The function \(f(x) = -2x^2 + 2x + 4\) is a quadratic equation. When graphed, quadratic equations form a distinctive U-shaped or inverted U-shaped curve known as a parabola. This analysis will explore the essential properties of this specific function, providing a clear understanding of its mathematical behavior and graphical representation.

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Essential Insights: Key Takeaways

• Downward Curve: The negative coefficient of the \(x^2\) term (\(a = -2\)) means the parabola opens downwards, indicating the function has a maximum value.
• Peak Performance: The vertex, representing the highest point of the parabola, is located at the coordinates \((0.5, 4.5)\).
• Crossing Points: The function intersects the x-axis (its roots) at \(x = -1\) and \(x = 2\), and it crosses the y-axis at \(y = 4\).

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Deconstructing the Quadratic Function \(f(x) = -2x^2 + 2x + 4\)

To fully understand this function, we begin by examining its fundamental components and form.

Standard Form and Coefficients

The given function \(f(x) = -2x^2 + 2x + 4\) is already in the standard form of a quadratic equation, which is:

\[ f(x) = ax^2 + bx + c \]

By comparing our function to the standard form, we can identify the coefficients:

• \(a = -2\) (the coefficient of \(x^2\))
• \(b = 2\) (the coefficient of \(x\))
• \(c = 4\) (the constant term)

These coefficients are crucial as they determine the shape, position, and orientation of the parabola.

Direction of the Parabola

The coefficient 'a' dictates whether the parabola opens upwards or downwards.

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), the parabola opens downwards.

In our case, \(a = -2\), which is negative. Therefore, the parabola for \(f(x) = -2x^2 + 2x + 4\) opens downwards. This implies that the parabola has a highest point, known as the maximum.

A typical parabolic curve, illustrating key features such as the vertex and axis of symmetry.

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Pinpointing Key Features of the Parabola

Several key features define the specific characteristics of the parabola associated with \(f(x) = -2x^2 + 2x + 4\). These include the vertex, axis of symmetry, and intercepts.

The Vertex: The Peak of the Curve

The vertex is the point where the parabola reaches its maximum (or minimum) value. Since our parabola opens downwards, the vertex represents its highest point.

Calculating the Vertex Coordinates

The x-coordinate of the vertex \((h, k)\) is found using the formula:

\[ h = -\frac{b}{2a} \]

Substituting our values \(a = -2\) and \(b = 2\):

\[ h = -\frac{2}{2(-2)} = -\frac{2}{-4} = 0.5 \]

To find the y-coordinate (\(k\)), we substitute this x-value back into the function \(f(x)\):

\[ k = f(0.5) = -2(0.5)^2 + 2(0.5) + 4 \] \[ k = -2(0.25) + 1 + 4 \] \[ k = -0.5 + 1 + 4 \] \[ k = 4.5 \]

Thus, the vertex of the parabola is at the point \((0.5, 4.5)\). This is the maximum value of the function.

Vertex Form of the Equation

A quadratic function can also be expressed in vertex form: \(f(x) = a(x-h)^2 + k\). Using our values for \(a\), \(h\), and \(k\):

\[ f(x) = -2(x - 0.5)^2 + 4.5 \]

This form directly shows the vertex coordinates and the direction of opening.

Axis of Symmetry: The Line of Reflection

The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. The equation for the axis of symmetry is given by \(x = h\).

For our function, the axis of symmetry is \(x = 0.5\).

Intercepts: Where the Parabola Meets the Axes

Intercepts are the points where the parabola crosses the x-axis and y-axis.

Y-Intercept

The y-intercept occurs when \(x = 0\). We can find it by substituting \(x=0\) into the function:

\[ f(0) = -2(0)^2 + 2(0) + 4 = 0 + 0 + 4 = 4 \]

So, the y-intercept is at the point \((0, 4)\).

X-Intercepts (Roots)

The x-intercepts, also known as the roots or zeros of the function, occur when \(f(x) = 0\). To find them, we need to solve the quadratic equation:

\[ -2x^2 + 2x + 4 = 0 \]

We can simplify this equation by dividing all terms by -2:

\[ x^2 - x - 2 = 0 \]

This simplified quadratic equation can be solved by factoring. We look for two numbers that multiply to -2 and add to -1. These numbers are -2 and +1.

\[ (x - 2)(x + 1) = 0 \]

Setting each factor to zero gives the solutions for x:

• \(x - 2 = 0 \implies x = 2\)
• \(x + 1 = 0 \implies x = -1\)

Therefore, the x-intercepts are at the points \((-1, 0)\) and \((2, 0)\).

