Here, \( a = 2 \), \( b = -3 \), and \( c = -5 \). - United Radiology

["Understanding Quadratic Forms: A Deep Dive into ( a = 2 ), ( b = -3 ), and ( c = -5 )", "When studying quadratic equations in the standard form ( ax^2 + bx + c ), the coefficients ( a ), ( b ), and ( c ) play pivotal roles in determining the shape, orientation, and behavior of the parabola represented by the equation. In this article, we explore the specific case where ( a = 2 ), ( b = -3 ), and ( c = -5 ), breaking down the meaning, graph behavior, and real-world applications tied to these values.", "### Breaking Down the Coefficients", "Let’s begin by identifying each coefficient clearly:", "- ( a = 2 ): The leading coefficient controls the parabola’s concavity and vertical stretch. Since ( a > 0 ), the parabola opens upward.
\n- ( b = -3 ): This coefficient influences the position of the vertex and affects symmetry. It contributes to the axis of symmetry and the value at which the vertex lies horizontally.
\n- ( c = -5 ): The constant term determines the y-intercept, the point where the parabola crosses the y-axis.", "### Vertex and Axis of Symmetry", "For a quadratic equation ( y = ax^2 + bx + c ), the vertex ( x )-coordinate is found using:", "[
\nx = -\frac{b}{2a}
\n]", "Plugging in ( a = 2 ) and ( b = -3 ):", "[
\nx = -\frac{-3}{2 \cdot 2} = \frac{3}{4}
\n]", "So, the vertex lies at ( x = \frac{3}{4} ). To find the corresponding ( y )-coordinate, substitute ( x = \frac{3}{4} ) into the original equation:", "[
\ny = 2\left(\frac{3}{4}\right)^2 + (-3)\left(\frac{3}{4}\right) - 5
\n]", "[
\ny = 2 \cdot \frac{9}{16} - \frac{9}{4} - 5 = \frac{18}{16} - \frac{36}{16} - \frac{80}{16} = \frac{18 - 36 - 80}{16} = \frac{-98}{16} = -\frac{49}{8}
\n]", "Thus, the vertex is located at ( \left( \frac{3}{4}, -\frac{49}{8} \right) ).", "### Axis of Symmetry", "The vertex defines the axis of symmetry—a vertical line passing through ( x = \frac{3}{4} ). This line divides the parabola into two symmetric halves.", "### Y-Intercept", "The constant ( c = -5 ) indicates the y-intercept. The parabola crosses the y-axis at ( (0, -5) ).", "### Direction of Opening", "Since ( a = 2 ) is positive, the parabola opens upward. This means:", "- The function has a minimum at the vertex.
\n- As ( x ) moves away from ( \frac{3}{4} ) in either direction, ( y ) increases without bound.", "### Discriminant and Roots", "The discriminant ( D = b^2 - 4ac ) tells us about the nature of the roots:", "[
\nD = (-3)^2 - 4(2)(-5) = 9 + 40 = 49
\n]", "Since ( D = 49 > 0 ), the quadratic has two distinct real roots, meaning the parabola intersects the x-axis at two points.", "### Graph Behavior", "- Vertex: Minimum point at ( \left( 0.75, -6.125 \right) )
\n- Axis of symmetry: ( x = 0.75 )
\n- Y-intercept: ( (0, -5) )
\n- X-intercepts: Solve ( 2x^2 - 3x - 5 = 0 ) using the quadratic formula:", "[
\nx = \frac{3 \pm \sqrt{49}}{4} = \frac{3 \pm 7}{4}
\n]", "So, ( x = \frac{10}{4} = 2.5 ) and ( x = -\frac{4}{4} = -1 )", "Thus, the parabola crosses the x-axis at ( x = -1 ) and ( x = 2.5 ).", "### Real-World Applications", "Understanding parabolas with specific coefficients helps model physical phenomena:", "- Projectile motion: The upward-opening parabola models trajectories under gravity when air resistance is negligible.
\n- Economic optimization: Quadratic functions describe cost or profit models near break-even points, where the vertex represents maximum or minimum value.
\n- Engineering design: Parabolic shapes optimize structural strength and structural aesthetics in arches and antennas.", "### Conclusion", "With ( a = 2 ), ( b = -3 ), and ( c = -5 ), the quadratic function ( y = 2x^2 - 3x - 5 ) forms an upward-opening parabola with a vertex at ( \left( \frac{3}{4}, -\frac{49}{8} \right) ), intercepts at ( (-1, 0) ) and ( (2.5, 0) ), and a positive discriminant confirming two real roots. Mastering such specific cases enhances comprehension of quadratic behavior and supports applications across science, engineering, and economics.", "---", "Keywords: quadratic function ( y = ax^2 + bx + c ), vertex formula, axis of symmetry, discriminant, graph of parabola, coefficients ( a=2 ), ( b=-3 ), ( c=-5 ), real roots, projectile motion, optimization."]