So \( f(x) = -x^2 - 2x + 4 \).

["# Understanding the Quadratic Function ( f(x) = -x^2 - 2x + 4 ): A Comprehensive Guide", "Explore the quadratic function ( f(x) = -x^2 - 2x + 4 ) in depth. This article breaks down its key features, graph, vertex, domain, range, applications, and tips for solving related equations. Perfect for students, educators, and math enthusiasts looking to master parabolas and their real-world relevance.", "## The Basic Structure of the Function", "The function ( f(x) = -x^2 - 2x + 4 ) is a quadratic equation in standard form:\n[\nf(x) = ax^2 + bx + c\n]\nHere, ( a = -1 ), ( b = -2 ), and ( c = 4 ). Since the coefficient of ( x^2 ) is negative (( a < 0 )), the parabola opens downward, indicating that the function has a maximum point (vertex) rather than a minimum.", "## Finding the Vertex: Maximum Point", "The vertex represents the peak of the parabola and is crucial for understanding the function’s maximum value. To find the vertex, use the formula:\n[\nx = -\frac{b}{2a}\n]\nSubstituting ( a = -1 ) and ( b = -2 ):\n[\nx = -\frac{-2}{2(-1)} = \frac{2}{-2} = -1\n]\nNow plug ( x = -1 ) back into the function to find ( f(-1) ):\n[\nf(-1) = -(-1)^2 - 2(-1) + 4 = -1 + 2 + 4 = 5\n]\nThus, the vertex is at ( (-1, 5) ).\n💡 Key Insight: Because ( a < 0 ), the point ( (-1, 5) ) is the maximum of the function — the highest value ( f(x) ) reaches.", "## Domain and Range", "### Domain\nSince ( f(x) ) is a polynomial with no restrictions on ( x ), the domain includes all real numbers:\n[\n\ ext{Domain: } (-\infty, \infty)\n]", "### Range\nWith ( a < 0 ), the parabola opens downward, so the maximum value at the vertex is the highest output. Therefore, the range is:\n[\n\ ext{Range: } (-\infty, 5]\n]\nValues under or equal to 5 are possible outputs.", "## Graphing the Quadratic: Sketching the Parabola", "To graph ( f(x) = -x^2 - 2x + 4 ), follow these steps:", "1. Plot the vertex: Mark ( (-1, 5) ).\n2. Plot additional points: Choose ( x )-values around the vertex—say ( x = -3, -2, 0, 1 )—and compute ( f(x) ):\n - ( f(-3) = -9 + 6 + 4 = 1 )\n - ( f(-2) = -4 + 4 + 4 = 4 )\n - ( f(0) = 0 + 0 + 4 = 4 )\n - ( f(1) = -1 -2 + 4 = 1 )\n3. Draw the curve: Connect the points smoothly—because ( a < 0 ), the parabola curves downward, forming a "smile" facing upward (since it opens down), peaking at ( (-1, 5) ).", "## Solving Equations and Inequalities", "### Finding Roots (Zeros)\nTo solve ( f(x) = 0 ):\n[\n-x^2 - 2x + 4 = 0\n]\nMultiply both sides by (-1):\n[\nx^2 + 2x - 4 = 0\n]\nUse the quadratic formula:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-4)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 16}}{2} = \frac{-2 \pm \sqrt{20}}{2}\n]\n[\nx = \frac{-2 \pm 2\sqrt{5}}{2} = -1 \pm \sqrt{5}\n]\nRoots are:\n[\nx = -1 + \sqrt{5} \approx 1.236 \quad \ ext{and} \quad x = -1 - \sqrt{5} \approx -3.236\n]\n🎯 Tip: Use these roots to sketch the x-intercepts.", "### Solving Inequalities\nTo find where ( f(x) \leq 5 ):\nSince the vertex is the maximum at 5, the function reaches or stays below 5 everywhere—so:\n[\nf(x) \leq 5 \quad \ ext{for all real } x\n]\nTo solve ( f(x) \leq 0 ), use the roots:\n( x \leq -1 - \sqrt{5} ) or ( x \geq -1 + \sqrt{5} )", "## Applications of the Quadratic Function", "Quadratic models like ( f(x) = -x^2 - 2x + 4 ) appear in various real-world scenarios:", "- Projectile motion: The height of an object thrown upward (within optimal range) follows a parabola opening downward.\n- Profit analysis: Maximizing revenue or minimizing cost often involves quadratic relationships.\n- Engineering design: Bridges, antennas, and satellite dishes use parabolic shapes to optimize strength and signal focus.", "Understanding such functions helps engineers, economists, and scientists model and optimize outcomes.", "## Final Thoughts: Mastering Quadratics", "The function ( f(x) = -x^2 - 2x + 4 ) exemplifies the elegance of quadratic functions. With a downward-facing parabola, clear vertex, and real-world utility, mastering it builds a foundation for calculus, physics, and applied mathematics. Whether graphing by hand or solving equations, this function is a powerful tool for students and professionals alike.", "🔍 Quick Summary:\n- Vertex at ( (-1, 5) ) (maximum point)\n- Opens downward (since ( a = -1 < 0 ))\n- Domain: all real numbers\n- Range: ( (-\infty, 5] )\n- Roots: ( x = -1 \pm \sqrt{5} )\n- Useful in modeling maxima and symmetric systems", "---", "Keywords for SEO:\n( f(x) = -x^2 - 2x + 4 ), quadratic function, vertex form, parabola graph, maximize function, quadratic equations, real-world applications, roots of quadratic, domain and range, parabolic motion, solving quadratics, math tutorial.", "Meta Description:\nDiscover the full analysis of ( f(x) = -x^2 - 2x + 4 )—its vertex, graph, domain, range, and practical uses. Perfect for understanding downward-opening parabolas."]