Tempestt Graphs a Function that has a Maximum Located at (–4, 2). Which could be Her Graph?

The question: Tempestt Graphs a Function that has a Maximum Located at (–4, 2). Which could be Her Graph? When working with complex mathematical concepts like graphing functions, finding the correct representation is key. Tempestt faces this exact challenge as she investigates which function could best represent a graph with a maximum located at (-4, 2).

This blog post will break down the details of graphing functions with maxima, explore the characteristics of potential graphs, and pinpoint the types of functions Tempestt should consider.

What Does It Mean for a Graph to Have a Maximum?

A graph with a maximum point represents the highest value of the function at a particular coordinate. For Tempestt’s case, the maximum is located at (-4, 2). This means:

• When \( x = -4 \), the corresponding \( y \)-value is 2, and no other value of \( y \) is larger than 2 within the graph.
• The graph will be downward-sloping on either side of \( x = -4 \) because this is the “peak” of the graph.

Key Characteristics of Functions with Maximums

• Downward Concavity: The graph opens downward, which is typically indicated by a negative leading coefficient in the function.
• Vertex Representation: The vertex of the graph represents the point (-4, 2).
• Behavior at Infinity: For parabolas that open downward, \( y \) decreases toward negative infinity as \( x \) moves away from the vertex in either direction.

Mathematical Concept of Maxima

Functions with a maximum at (-4, 2) must satisfy specific conditions, such as:

• The Vertex Form of a Quadratic Function:

  A quadratic function in vertex form is expressed as:

  \[

  f(x) = a(x – h)^2 + k

  \]

  Here, \( (h, k) \) represents the vertex. For Tempestt’s graph:

• \( h = -4 \)
• \( k = 2 \)
• \( a \) must be negative to ensure a downward opening.

• Negative Second Derivative:

  If the second derivative of the function is negative at \( x = -4 \), it confirms the presence of a maximum.

Possible Functions for Tempestt’s Graph

Based on the given maximum point (-4, 2), here are a few functions Tempestt could work with:

1. Quadratic Function

The simplest function that fits the description is a quadratic function:

\[

f(x) = -(x+4)^2 + 2

\]

• Why It Fits:

• • • The negative sign ensures the parabola opens downward.
    • The vertex is located at (-4, 2), as \( h = -4 \) and \( k = 2 \).

• Behavior:

• • As \( x \) moves away from -4, the function decreases symmetrically, approaching negative infinity in both directions.

2. Cubic Function

Another possible option is a cubic function:

\[

f(x) = -(x+4)^3 + 2

\]

• Why It Fits:

• • • The maximum is still at (-4, 2).
    • However, cubic functions have a more pronounced curve compared to parabolas.

• Behavior:

• • The cubic function increases rapidly before \( x = -4 \), reaches the peak at (-4, 2), and decreases rapidly afterward.

3. Quartic Function

A slightly more complex option is a quartic function:

\[

f(x) = -(x+4)^4 + 2

\]

• Why It Fits:

• • • The maximum is still at (-4, 2).
    • Quartic functions can exhibit additional turning points, making the graph more detailed.

• Behavior:

• • Like the quadratic and cubic functions, the quartic graph opens downward and approaches negative infinity as \( x \) moves away from the vertex.

4. Trigonometric Function

For a more unique approach, a sine or cosine function shifted accordingly can also work:

\[

f(x) = 2\cos(x+4)

\]

• Why It Fits:

• • Trigonometric functions can have maximum points based on their periodic nature.
  • The maximum at (-4, 2) can be achieved by shifting the graph horizontally and scaling it vertically.

Verifying the Graphs’ Behavior

To confirm which graph best suits Tempestt’s criteria, one must:

• Analyze the Derivatives:

• Check the first derivative to confirm the slope is zero at (-4, 2).
• Check the second derivative to ensure concavity is negative at (-4, 2).

• Plot the Functions:

• Use graphing software or tools to visualize how these equations behave near (-4, 2).

