A function \( f(x) = ax^2 + bx + c \) has a vertex at \( (2, -3) \) and passes through the point \( (1, 0) \). Find \( a \), \( b \), and \( c \).
["# How to Find Coefficients ( a ), ( b ), and ( c ) for a Quadratic Function Given Its Vertex and a Point", "Understanding how to determine the coefficients of a quadratic function ( f(x) = ax^2 + bx + c ) is essential in algebra. In this article, we’ll explore how to find ( a ), ( b ), and ( c ) when the vertex is known and the curve passes through a specific point. We focus on the quadratic function with a vertex at ( (2, -3) ) and a point ( (1, 0) ) lying on its graph.", "---", "## Step 1: Use the Vertex Form of a Quadratic Function", "Since the quadratic has a vertex at ( (2, -3) ), it’s useful to start with the vertex form of a parabola:", "[\nf(x) = a(x - h)^2 + k\n]", "where ( (h, k) ) is the vertex. Substituting ( h = 2 ) and ( k = -3 ):", "[\nf(x) = a(x - 2)^2 - 3\n]", "This expression contains one unknown, ( a ), which we can determine using the known point ( (1, 0) ).", "---", "## Step 2: Plug in the Point to Solve for ( a )", "We know ( f(1) = 0 ). Substitute ( x = 1 ), ( f(1) = 0 ), ( h = 2 ), and ( k = -3 ) into the vertex form:", "[\n0 = a(1 - 2)^2 - 3\n]", "[\n0 = a(-1)^2 - 3 \quad \Rightarrow \quad 0 = a(1) - 3 \quad \Rightarrow \quad a = 3\n]", "---", "## Step 3: Expand to Standard Form ( f(x) = ax^2 + bx + c )", "Now that ( a = 3 ), substitute back into the vertex form:", "[\nf(x) = 3(x - 2)^2 - 3\n]", "Expand the square:", "[\nf(x) = 3(x^2 - 4x + 4) - 3 = 3x^2 - 12x + 12 - 3 = 3x^2 - 12x + 9\n]", "---", "## Step 4: Extract Coefficients", "Comparing with ( f(x) = ax^2 + bx + c ), we identify:", "- ( a = 3 )\n- ( b = -12 )\n- ( c = 9 )", "---", "## Final Verification", "Check that the vertex is indeed ( (2, -3) ) and that ( f(1) = 0 ):", "- Vertex: At ( x = 2 ), ( f(2) = 3(2)^2 - 12(2) + 9 = 12 - 24 + 9 = -3 ) ✔️\n- Point: ( f(1) = 3(1)^2 - 12(1) + 9 = 3 - 12 + 9 = 0 ) ✔️", "---", "## Summary", "Given a quadratic function with vertex ( (2, -3) ) and passing through ( (1, 0) ), the coefficients are:", "[\na = 3, \quad b = -12, \quad c = 9\n]", "This demonstrates how vertex form simplifies identifying ( a ), then solving for ( b ) and ( c ) via expansion and substitution — a powerful technique for modeling quadratic relationships in algebra and real-world applications.", "---", "## SEO Keywords:\nquadratic function coefficients, find a from vertex and point, solve quadratic using vertex form, algebra quadratic vertex and point, calculate a, b, c for f(x) = ax² + bx + c"]
