["# Understanding the Quadratic Function ( f(x) = 2x^2 - x - 5 ) for All Real ( x )", "The quadratic function ( f(x) = 2x^2 - x - 5 ) is a fundamental expression in algebra with widespread applications in mathematics, physics, engineering, and economics. Whether you're studying quadratic equations, graphing, optimization, or modeling real-world phenomena, understanding this function is essential. In this comprehensive guide, we explore ( f(x) = 2x^2 - x - 5 ) in detail, covering its key properties, graph behavior, methods of analysis, and practical applications.", "---", "## What Is ( f(x) = 2x^2 - x - 5 )?", "The function ( f(x) = 2x^2 - x - 5 ) is a quadratic polynomial because its highest power of ( x ) is 2. It takes the standard form:", "[
\nf(x) = ax^2 + bx + c
\n]", "where:
\n- ( a = 2 )
\n- ( b = -1 )
\n- ( c = -5 )", "Since ( a > 0 ), the parabola opens upwards, indicating a minimum point at its vertex.", "---", "## Key Characteristics of the Function", "### 1. Degree and Shape", "- Degree: 2 (quadratic)
\n- Axis of Symmetry: The vertical line ( x = -\frac{b}{2a} )
\n- Vertex: The turning point (minimum in this case)
\n- Y-Intercept: ( f(0) = -5 )
\n- X-Intercepts (Roots): Solutions to ( 2x^2 - x - 5 = 0 )", "### 2. Vertex Calculation", "The vertex ( (h, k) ) lies at:", "[
\nh = -\frac{b}{2a} = -\left(\frac{-1}{2 \cdot 2}\right) = \frac{1}{4}
\n]", "Now compute ( k = f\left(\frac{1}{4}\right) ):", "[
\nf\left(\frac{1}{4}\right) = 2\left(\frac{1}{4}\right)^2 - \left(\frac{1}{4}\right) - 5 = 2 \cdot \frac{1}{16} - \frac{1}{4} - 5 = \frac{1}{8} - \frac{1}{4} - 5 = -\frac{1}{8} - 5 = -\frac{41}{8}
\n]", "So, the vertex is at ( \left(\frac{1}{4}, -\frac{41}{8}\right) ).", "---", "## Finding the X-Intercepts (Roots)", "To find where ( f(x) = 0 ):", "[
\n2x^2 - x - 5 = 0
\n]", "Use the quadratic formula:", "[
\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(2)(-5)}}{2 \cdot 2} = \frac{1 \pm \sqrt{1 + 40}}{4} = \frac{1 \pm \sqrt{41}}{4}
\n]", "So the roots are:", "[
\nx = \frac{1 + \sqrt{41}}{4} \quad \ ext{and} \quad x = \frac{1 - \sqrt{41}}{4}
\n]", "Since ( \sqrt{41} \approx 6.403 ), both roots are real and distinct, confirming two x-intercepts.", "---", "## Graphing the Function", "To sketch ( f(x) = 2x^2 - x - 5 ):", "- Start by plotting the vertex at ( \left(\frac{1}{4}, -\frac{41}{8}\right) \approx (0.25, -5.125) ).
\n- Draw the axis of symmetry ( x = 0.25 ).
\n- Plot the two x-intercepts approximately at ( x \approx -1.76 ) and ( x \approx 2.16 ).
\n- Plot the y-intercept at ( (0, -5) ).
\n- Sketch the smooth parabola opening upwards through these points.", "Using graphing technology or exact plotter tools helps visualize how the function behaves across all real ( x ).", "---", "## Analyzing Function Behavior", "### Domain", "Since ( f(x) ) is a polynomial, its domain is all real numbers:", "[
\n\ ext{Domain: } (-\infty, \infty)
\n]", "### Range", "Because the parabola opens upward and has a minimum value at the vertex, the range is:", "[
\n\ ext{Range: } \left[ -\frac{41}{8}, \infty \right)
\n]", "---", "## Applications of ( f(x) = 2x^2 - x - 5 )", "This quadratic function appears in various real-world scenarios:", "- Physics: Modeling projectile motion where the path is parabolic.
\n- Engineering: Optimization problems involving cost, strength, or efficiency.
\n- Economics: Representing profit or loss curves subject to quadratic cost functions.
\n- Mathematics: Solving quadratic equations and inequalities.", "Understanding ( f(x) ) helps students and professionals apply algebraic concepts in modeling practical situations.", "---", "## Solving Inequalities with ( f(x) )", "To solve ( f(x) \geq 0 ), locate where the parabola lies above or on the x-axis. Since the roots divide the x-axis into intervals:", "- ( x < \frac{1 - \sqrt{41}}{4} ): ( f(x) > 0 )
\n- ( x = \frac{1 - \sqrt{41}}{4} ): ( f(x) = 0 )
\n- ( \frac{1 - \sqrt{41}}{4} < x < \frac{1 + \sqrt{41}}{4} ): ( f(x) < 0 )
\n- ( x > \frac{1 + \sqrt{41}}{4} ): ( f(x) > 0 )", "Therefore:", "[
\nf(x) \geq 0 \quad \ ext{when} \quad x \leq \frac{1 - \sqrt{41}}{4} \quad \ ext{or} \quad x \geq \frac{1 + \sqrt{41}}{4}
\n]", "---", "## Conclusion", "The quadratic function ( f(x) = 2x^2 - x - 5 ) exemplifies the power of algebraic functions in modeling and problem-solving. Its straightforward form enables easy analysis of key properties including vertex location, intercepts, and inequality solutions. Mastering such functions strengthens foundational math skills essential for advanced studies in calculus, linear algebra, and applied sciences.", "By understanding how ( f(x) = 2x^2 - x - 5 ) behaves, students and practitioners gain valuable insight into parabolic functions and their significance across disciplines.", "---", "## Further Exploration", "- Use technology tools like Desmos or GeoGebra to visualize ( f(x) )
\n- Compute derivatives to analyze rates of change
\n- Explore transformations by comparing ( f(x) ) with ( g(x) = 2x^2 - x ) shifted horizontally
\n- Apply the function to model real-life problems involving quadratic relationships", "---", "# Key Terms for SEO Optimization", "- quadratic function
\n- f(x) = 2x² - x - 5
\n- parabola
\n- vertex of f(x)
\n- x-intercepts of f(x)
\n- real roots
\n- quadratic equations
\n- graph quadratic function
\n- inequalities with quadratics
\n- function analysis
\n- algebra quadratic functions", "Tailor content with these keywords to boost search visibility and provide comprehensive value to readers exploring ( f(x) = 2x^2 - x - 5 ).", "---", "Let us know if you'd like a linked interactive graph or step-by-step calculation examples for deeper understanding!"]