• Examine Symmetry:

• Quadratics are symmetric, but other functions may exhibit asymmetry, depending on their degree.

Why Understanding Maxima is Important

The concept of maxima is essential not only in mathematics but in various real-world applications:

• Economics: Identifying profit-maximizing production levels.
• Physics: Analyzing trajectories to find maximum heights.
• Engineering: Optimizing designs for structural integrity and performance.

For Tempestt, understanding the behavior of functions with maxima will not only help her plot the right graph but also enhance her problem-solving and analytical skills in these broader contexts.

The Takeaway for Tempestt

For a graph with a maximum at (-4, 2):

• The simplest and most straightforward option is the quadratic function:

 \[

 f(x) = -(x+4)^2 + 2

 \]

• If more complexity or detail is needed, cubic, quartic, or even trigonometric functions can also represent this behavior, depending on the scenario.

Understanding how to identify and verify such graphs equips Tempestt (and anyone tackling similar mathematical challenges) with the tools to analyze and visualize functions effectively.

The Other Solution is:

When analyzing functions with specific maximum points, understanding the characteristics of different graph types becomes crucial. Tempestt’s graph, which has a maximum located at (-4, 2), could belong to multiple function families. Let’s explore the most likely candidates and their graphical representations.

Understanding Maximum Points in Functions

A maximum point represents the highest value a function reaches within a specific domain. For Tempestt’s case, we focus on absolute maxima (the highest point on the entire graph) rather than local maxima.

Key Characteristics of Tempestt’s Graph

1. Vertex location: (-4, 2) must be the peak
2. Downward concavity: The graph must open downward
3. Symmetric properties: For quadratic functions, symmetry about the vertical line x = -4

Quadratic Function Analysis

The most straightforward solution comes from parabolic functions in vertex form:

Function formula:

 

f(x)=−(x+4)2+2

f(x)=−(x+4)

2

+2

Breaking Down the Components

Component	Value	Significance
a a	-1	Determines downward opening
h h	-4	Horizontal shift
k k	2	Vertical shift

This configuration creates:

• A peak at (-4, 2)
• Symmetrical arms decreasing equally on both sides
• Y-intercept at 
• f(0)=−(−4)2+2=−14
• f(0)=−(−4)
• 2
• +2=−14

Graphical features:

• Passes through (-6, -2) and (-2, -2)

• Smooth U-shaped curve
• No secondary turning points

Cubic Function Possibility

While less common, cubic functions can also produce graphs with single maxima:

Alternative function:

 

f(x)=−(x+4)3+2

f(x)=−(x+4)

3

+2

Comparison With Quadratic Graph

Feature	Quadratic	Cubic
End behavior	Both arms ↓	Left ↑, Right ↓
Turning points	1	2
Symmetry	Yes	No
Rate of descent	Gradual	Rapid

The cubic option creates:

• A steeper decline past the maximum point
• An inflection point after (-4, 2)
• Asymmetric graph structure

Verifying the Correct Graph

When evaluating graph options:

Elimination criteria:

• Upward-opening parabolas → Immediately invalid (would show minima)

• Vertex mismatch → Any peak not at (-4, 2) gets rejected

1. Multiple maxima → Disqualifies polynomial functions of degree 4+

Confirmation method:

1. 1. Calculate 
   2. f(−4)

• f(−4) → Must equal 2

1. Check concavity → Second derivative should be negative
2. Verify endpoint behavior → Should trend downward infinitely

From the analyzed options, the correct graph shows:

• • Clear downward-opening parabola
  • Vertex precisely at (-4, 2)

• Passing through (-6, -2) and (-2, -2)

This analysis confirms that while multiple functions could theoretically have this maximum, the most likely graph matches the quadratic function 

f(x)=−(x+4)2+2

f(x)=−(x+4)

2

+2 with the specified characteristics. Understanding these graphical properties helps students quickly identify function types and verify maximum/minimum points in algebraic problems.

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