (\(F( x) = ax^ 4 + bx^ 3 + cx^ 2 + dx + e\)):

- Shape: W-shaped or M-shaped curve (if \(a > 0\)) or inverted W/M (if \(a < 0\))

- Can have up to three turning points

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