Alternatively, the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) could be used on the original equation \(-2x^2 + 2x + 4 = 0\):

\[ x = \frac{-2 \pm \sqrt{2^2 - 4(-2)(4)}}{2(-2)} \] \[ x = \frac{-2 \pm \sqrt{4 + 32}}{-4} \] \[ x = \frac{-2 \pm \sqrt{36}}{-4} \] \[ x = \frac{-2 \pm 6}{-4} \]

This gives two solutions:

\[ x_1 = \frac{-2 + 6}{-4} = \frac{4}{-4} = -1 \] \[ x_2 = \frac{-2 - 6}{-4} = \frac{-8}{-4} = 2 \]

These results confirm the x-intercepts found by factoring.

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Visualizing the Parabola: Graph and Characteristics

Sketching the Graph

To sketch the graph of \(f(x) = -2x^2 + 2x + 4\), we plot the key points we've found:

• Vertex: \((0.5, 4.5)\)
• Y-intercept: \((0, 4)\)
• X-intercepts: \((-1, 0)\) and \((2, 0)\)

Knowing the parabola opens downwards and is symmetric about the line \(x=0.5\), we can sketch a smooth curve through these points. For instance, the point \((0,4)\) is 0.5 units to the left of the axis of symmetry. Due to symmetry, there's a corresponding point 0.5 units to the right, at \(x = 0.5 + 0.5 = 1\). The y-value at \(x=1\) is \(f(1) = -2(1)^2 + 2(1) + 4 = -2 + 2 + 4 = 4\). So, \((1,4)\) is another point on the parabola.

Domain and Range

The domain and range describe the set of all possible input (x) and output (y) values for the function.

• Domain: Quadratic functions are defined for all real numbers. Thus, the domain of \(f(x)\) is \((-\infty, \infty)\).
• Range: Since the parabola opens downwards and its maximum y-value is at the vertex (\(y=4.5\)), the function can take any y-value less than or equal to 4.5. Thus, the range of \(f(x)\) is \((-\infty, 4.5]\).

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Summary of Parabola Properties

The table below consolidates the key characteristics of the quadratic function \(f(x) = -2x^2 + 2x + 4\):

Feature	Value/Description
Equation	\(f(x) = -2x^2 + 2x + 4\)
Type of Function	Quadratic
Shape of Graph	Parabola
Direction of Opening	Downwards (since \(a = -2 < 0\))
Vertex \((h, k)\)	\((0.5, 4.5)\) (Maximum point)
Axis of Symmetry	\(x = 0.5\)
Y-intercept	\((0, 4)\)
X-intercepts (Roots)	\((-1, 0)\) and \((2, 0)\)
Domain	All real numbers, \((-\infty, \infty)\)
Range	\(f(x) \le 4.5\), or \((-\infty, 4.5]\)

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Analytical Profile Radar Chart

This radar chart provides a conceptual overview of the relative ease and clarity associated with analyzing different aspects of the quadratic function \(f(x) = -2x^2 + 2x + 4\). Higher scores (on a scale potentially up to 10) indicate greater straightforwardness in determination or understanding. The values are subjective analytical assessments.

The chart visually represents that identifying the Y-intercept is typically the most straightforward, while finding roots and sketching the complete graph might involve a few more steps, though still quite manageable for this function.

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Interactive Exploration: Graphing Quadratic Functions

Quadratic functions, like the one we've analyzed, are fundamental in algebra. Understanding how to graph them, particularly by identifying the vertex and intercepts, is a key skill. The following video provides a helpful overview of graphing quadratic functions, focusing on the vertex form, which is closely related to the standard form characteristics we've discussed.

This video complements our analysis by visually demonstrating the graphing process and reinforcing concepts such as the vertex's role in determining the parabola's shape and position. While our function \(f(x) = -2x^2 + 2x + 4\) was analyzed from its standard form, converting it to vertex form (\(f(x) = -2(x - 0.5)^2 + 4.5\)) directly reveals the vertex \((0.5, 4.5)\), as discussed earlier.

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Mindmap: Key Aspects of \(f(x) = -2x^2 + 2x + 4\)

This mindmap provides a visual summary of the core components and characteristics of the quadratic function \(f(x) = -2x^2 + 2x + 4\), branching out from the main function to its type, graphical features, and key calculated values.

This visual tool helps to connect the various pieces of information discussed, offering a holistic view of the function's properties.

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Frequently Asked Questions (FAQ)

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Recommended Further Exploration

To deepen your understanding of quadratic functions and related concepts, consider exploring these topics:

• How does changing the 'a' value affect the shape and width of a parabola?
• What are some real-world applications of quadratic functions and parabolas?
• How can you convert a quadratic function from standard form to vertex form using the completing the square method?
• What is the discriminant of a quadratic equation and what does it reveal about the nature of its roots?

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References